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Chapter 5 Eisenstein Systems
CHAPTER 5
Eisenstein Systems
In this chapter, we shall give an account of the Langlands theory of Eisenstein systems.This is difficult mathematics, compounded additionally by the apparent lack ofa central theme. For the purposes of the present work, the theory has two objectives (closely related to one another).
( I ) Define the spaces L$,(G/T; 6, O o ;W) and obtain the orthogonal decomposition
L:,(G/r; 6 , o o ) =
1 0 L$,(G/r; 6,@,; W).
Y>,WO
(2) Develop the theory of Eisenstein series and c-functions for residual forms. Eisenstein systems provide an inductive vehicle for arriving at these goals. The reader will simply have to take this on faith: The ends justify the means. Langlands’s discussion of these matters is virtually impossible to read. [He himself has admitted that it is almost impenetrable.] Furthermore, points (1) and (2) are not developed at all systematically, even though they are absolutely essential for the trace formula. It is for this reason that we have decided to give a complete and detailed account of the theory. Our version differs in many respects from that of Langlands; but there is really nothing to be gained by making a detailed comparison here. Modulo what we consider to be elementary or obvious, our proofs, we hope, will be found accessible. 122
EISENSTEIN SYSTEMS
123
Nevertheless, the reader is warned in advance that there are many definitions to be absorbed, notation to be kept in mind, and intricate constructions to be contemplated. With perseverance, however, these difficulties should move to the background. The main result is Theorem 5.12. Its statement can be understood after digesting the definition of Eisenstein system and some of the accompanying axioms:This turns out to be the source of (1). Proposition 5.4 is of crucial importance for (2). Before defining the notion of Eisenstein system, it will be necessary to set up some notation and state a few simple facts. Let (P, S) be a r-cuspidal split parabolic subgroup of G with split component A . Let X be an affine subspace of the complexification of &then we shall say that X is admissible if X can be represented as an intersection of hyperplanes of the form 1= c(1 E C,(g, 6)). Fix an admissible affine subspace X. If
x = n (1= c),
then X admits a unique decomposition X=X"@X,
where
3-
=n(x=o)
and X is a vector which is orthogonal to X-,the normal translation in 3. We shallwriteX1fortheorthogonalcomplementofX-. ByS,,wethenunderstand the symmetric algebra over XI.There is a unique conjugate linear isomorphism *: s, {u
-+
s,
H u*
of S, with itself such that A* = -12 if A belongs to XI. Let V be a finite-dimensional complex Hilbert space. Form
{S;oidm&,
V).
There is a natural pairing S, 0 V x Hom(Sx, V ) + C
which is linear in the first variable and conjugate linear in the second variable, characterized by the condition (u 0 u, T ) = (0,T(u*)).
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CHAPTER 5
Form
There is a natural pairing
S, Q
v x Hom(S,,
V ) -,C
which is linear in both variables, characterized by the condition (U Q
5, T) = (a, T(u)).
Lemma Suppose that A is a linear function on S , Q V-then there is a TAin Hom(S,, V ) such that
NQ)=
(QI
‘GI
(Q E S, 0 V ) .
Lemma @is) Suppose that A is a linearfunction on S, 0 V-rhen there is a TA in Hom(S,, V ) such that
N Q ) = ( Q , ‘M
(Q E S, 0
v).
One can identify Hom(S,, V ) with the space of formal power series over X I with coefficients in V. In this interpretation, we can speak of the order of an element T: ord(T) is the order of the term of lowest degree which actually occurs in the power series expansion of T. This being so, a linear transformation from Hom(S,, V ) to another vector space is said to be of degree n iff it annihilates all the terms of order >n but does not annihilate every term of order =n.
Lemma Suppose that A is a linearfunction on Hom(S,, V)-then A is of jinite degree i8there is a QA in S , Q V such that ( T EHom(S,, V ) ) .
NT) =
Lemma (bis) Suppose that A is a linearfunction on Hom(S,, V)-then A is ofjinite degree Iffthere is a Q A in S , @ such that A(T) = ( Q A , T )
( T E Hom(S,, V ) ) .
Remark Suppose given
( P ,S‘; A ‘ ) (p“,S”;A ” )
with associated data
125
EISENSTEIN SYSTEMS
Let
V: Hom(S,,, V') -+ S,,, 0 V" be a linear transformation of finite degree-then there exists a unique linear transformation
V*: Hom(S,.., V " ) + S,. 0 V' of finite degree such that (VT', T " ) = (V*T", T ' )
for all
{
T' E Hom(S,, , V ' ) T" E Horn(&, V").
We shall, accordingly, refer to V* as the adjoint of V. Let 'I) be an admissible affine subspace which is contained in X, so that 'I)='I)"@Y with 'I)" contained in X ". Call S,complement of 3' in gl-then S,
the symmetricalgebra over theorthogonal =
Sa-, Q S,.
If U O E S,-E
T E Hom(S,, V ) ,
then there is determined an element uo v T E Hom(S,, V )
characterized by the condition that One has
(uo v T)(u) = T(u0 0 u )
(u E S,).
(Q, uo v T ) = (4 0 Q, T ) (Q, uo v T ) = ( u o Q Q, T )
(Q E S, Q
i
v)
(QESX Q
v).
The discussion thus far has centered on (6, X, V ) . We shall now inject another ingredient into our considerations. Let x be a linear subspace of 6then x is said to be X-admissible if x is contained in X". Fix an X-admissible subspace x. Obviously: X' c x'.
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CHAPTER 5
Lemma There is a canonical injection u Ha, morphic diflerential operators on x1 such that d,f(Ao) =
df
(A0
S , into the algebra of holo-
of
+ tA)
lt=o
for all A in X I . Fix a point A. in x1 and an open neighborhood N o of A,. Let @:N0+V
be a holomorphic function. By d@(Ao)we mean that element of Hom(S,, V ) specified by the rule d@(Ao)(u)= a,@(Ao)
( U E Sx).
In the formal power series picture, d@(Ao)is obtained by expanding
+ A))
= @(AO
@A0
about the origin. Observe that d@ may be thought of as a map d o : N o + Hom(S,, V ) with the property that for every u E S,, d@(?: u): No
+
V
is holomorphic on N o . Let F: N o + Hom(S,, V ) be a map with the property that for every u E S,, F(?: u ) : No + V is holomorphic on N o .As before, let '1) be an admissit.,: affine subspace v, ..ich is contained in 3. Assume x is 9-admissible. We shall then denote by do F(Ao) that element of Hom(SB,,V ) given by the prescription doF(uo @ u) = duoF(Ao:u).
Remark It is easy to check that dO(dWA0) = d@(Ao)* The notation, however, is deceptive. On the left, d@ is viewed as an element of Horn(&, V ) while, on the right, d@ is viewed as an element of Hom(SB,,V ) . In the sequel, we shall let the context dictate the appropriate interpretation.
127
EISENSTEIN SYSTEMS
Let Vl and V, be finite-dimensional complex Hilbert spaces-then Hom(Vl, V,) is again a finite-dimensional complex Hilbert space. Let
f : N o -,Hom( V,, V,) be a holomorphic function. One then has that df(A0)E Hom(S,, Hom(V1, V,)),
I.e., df(A0) E
0 Vl,
V2),
I.e., df(A0) E Horn( Vl,
3
V2N.
The composition of any element in Hom(S,, V,)with df(Ao)provides us with an element of Horn(&
3
9
V2)h
or still, with an element of Hom(S, 0 S,, Vd. The Hopf algebra map
s,
-+
s, 0 s,
then induces a morphism of restriction Hom(S, @ S,, V,)
-, Hom(S,,
V,).
There is, therefore, a natural map Hom(S,, Vl) -, Hom(S,, V2), thereby determining an element dHom
f(A01 E Hom(Hom(S,
7
Vl)? Horn(&
7
V2)).
Let us now go back to the beginning of this discussion. Starting from the fact that df(A0) E Hom(S,, Hom(V1, V2N7
I.e., df(A0)
E
0
Vl,
V,),
we obtain an element in Horn(& 0 S, 0 Vl, S, .@ V2)
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CHAPTER 5
by tensoring df (Ao) on the left with the identity map on S,. Tensoring the Hopf algebra map
s, s, 6 s, +
on the right with the identity map on V, yields an element in Combine these two maps under the composition on thereby determining an element In general, any has an adjoint D* E Hom(Hom(S,, V2),Hom(S,, V,)) characterized by the condition
( N u 0 VJ, Lemma Let
T2) = (u 6 V1, D*T2).
-,Hom(l/,, V2) g: N o-+ Hom(V2, V,)
be holomorphic functions. Suppose that Then
[Needless to say, it is necessary to impose an obvious condition on No.] There is one final point in this circle of ideas which should be mentioned.
Lemma
There is a canonical injection u Hpu of S , into the algebra of polynomial functions on i1 such that PA
=A
for all A in . ' 3
We shall now take up the definition of Eisenstein system. The definition, in toto, is rather lengthy, so we shall break it up into a series of assumptions, developing the necessary preliminaries as we go along.
129
EISENSTEIN SYSTEMS
Let % be an association class of r-cuspidal split parabolic subgroups of G, gia G-conjugacy class in W. Let (Pl, s,;A , ) ( P 2 , s2;A , )
{
be members of qi.We then define an element W2IA2:
P1IAl)E
WA2,
,411
as follows. Select x in G with the property that x(P,,
s,:A,)x-'
=
( P , , s,; A,).
Put W 2 l A 2 :
PlIA,) = Int(x)IA,,
a definition independent of the choice of x. There are certain elementary properties inherent in this construction, e.g., transitivity. Less elementary but still easy are the conditions of descent.
Lemma Let %], %, be association classes of r-cuspidal split parabolic subgroups oj' G, W i t ,W i 2 G-conjugacy classes in %, , %,. Let
i
fP2,s;; A ; ) (pi, s;; A ; )
s;;A ; ) E W i P
(Ply
( P ; , s;; A ; )
with
i
E Wi2
s;;A ; ) 3 v 2 7 s;; A ; )
(Pl,
(Pi, s;; A ; ) 3 (Pi,s;; A';).
Then
I ( P y A ' ; : P;IA;)IA;= I(P;lA;:P,IA;). Lemma Let g1,%, be association classes of r-cuspidal split parabolic subgroups of G, gil,Vi2G-conjugacy classes in V1, %',. Let
with
Then
I(Pi I A'; : P,I A ; ) I 'A2
=
I("P, I " A , : ' P , 1 'A,).
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CHAPTER 5
Both of these facts will be used without comment in what follows. Let V be an association class of r-cuspidal split parabolic subgroups of G, V i and V j G-conjugacy classes in V. By the symbol
W(wj,wi) we understand the set of equivalence classes in
where
i
W ( A J ,A;) wyi E W(AY, A;)
Wji E
are declared equivalent iff
w'!. = I(P'!I A'!: P'. I A'.) 0 w'.. 0 I(p;I A ; : PI' I A;). J1 J J J J Jl
Observe that if
then wji n W ( A j ,A,) is a singleton; call it wji(PjIAj; P,IA,) (or wji). Let V, V, be association classes of r-cuspidal split parabolic subgroups of G, V k ,Vio,and VjoG-conjugacy classes in V, Vo. Suppose that
This being so, let
131
EISENSTEIN SYSTEMS
be the subset of W ( A j o ,Aio)comprised of those wJoiosuch that I ( P j I A j :PiIAi).
WJoioIAi =
Let then wtk(@j0 9 @iO)
be the subset of W(Kj0,qi,,)formed of those wJoiowhich factor through the triangle
- _- . _._
W ( q j 0 ,y i o )
- - _-. *-*+
W(Aj0, AiJ
t
WJk(Ajo9
for ail choices on the right.
Remark The data (Pi7 S i ;
Ail
( P j , S j ; Aj)
P (Pio,S i o ; AiJ
+ (Pjo,Sjo; AjJ
determines, in the usual way, triples
Because M i and M j may be distinct groups, one cannot, in general, ask whether (P!07St’,)and (PJo,SJo)are associate. It is for this reason that one is forced to introduce the set W t , ( A j o ,Aio). If, however,
i
(Pi0 Sio ; AiJ (Pjo, Sjo; AjJ 9
admit a common dominant in V k ,say (P, S ; A ) , then W&,(Aj0,Aio)can be identified with W(AJo7A!o), a set, though, which may be empty (since the relation of association is not necessarily preserved under the daggering procedure). Let “be an association class of r-cuspidal split parabolic subgroups of G , V i a G-conjugacy class in W. By an equivariant system X of admissible affine subspaces attached to W i we mean a map X which assigns to each
(P,s; A ) E qi
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CHAPTER 5
a nonempty admissible affine subspace X(P,A ) of the complexification of 6 subject to the following compatibility condition: For all
{
(P1,sl; A1)egiT (P29S2; A 2 1 ~ 9 7 i 7
w 2 ,
A2) = W
2 I A 2 :
PlIAl)X(Pl, A l l .
An X-admissible subspace x of X is an equivariant system of linear subspaces attached to V i such that for each
( P , s;A ) € 9 7 i ,
x(P, A ) is X(P,A)-admissible. The terms
{
“dimension of X” “dimension of 1’’
are to be used in the obvious way. I f q j is another G-conjugacy class in V, then one can associate with each element wji E W(Vj, gi)an equivariant system X .I, of admissible affine subspaces attached to V j by requiring that X,,,(Pj, A j ) = - w j i ( P j [ A j :PiIAi)X(P,, A‘).
Fix an equivariant system 3 of admissible affine subspaces together with an X-admissible subspace I. Let
( P , s;A ) E % i ,
The conventions and notation set down above are then applicable here. In particular, we can write X(P, A )
=
X(P,A)“ 0 X ( P , A )
and consider sX(P, A ) .
Let 6 be a K-type and 0 an M-type-then 6,,,(6, 0) is a finite-dimensional Hilbert space. Specializing the earlier discussion, we can say that there is a natural pairing that is linear in the first variable and conjugate linear in the second variable, characterized by the condition (u
0 0,T ) =
( 0 7
T(u*)),
and a natural pairing
s,,,4) 0
0)’ x Hom(S,(P, A ) , 6,,,(4 0))
~clIs(63
+
c
EISENSTEIN SYSTEMS
133
which is linear in both variables, characterized by the condition (u
0 6,T ) = (6,T ( u ) ) .
Since a polynomial function on a subspace of a may always be regarded as a function on G, the presence ofthe X(P, A)-admissible subspace x(P, A ) implies that there is a map
-
S,(P,A,
0 ~,",(6,0) Fnc(G) +
sending u 0 @ to p , @. We hardly need point out that this map depends on
W, A).
We shall now change our notation slightly and freeze the data. Fix an association class W0 of r-cuspidal split parabolic subgroups of G. Let W i , be a G-conjugacy class in W 0 , X an equivariant system of admissible affine subspaces attached to %.,; Suppose that %? WiO-then the set of all (P, s; Ai) in %? for which there exists a (PiorSio;Aio)in Wi0 such that
+
(Py S ; Ail
+ (Pi,, Sio; AiJ
is itself a union of G-conjugacy classes in %?; call it %(io). Let W kbe a G-conOne may then define an equijugacy class in % which is contained in %?(io). variant system x y k of linear subspaces attached to Wi0 as follows. Let (Pio, Sio;Aio) be a member of Vi,-then there exists a unique element ( P , S ; A i ) in %?k such that (P, S ; Ail (Pior S i o ; Aio).
+
We write X(Pio,AiC,)+for the orthogonal projection of X(Pio,Aio) onto the complexification of a/,. This said, let xvk(Pio,AiJ = 6i.
It is easy to check that x y k does in fact have the required properties. Write
%Go; X) for those G-conjugacy classes V kin %(io) for which xy, is X-admissible. Of course, V(io; X) could be empty. Let Wj0 be another G-conjugacy class in V 0 . Suppose that there exists gkin %?(io;3)such that %?k 3 WjO-then we can form Wtk(Wj,, Via), which, as we recall, is a subset of W(qj,, gi,).Each wJfoioin WLk(Wj0,Via) thus takes X to another equivariant system XW~,,,.If we agree that two elements of W?,(Wj,, Wi0)are to be regarded as equivalent when their action on X coincides, then we obtain a set of equivalence classes
WLk(X; gjo gio). 9
Let
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CHAPTER 5
Denote by
wtk(X;A j o , Aio) the set of distinct linear transformations from 3E(Pio,Aio) into the complexification ofEijoobtainedbyrestrictingtheelementsof W$,(Aj0,Aio)toX(Pio,Aio)then there is a canonical bijection
wtk(X;qjo,gio)-,Wtk(3E;A j o , AiJ We write ~ ( p j ,A, ~ ~ ) , J=~ ~-~WJoio Wpio
9
~
i
~
<
)WJoio E
Wtk(X ; A j o
9
Remark Suppose that ( P , S; Ai)
>r (Pio,S i o ; AiJ
The daggering procedure then determines a split parabolic subgroup (P,’,, Si’,) of Mi with split component A!o which, of course, is r,,-cuspidal. There are orthogonal decompositions
{hie
aio =
@ a,
=
tiit, 0 E i i .
X(Pio,Aio)tis an admissible affine subspace of the complexification of Ei!<) with normal translation X(Pio, Aio).Conventionally, an admissible hyperplane in X(Pi,, Aio)t is an intersection of the form ( I t E Cp?o(mi,a!,).
x(P,,, A ~ , ) +n ( I t = c )
Consider, then, those association classes %? such that (i) (ii) %‘(io;
+ X) # 0. q i o y
Fix a K-type 6 and an orbit type O0.The assignment to all
i
(P. S ; A i ) E W(io; X) (Pio,S i o ; Ai,J E q i o
standing in the relation
( P , S; A i l of a complex-valued function
+
(Pi09
s i o ; AiJ
E(X: P I A , : P i o ) A i oTo: :
x)
on Hom(Sx~pi,,Ai,),
Qio))
x X(Pi0, AiJ+ x G
135
EISENSTEIN SYSTEMS
'E(3: P ( A i :PioIAio:c,T,b + c,T,Zo:A!o: x) = c,E(X: P I A , : PioIAi,: Tib: A!o: X) c , E ( X : PIA,: PioIAio:T i : x)
+
E(3: P I A i : P i o ) A i oT : o :AJo:X) =0 if ord(T,,) 9 0 (uniformly in (A!o, x))
E(X: PIA,: P i o ( A i oTo: : A!o: xya,n) = E(X: P I A i : P i o I A i o :To:A!o:x)
HO~(SX,P A ,, ~~) ,a c u s ( h , 3
(ya,nE(rn P ) . A , . N )
OiJ)
to SX(P,(,.4,,)>~;o,o 0 a c u s ( h , OjJ such that
(V(3: P I ( A j , Ai):PjolAjo:PioIAio:wJoi0:A!o)To, Ti,) if
i
=
0
ord(To) 9 0 ord(qo) 9 0
is said to be an Eisenstein system belonging to infra are met in a nontrivial way.
X provided the assumptions
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CHAPTER 5
Assumption 1 (E) (Meromorphicity) Suppose that (PI S; Ai) 3 (Pi,, Sio; AiJ We shall then assume that
E ( 3 : P J A , :P i o ) A i , :To:A!,: x) is a meromorphic function of A!, in X(P,,, admissible hyperplanes. Let
whose singularities lie along
D(X:E : P I A , : PioIAi,) be the set of all A,toin X(Pi,, A,,)+ which admit a neighborhood N:, with the property that
E ( X : P ( A , :PioJAio: To:?: x)
Ti,, and all x. We shall then assume that E(X: P J A , :P i o ( A i o To: : A/,: x)
is holomorphic on M!, for all
is a differentiable function of (A!,, x) on D(X: E : PJA,:PiolAio)x G.
Let G M ,be a Siege1 domain associated with a r,,-percuspidal subgroup P M ,of M i . We shall then assume that
parabolic
IE(X: PIA,: PioIAio:To:A!,: hi)[
I C . II Toll* zpM,(mi)l
((kmi) E K
x G.M,),
C and r being uniform on cornpacta in D(X: E : P I A i : Pi, I Ai,).
Assumption 1 (V) (Meromorphicity) Suppose that
{
( P , S; Ai) 3 (Pi,, S i o ; AiJ ( P , S; A j ) 3 (Pj,, Sjo; AjJ We shall then assume that
(V(X: P J ( A j ,Ai): P j o J A j oP: i o J A i , wIoio: : A!,)T0, Tj,) is a meromorphic function of A!, in X(P,,, A,,)t whose singularities lie along admissible hyperplanes. Let D(X: V: P J A , :PiolAio) be the set of all A!, in X(Pi,, which admit a neighborhood N!, with the property that (V(3E: P I ( A j , A,): PjoIAj,: PioIAi,: wJoio:?)To, Tj,) is holomorphic on N!, for all wJoioand all To,Ti,.
137
EISENSTEIN SYSTEMS
Assumption 2 (E) (Equivariance) Suppose that ( P , S; Ail
3
(Pi09
Sio; AiJ
Let ni, E Nio-then we shall assume that
E(X: PI ni, Aini, ' : Pi, I n,, A i , ni, ' : n,, To:n,, . A!,: x) = E ( X : P I A , : PioIAi,:T o :A!,:
x).
Assumption 2 (V) (Equivariance) Suppose that
i s;
(P, Ail 3 (Pi,, si,; 4,) ( P , S; A j ) 3 (Pj,, S j o ; Ajo)*
Let ni,
E Ni,,
njoE Nj,-then
', '
we shall assume that
'
V(X : PI (njo~ ~ n ni, j ,~ ~ n'):i p, j , I njoA ~n,;, : p i , I ni, Ai,n,; 1.. njo~J,i,ni, : ni, * A!,)ni, * To = n j o .(V(X: P I ( A j , A i ) :P j o l A j o :PioIAi,: wjoi0: A!,)T0).
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CHAPTER 5
their common value being denoted by the symbol yPy-'. This said, we shall then assume that V ( X : ypy- I ( y j A j y j l , y i Aiy;
1):
yj
P , y;
l
IyjAjoyj 1 : y,P,,yr:
I YiA,,y;
:
YjwfOioy; : y i . Ait,)(dHornRytr(Aito)(To)) 0 7; * (~,R~,(w~,~,A,!,)(V P I( (XA: j , Ai): Pjol-4jo:Pi,( A,,: wJ,~,:AI,)IT;.,)).
= yj
Assumption 4 (E) (Transitivity) Suppose that ( P , S ; A,) with
+ (f",s';A;,) 3 (Pio,Sio; A,,)
X(P,,,A,,)"
3
a;,
3
hi.
hi,
=
h,td @ a;.
hi,
hit, @ 6, = 'hi, @ h i ,
There are orthogonal decompositions aio = aro t'
a;,
a,, = a!, @ a, a;, = 'a,. @ a,
so that
{
a;.
=
X(Pio,AiJ' = 'X(Pio, Ai,) O X(Pio,Aio)t'v
where, of course, %(Pi,, A,,) is simply the complexification of 'hi,. Let 9 ( X : E : PIA,: P'IA;,:P i , ( A i o )
be the set of all points A!o in 3 ( P i , , A,,)+ which admit acompact neighborhood N!,in X(P,,,A,,)+ such that (a) The projection A"!: onto X(P,,,A,,)" is contained in D ( X : E : P' I A;.: Pi, IA,,); (b) The series
1
Y E rnP/rnP'
~f,(~y)"*"-""'.E(3: P ' I A : . : PioIAi,:To:A/;: x?)
is uniformly convergent for A/, = 'A,, of G (any IT;.,). One can prove that
+ A&'in A"!, and x in a compact subset
9 ( X : E : PIA,: P'IAf.:PiolAio)
is a nonempty open connected subset of X(Pi,, A,,)' whose projection onto X(Pi,, is exactly D(X:E : P'IA;.:Pi,IAio). This said, we shall then assume that E ( X : P ( A , :Pi,(Aio:IT;.,: A/,: x) =
C
U~,(xy)"Ai'-'~~''.E(X: P'IA;,:PioIAi,:To:A!:
y~rnPlrnP'
for all A!,E~(X: E : PIA,: P'IA;,:PiolAio).
XY)
139
EISENSTEIN SYSTEMS
Assumption 4 (V)
I
(Transitivity) Suppose that
( P , s; Ai) 3 (Pj,, sj,;A;.) 3 (Pi,, si,; Ai,) ( P , s; A,) 3 (Pi,, sir;A;.) (P,,, s,,; A,,)
with
*
X(P,,,A,,)"
3
6;s 3 6,.
There is a canonical injection W&'L(X;A,,, AiJ
4
wtk(X;A j o , AiJ
This said, we shall then assume that
V(X: P I ( A j , Ai): P j o ~ A j oPioIAi,: : wJ:~,: A!,) = V(X: P'l(Ai,, A;,):PjolAjo:PioIAi,: wJdi0:A/:) for all WJiioE
W&'& A,,, Ai,)
if PI. = Pi. = P', say, but that
V(X: PI(A,, Ai): P j o ~ A j oPio)Ai,: : wJii0:A/,)
=0
for all WJii0
E
W&'& A,,, Ai,)
if Pi, and Pi. are not r-conjugate.
Assumption 5 (E-V)
(Negligibility) Suppose that
i
( p ,S ; Ail
3 (Pi,, Si,;
(f', S; Aj) 3 (Pj,,
Sjo;
AiJ A,,)*
We shall then assume that
E K x p j , ( X :PIA,: PioIAio:To:A!,: km,) WfOiO€
x
c
Wt.,(X;A,,.Aio)
aJo(mj)wfo~O"fo(V(X:
PI(A,, A i ) :PjoIAj,:P i o ( A i , :wloio:A&)&,)(kmj).
On the other hand, suppose that (P, S; A) is a dominant successor of some other r-cuspidal split parabolic subgroup of G which is not in W0,say
( P , s; A ) 3 (P',s';A').
We shall then assume that
E K x , p ( X :P I A : PioIAi,:To:AL: km).., 0.
140
CHAPTER 5
[An empty sum will, by convention, be set equal to zero. In this connection, observe that (P]o,S!o)and (PI,, S:,) could very well lie in distinct association classes when projected into P / A , N = P / A , N ; but this happens iff Wt,,(X; A,,, Aio) is empty.]
There is one additional assumption which must be made. Its role in the following developments is somewhat surreptitious, which is why we have chosen to isolate it. Assumption (Rep) There are two parts. (i)
Let k E K-then we shall assume that E ( 3 : PIA,: PioIAio:To:A!,: k x ) = E ( X : PIA,: Pio(Aio:L, To:A!o: x). 0
(ii) Let u E I,"(G) or C:(G;
a), so that
IndLio.Ai,.Ni,((oio A!Jcus) (a) E Hom(bcus(a, o i o ) , Scus(a, oio))* We shall then assume that
L,,,(a)E(X: PIA,: P i o ) A i , :To:A!o: ?)(x) = E ( X : PIAi: Pi01 Aio: &om Ind~,O.A~O.N~O((OiO, A!o)cus)(a)(To):A/o: x). [It is a question here, therefore, of E-functions; there is no direct counterpart for V-functions.]
Remark To say that an Eisenstein system belonging to X is nontrivial means exactly this: There are triples
I
( P , S ; Ail (Pi09 s i o ;
standing in the relation for which
Aio)
(PI S ; Ai) 3 (Pi09 S i o ; Aio) E ( 3 : P ( A , :Pio(Aio:?: ?: ?) # 0.
As will become apparent, it is necessary to make such an assumption in order to draw certain geometric and group-theoretic conclusions.In this connection, observe that if Wk 3 Wio,sWkbeing &admissible, then for some (P, S ; A,) in %?k and some (Pie, Sio; Aio) in W,,, with
it is true that
(PIS ; Ail 3 (Pi09 S i o ; AiJ E(X: P J A i :PioIAio:?: ?: ?) # 0.
141
EISENSTEIN SYSTEMS
In fact, if this were not the case, then for every (Pie, Sio;Aio)in W i o ,one would have
E(X:GI{l}: PioIAio: ?: ?: ?)
=0
(Cf. 4(E));
thus
V(X: GI({1}, {I}): Pjolz4jo:PioIAio:?: ?) = 0
(Cf.
5(E-V)),
so that
V(X: P ' J ( A ; ,A;,): , PjolAjo:PioIAio:?:?)
=0
(cf. 4(V)),
implying E(X: P'IA;,:PioIAio:?: ?: ?) = 0
(Cf. 5(E-V)),
which contradicts the nontriviality supposition. It should also be noted that the vanishing or nonvanishing of
E(X:PIA,: P i o ( A i o?:: ?: ?) is a function of the r-conjugacy class of P in Wk (cf. 3(E)). The assumptions which define an Eisenstein system are, of course, abstracted from those encountered in the study of Eisenstein series associated with cusp forms (cf. Chapter 4).The ambient group here is K x M i (or M i ) instead of just G itself. One needs this degree of generality for the inductive arguments which are to follow, although, ultimately, the final results will be phrased in terms of G alone. Remark Let {E, V} be an Eisenstein system belonging to X-then our assumptions imply that the E-functions, viewed on K x M i (or Mi), are (or rM,). automorphic forms per { 1} x rM, Example (The Canonical Eisenstein System) Let 3 be the equivariant system of admissible affine subspaces attached to Wio via the prescription X(Pio,Aio)= complexification of ai0.
In this case, then, W(io; X ) = W(io). Moreover, Sx(pio, A,o) = scalars, so that the map
ToH To(1) sets up an isomorphism H~m(Sxcp,~, Ado)
This said, suppose that
3
&cus(a,
oio))
+
&cus(a,
@iol
142
CHAPTER 5
Put
E(X:PIA,: PioIAio:To:A!o: x ) =
1
y E rn P / r nP i ,
uio(xy)(~~o-pfO) * T0(l)(xy).
This function has been encountered earlier. In fact, referring to the discussion prefacing the statement of Lemma 4.1, we see that E(X: PIA,: PioIAio:To:A!o: km,) ,
=
1
Y E r M l / r M , n p?,
- To(l)(kmiy),
uf,(mi)("?o-Pto)
the Eisenstein series
or still
O(kmi) = ( C ~ , , ~ ( PP!oIA!o: ~ ~ ~ wJoio: A ~ ~AJoio) : To(l))(kmi).
That these data do in fact constitute an Eisenstein system is a consequence of what has been said in Chapter 4.
EISENSTEIN SYSTEMS
143
It will become apparent in the end that all the Eisenstein systems of interest to us will arise from the canonical Eisenstein system by means of a residue taking process. Even though this procedure will not be of any immediate applicability, it seems appropriate to discuss it now since it serves to provide one with a deeper understanding of Eisenstein systems in general. Suppose, for the moment, that X is an affine subspace of C". Let f be a meromorphic function on X whose singularities lie along hyperplanes. Let fi be a hyperplane in X defined over R (which, though, need not be a singular hyperplane off). One can then define a meromorphic function resdf 1 on Sj as follows. Pick a real unit normal zl to $5. Set 1
P
6 being a small contour around the origin in the complex plane, so chosen that .f is holomorphic at z (zl. It is clear that resB(f) is in fact a meromorphic function on Sj whose singularities, moreover, lie along those hyperplanes obtained by intersecting 6with the singular hyperplanes off different from fi.
+
Remark The definition of residue depends upon the choice of a real unit normal. Changing the normal, however, has the effect of only changing the sign of the residue. As before, let X be an equivariant system of admissible affine subspaces attached to Xi,.By a hyperplane fi in X we understand an equivariant system of admissible affine subspaces attached to Wio with the following property: Forall(Pin,Si,;Ai,,)E %'in,fi(Pi,,Ain)isanadmissible hyperplaneinX(Pio,Ain). Fix a hyperplane Sj in X. Suppose given an Eisenstein system ( E , V} belonging to X. We shall now define an Eisenstein system { %e5,&E), %ee,(V)} belonging to Sj. This, as should be expected, requires some preparation. Fix an association class W, dominating Wio, such that W(io; fi) # 125then, of course, W(io : 3) # (21. Again, generically, vk stands for a G-conjugacy Sio;Aio)(or(P,,, Sjo; A,)) is a typical member of class in %?(io;B),while (Pi,, Wio (or W j o )admitting ( P , S; Ai) (or ( P , S; A,)) in %?k as its dominant successor. Suppose, therefore, that
and
144
CHAPTER 5
We must then construct functions %esB(E)(X: PIA,: PioIAi,:To:A!,: x) %es6(V)(X: P I ( A j , A , ) : P,,IAjo: PioIAi,:wJoi0: A!,)
i
that satisfy the various assumptions supra. Of course, the corresponding data in terms of Sj alone would be written
i
E ( 8 : P ( A , :PioIAi,:To:A!,: x) V(Sj: PI(Ai, A j ) : PjolAjo:PioIAi,:wJoio:A!,).
According to our notational principles, we can write 'B(PiosAi0) - 'B(Pco.Ai,)-X(Pi,,Aio) @ SX(Pio,Aio)' Pick a real unit normal AL to Sj(Pio,A,,) in X(P,,,A,,)-then A 1 is a real unit normal to @(Pi,, in 3 ( P i , , A d t . Let f be a meromorphic function on X(Pi0,AiO)' whose singularities lie along hyperplanes-then, on $(Pi,, the Laurent expansion gives
f(A!o + CAI) =
m
C an(A!JC", n= -N
the a, being meromorphic functions on Sj(Pio, Ai,)+ whose singularities lie along hyperplanes. This said, put
The definition is suggested by the following considerations. Let F be a holomorphic function on $(Pi,, A,,)'. Write
F(A,',
+ CAD = n
a l
=O
AAA!Jr",
the A, being holomorphic functions on $(Pi,, resb(Pi,. Aio)t N- 1
=
(fF)
1 u-,-
n=O
(A/o)
I(A!,)* AXA!,)
Then
145
EISENSTEIN SYSTEMS
More generally, let V be any complete LF-space (e.g., the symmetric algebra over a finite-dimensional complex vector space in its canonical topology). Let J' be a weakly meromorphic function on X(Pi,, Ai,,)' with values in V whose singularities lie along hyperplanes-then, on $(Pi,, the Laurent expansion gives m
the a,, being weakly meromorphic functions on &(Pi,, Aio)t with values in V whose singularities lie along hyperplanes. Explicitly,
This said, put
I
ReS&(Pro,Aio)t(f
A!,
N-1 H
1
1 -.n!
n=O
1:
8
S&(P~o,A~,)-~(P~o,A~,)
(Al)'' 6 u-,,- l(A/,).
We shall now define %ee,(E)(X: PIA,: PioIAio:Ti,: A/,: x).
For this purpose, note that E(X: PIA,: PioIAi,:?:
x)
may be regarded as an element of the dual of Hom(Sx(~,,,,A , ~ )& c u d s , 9
oiJX
so that, being of finite degree, it is represented by an element Q(X: P I A i : P i o ( A i , :A/o: x)
in
146
CHAPTER 5
It is clear that Q(X: P ) A i :PidlAio:?: x) is a weakly meromorphic function on X(Pio,AiO)’with values in SX(P*,,A,,)8 &cus(6,
oio)*,
a complete LF-space, whose singularities lie along admissible hyperplanes. Form Res,(,,o,A,o)t(Q(x: PIAil PioIAi,,: ?: x)), which, by definition, is an element of S B ( P ~ ~ x(P~,. . A ~A,,) ~ )8 - SX(P~,. ~ i ~ l Ccus(6, @ - SB(P,~.A,,) 8 Ccud6, oi,Jv* This being the case, let
oio)”
%esB(E)(X: P I A , : PioIAio:To:A!o: x) be the complex-valued function on H O ~ ( S ~ ( P , ~Ccus(6, , A , ~ )OiJ) , x $(Pi,, A d t x G defined by the prescription
%esB(E)(3: PIAi: P i o ( A i oTo: : A!,: x) = (Resg(~,~, ,,,,)t(Q(X: PIAi: PioIAio: ?: x))(AI0), To). On $(Pi,, Ai0)’, the Laurent expansion gives
E(X: PIAi: PioIAio:?: A/,
m
+ CAI: X) = 1 En(?:AJ0:X)Y, -N n=
the En(?: Ab: x) being meromorphic functions of A t in $(Pi,, Aio)t whose singularitieslie along admissible hyperplanes and differentiable functions of x in G. We have
%es,(E)(X: P I A , : PiolAio:To:AT,:
X)
Therefore %es5(E)(X:PIAi:PioIAio:To:Ato: x)
147
EISENSTEIN SYSTEMS
is a meromorphic function of A!o in B(Pi0,Ai0)+whose singularities lie along admissible hyperplanes.
Remark Suppose that is a point in $(Pi,, A d t . Let be a holomorphic function on some neighborhood of A!o in the complexification of ti!o with values in dcus(6, 0,)-then the definitions imply that P J A , : PioIAio:d@~o(A!o):A!o: x) resb(p,o,A,o)t(E(X: PIA,: PioIAio:dolo(?): ?: x))(Ait0).
%ee,(E)(X: -
It remains to define
%es,(V)(X:
P I ( A j , Ai): P j o l A j o :P i , ) A i o :wJoio:A!o).
There is a canonical injection from SX(P,,.Ai,) @ gcus(6, 0io)" @ S x ( p , , , Ajo),YjoioO acus(8, 0,)
into Hom(Hom(Sx(Pio,A,,)
9
gcus(8,
0iJX S x ( P j o ,Ajo),..:,,, 8 g c u s ( 6 , 0jJ),
the range of which is the linear transformations of finite degree. In particular, therefore,
V(X: P I ( A j ,AJ: PjolAjo:PioIAio:wJoio:?) can be viewed as a weakly meromorphic function on X(Pio,Ai0)' with values in S X ( P ~ ~ ,@ A ~'cus(8, ,)
OiJv O S X ( P j o . A j o ) j w ~ oO , O gcus(6,
OjoL
a complete LF-space (even though it is a question of algebraic tensor products) whose singularities lie along admissible hyperplanes. Form ReS6(Pio.A,o)t(V(X: P l ( A j , Ai): P j o I A j O :
P i o I A i o : Wjoio:
which, by definition, is an element of the tensor product of S 6 ( P i 0 .4
0 )
- X(Pi,. 4,)
with S X ( P ~ , . A6 ~ , gcus(6, ) OiJv @ S x ( P j o ,AjO),,,J,,,, 0 &'cus(d, 0jJ* The Hopf algebra morphism 's(pio. Aio)-x(pios ~
i o+ )
6S
S ~ ( p i A0 i ~o ) - x ( P i o , ~ i o )
~ p i Aoi o~) - x ( P i o * ~ i o ,
followed by the morphism ' B ( p i o . ~ , o ) - x ( ~ i~oi , o @ ) SS(Pio,Aio)-X(Pio.Aio) +
' B ( p i o , A i o ) - z ( p i o , Aio)
@ 'b(Pj0,
Ajo),,.J,,,- x ( P j 0 . AjO)rvjoio
148
CHAPTER 5
induced by ID 8 wJoio then leads, in the obvious way, to a map from the tensor product of
to
by the symbol %es~(V)(wJoio:?).
On fi(Pio,Aio)t, the Laurent expansion gives V(3: P I ( A j , Ai): PjoIAjo:PioIAio:wJoio:A!o
+ CAI)
the’V, being weakly meromorphic functions on fi(Pio,Aio)t with values in
and so
149
EISENSTEIN SYSTEMS
is a meromorphic function of Al0 in $(Pi,, Ai0)+whose singularities lie along admissible hyperplanes. There is a surjective morphism of restriction ResX-6: W&k(X;A j o , Aio) This being the case, let
+
W&,(B; Ajo,Aio)*
%es5(V)(X: PI(A,, A i ) : PjolAjo:PioIAio:wJoio:,A!o)
be the Hom(Hom(Sg(~i~, Ai,) valued function on
9
g c u s ( 6 , o i o ) ) , S 5 ( p j 0 , A , ~ ) , . . : ~O ~ , gcus(8,
W&k(sj;A j o , AiJ x defined by the prescription
$(Pi03
ojJ)
AiJt
%ee5(V)(X: PI(A,, A i ) : PjolAjo:PioIAio:wjoio:A!o) %es5(V)(*: A!o).
= Resx
We have
$(*) = w:oio
(%es5(V)(X: PI(A,, A i ) : P j o l A j o :PioIAio:wtjoio:A!o)To, To)
*(V-,-1(*: A!o)((Al)k v
To),(*AI)"-k v To).
Therefore (%es,(V)(X: PI(A,, Ai): PjolAjo:PioIAio:w : ~ AJo)T0, ~ ~ : qo) is a meromorphic function of A!o in @(Pi,, Ai0)+whose singularities lie along admissible hyperplanes. Remark Suppose that A!o is a point in $(Pi,, Let be a holomorphic function on some neighborhood of A!" in the complexification of 6!o with values in ECu,(6,Oio);let Y :o be a holomorphic function on some open set in the complexification of tijocontaining { - wjoio A!o : wjoio E w&k(sj; A j o AiJI 7
with values in 8,,,(h, Ojo)-then the definitions imply that (%es,(V)(X:
PI(A,, Ai): PjolAjo:PioIAio:wjoi0:
d@!o(A!o), dY~o(-~~oio~~o)), when summed over the wjoioin W&,(sj;A,,, Aio),is equal to
PI(Aj, Ai): PjolAjo: PioIAio:wJfoi0:?) d@,!,(?),dYJ0(- ~ j ~ i ~ ~ ) ) ) ( A ! ~ ) ~~S~(P,,,A,,)+((V(X:
when summed over the wJoioin W&,(X; Ajo,Aio).
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CHAPTER 5
Using the explicit formulas developed above, it is a straightforward exercise to verify that { %eeB(E),%es5(V)}is an Eisenstein system belonging to !+ (provided j the data are nontrivial). There is nothing to be gained by providing the details. On the basis of the assumptions alone, it is possible to relate the singularities of the E-functions with those of the V-functions. In this connection, recall that we have introduced sets
i
D ( X : E : PIAi: Pi,IAi,) D ( X : v : P I A , : PiolAio).
Our assumptions imply that both of these sets are open and dense in X(Pi,, A d t and that the complement of either one is a locally finite set of hyperplanes, which will be referred to as the singular hyperplanes of E or V, as the case may be. Proposition 5.1 Let { E , V} be an Eisenstein system belonging to X-then the singularities of E are contained in those of V. In precise terms, suppose that
(P, S; Ai)
+
(pi07 s i o ;
Aio).
Let H , be a singular hyperplane of E(X: PIA,: PioIAi,:To:A!,: x).
Then there exists a triple (P,,, S j o ; Ajo),with
( P , S ; Aj)
+
(Pj,, s j o ;
Ajo),
and an element wjoi0E W&,(X; A,,, Aio)such that H , is a singular hyperplane of
V(X: P ( ( A j ,Ai): P j o ~ A j PioIAi,: o: wjoio:A!,).
Corollary Let the notations and hypotheses be as above. Then D(X: v : P I A , : P i o ~ A i oc) D ( X : E : P I A , : Pi,IAi").
The proof of this proposition depends in an essential way on the proposition in the Appendix to Chapter 4,whose notation we shall feel free to use. There are but finitely many triples ( P o - l , S o - l ; A 0 - J ( I E I ) , standing in the relation
( P , S; Ai) 3 ( P o - , , So-1; AO-J, whose daggered parabolics are distinct modulo I-,,-conjugacy. [Observe that for distinct indices I' and I",
151
ElSENSTElN SYSTEMS
may very well lie in distinct association classes in M i . This, however, is irrelevant, our point being only that the ( P o - , , So-,;Ao-,), when daggered, describe a set of representatives for the r,,-conjugacy classes in the various association classes which do arise from goper the daggering procedure.] Now take
i
X = Cm(G/(T n P ) A i . N ) Y = fl Cm(K x M i ) (#(l)copies)
i
T:X-+Y
f H { f K x P b - , :
'EO,
X and Y being supplied with their usual topologies. Let Yo* be the weak*dense subspace of Y* consisting of the finite linear combinations of Dirac measures. Choose for U the set D(X: E : PI A i : PiolAio).Put
0 = E(X:PIA,: PioIAio:To). Then, as a map from U to X , 0 is weakly holomorphic, since, by assumption,
E(X: P I A , : PioIAio:To:A!o: x) is a differentiable function of (A!o, x) on D(X: E : P I A , : PioIAio)x G.
ProofofProposition 5.2 Let At be a point on H , which, without loss of generality, may be assumed simple in the sense that it does not lie on any other singular hyperplane of E ( X : PIA,: PioIAio:?: A!o: ?).
To arrive at a contradiction, let us suppose in addition that A: does not belong to any singular hyperplane of the V(X: PI(Ai, A,): P o - , I A o - , : PioIAio:W
A - , ~ ~Ai',). :
Let A be a polydisk around A: whose closure misses all the singular hyperplanes of
E(X: PIA,: P i o ) A i o?:: AJo:?), except for H,, and all the singular hyperplanes of the
V(X: P I ( A i ,Ai): P o - , l A o - , :PiolAio:w;-,~,:A!o), but whose silov boundary &A misses H,.Obviously, then, & A is contained
in U . If we could show that the hypotheses of the proposition referred to above were satisfied by our data, then we would be able to say that
0 = E ( X : PIA,.:PioIAio:To)
152
CHAPTER 5
extends to a weakly holomorphic function on U u A, which is an impossibility. There are, therefore, two things to be established. First, let C be a compact subset of U-then we must verify that T is one-to-one on alg-span{cTn(@(C))}. Thanks to Theorem 3.6, this is easy to do. If in fact
f E alg-span{5n(@(C))}, then f is a slowly increasing differentiable function on K x M i / { 1 ) x rMi (as follows from the growth condition postulated in Assumption l(E)). Assuming now that T(f 1 = 0, Assumption 5(E-V) then implies that the constant term off along any { l } x r,,-cuspidal parabolic subgroup of K x M i is negligible, hence that f = 0. Therefore T is one-to-one. To finish up, let
{ ( k ,miI): I E I} be a collection of points in ( K x M i ) (#(I)copies)-then we must show that E K X p j - , ( XPIA,: : PioIAio:To: k,mi,)
1 I
extends to a holomorphic function on U u A, which, however, is clear in view of our definitions and Assumption 5(E-V). // By a singular hyperplane of W&,(X; Aio,Aio) in X(Pio,Aio)t we mean a set of the form
H, = {A!o :w / ~ ~=~ A/o A (/ w ~
/ E~ W&,(X; ~ ~ Aio,Aio),wibi0 # 1)).
Let
D(X: W: PIA,: P i o [ A i o ) be the complement of the union of the singular hyperplanes of W&,(X; Aio,Aio) in %(Pi,, Aio)+. Proposition 5.2 Let { E , V } be an Eisenstein system belonging to X-then the singularities of V are contained in those of E and W. In precise terms, suppose that ( P , S ; Ail (Pi03s i o ; Aio) ( P , S ; Aj) 3 (Pjo, S j o ; AjJ
{
+
Let H,be a singular hyperplane of V ( X : PI(Aj, Ai): PjolAjo:PioIAio:wloio: A!o)To.
153
EISENSTEIN SYSTEMS
Then H , is either a singular hyperplane of
E(X: P I A , : PioIAio:To:Aio: x) or else is a singular hyperplane of W&,(X; Aio,Aio).
Corollary Let the notations and hypotheses be as above. Then
D(3:E : P ( A , :PioIAio)n D(X:W : PIA,: PiolAio) is contained in
D(X:V: PIA,: PioIAio). The proof of this proposition depends in an essential way on the proposition in the Appendix to Chapter 4, whose notation we shall feel free to use. One has
i
M j = K,, PJo Pjo = MIo. Ajo * iVjo
This being so, take
I
(MIo = M j o ) .
X = C"(K x M j o x Ajo)
Y= K
n Cm(Ajo) Mjo
T:X+ Y
f H {f(kmjo ?) :( k mjo)E K
x Mjol,
X and Y being supplied with their usual topologies. Let T,* be the weak*dense subspace of Y* consisting of the finite linear combinations of the maps which are evaluation on one of the entries of the product. Choose for U the set D(X: V: P I A , : PioIAio).Let @:U-,X
be the map given by the rule
WA!o)(k m j o , a f > = (aJo)w~ono"~o. (V(3E: P I ( A j , A i ) : PjolAjo:PioIAi,:wjoio:A ~ o ) ~ o ) ( k m j o a ~ o ) . It is clear that 0 is weakly holomorphic. ProofofProposition 5.2 Let A! be a point on H , which, without loss of generality, may be assumed simple in the sense that it does not lie on any other singular hyperplane of
V(3: P I ( A j , Ai): PjolAjo:PioIAio:?: AiJ?.
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CHAPTER 5
To arrive at a contradiction, let us suppose in addition that Af does not belong to any singular hyperplane of E(X:PIA,: PioIAi,: To:A!,: x) and that Ad does not belong to any singular hyperplane of W&,(X;Ai,, Aio). Let A be a polydisk around Af whose closure misses all singular hyperplanes of V(X: PI(A,, A i ) : PjoIAj,:PioIAi,:?: A!,)?, except for H,,and all the singular hyperplanes of E(X: PIA,: PioIAi,:To:A!,: x)
and all the singular hyperplanes of W$,(X ; Aio,Aio),but whose silov boundary dsA misses H,.Obviously, then, dsA is contained in U . If we could show that the hypotheses of the proposition referred to above were satisfied by our data, then we would be able to say that 0extends to a weakly holomorphic function on U u A, which is an impossibility. Since T is certainly one-to-one, we need only show that
extends to a holomorphic function on U u A. Owing to Assumption 5(E-V), one can express
EKxpjo(X: PIA,: PioIAio:To:A!,: k m j 0 a ~ , ) on A
H,as a sum over W$,(X; A,,, A s ) of certain terms, one of which is ( a ~ o ) w ~ o ~.~(V(X A ~ o: P I( A Ai) : PjoI A : Pi, I Aio: wjoio: A/,)T,) (kmj,ujo). Now fix ( k , m,,) E K x Mj,-then, on A - H,, E K x p j 0 ( X P: I A , : PioIAi,:To:A!,: km,,?) is simply an exponential polynomial qua a function of uj, E AJo.Because a -
,,
,,
given A!, E A lies on no singular hyperplane of W&,(X;Aio,Aio),distinct elements of W&,(X; A,,, Aio),when applied to A!o, have distinct images. Since exponentials of distinct linear functions are linearly independent over the ring of polynomials, it is possible to pick off each term of EKXp3,(X:PIA,: PiolAio:To:A!,: km,,?) in a holomorphic way, so long as A!, construct a map
EA -
-
H,.More carefully put, it is easy to
D(wjoi0:A!,): Cm(Ajo) Cm(AIo) (A!, E A), depending holomorphically on A!,, which, when applied to EKx,,j0(X:PIA,: PioIAio:To:A!,: km,,?),
155
ElSENSTElN SYSTEMS
gives t .
t
(?)wJo@’io.
on A
-
( V ( I : P I ( A j , Ai): PjolAjo:PioIAio:wJoio:A ~ o ) ~ o ) ( k m j o ? )
H , . The entity D(wjoio:AIo)(EKXpj$X:P J A i PioJAio: : To:A6: kmj0?))
then furnishes the desired continuation across H,. // It will be necessary to return to the subject of singularities and residues later on. For the time being, however, we shall study Eisenstein systems in general on an essentially “axiomatic” basis in the sense that we shall set forth a series of conditions and then deduce certain consequences when some or all of these conditions are in force, the interdependence of the various conditions also being a part of the investigation. Keeping to the earlier assumptions and notation, write
v,
= io
via.
Suppose there is attached to each Wio a collection xi0
=
{XI
of equivariant systems of admissible affine subspaces of dimension b. One sets
x(v0) = JJxi,. io
Now let %? be another association class. If, as usual, %?k is a G-conjugacy class in %?,then we put xio(%?k)
=
{X :%k c %?(iO; X))
and form x(Wk;
%O)
=
xio(vk>, io
a subset of I(%o) which could, of course, be empty. It will be convenient to introduce an equivalencerelation in x ( % k ; W),. For this purpose, suppose that %?k
Let
c
%?(io;
I)n W ( j o ; ‘I))
with
X per Wio ‘I) per g j o .
156
CHAPTER 5
be the subset of W&,(X;Wjo, %), consisting of those transformations w],,, such that X,jOio = 9.In terms of specific data, if (Pi07
sio;
Aio) E S i o
(Pjo, s j o ; A j J E g j o and if Ajo, AiJ
W&k(YI,
is the subset of W&,(X; A j o ,A,,) consisting of those transformations w],~, such that 3 ( P j o , A j J w J o i o = 9(Pjo*A j J
then there is a canonical bijection
wJk(YI, fi;
wjgjo, %iJ +
W&k(?l, Aj,, AiJ
This said, elements
{9 X
per Wio perwjj,
in x(%k; go)are declared equivalent iff
W&k(FD,X; w j o , ViJ is nonempty. As it stands, it is not clear that this relation is in fact an equivalence relation. Basically, what one needs to know is that
Wtk(3,X; w i o , giJ is nonempty. For the time being, we shall simply assume that this is so through the formulation and proof of the next proposition. Let X be an equivariant system of admissible affine subspaces attached to %,,-then by Dis(X) we understand the X-admissible subspace of X defined as follows. Let (Pie, Si,; A,,) be a member of %,,-then Dis(X)(Pio, AiJ is the distinguished subspace of a,, of maximal dimension which is contained in X(P,,, A,,)". In the event that the complexification of
(Pi, At,) 9
is actually equal to X(Pi,, A d " , then X is termed principal. There corresponds to Dis(X), in a natural way, an association class V(Dis(X)), containing a G-conjugacy class Wk(Dis(X))such that %?k(DiS(X))3 Wi0.
157
EISENSTEIN SYSTEMS
One can write
hi,
=
Dis(X)(Pi,, Ai,)t 0 Dis(SE)(Pio,Aio).
Let (PDis(X)
9
SDis(X); A b i s ( X ) )
denote the parabolic data determined by Dis(X)(P,,, Aio).We then have (PDis(X), SDis(X); A b i s ( X ) )
@
S i o ; Aio),
so that the daggering procedure determines parabolic data (Pbis(X), Sbis(X); Abis(XJ
in M b i s ( X ) . What follows is a list of possible geometric conditions which could be imposed on Xi,.
Geom: I (Local Finiteness) Xi, is locally finite in the sense that for all (Pi,, Sio;Aio)in Wio and every compact subset oioof the complexification of hi,, the set {3E E Xi0 X(Pio,AiJ n mi0 Z
01
is finite. Geom: I1 (Real Compactness) Xi, is real compact in the sense that for all ( P i , , Sio;A,,) in Wio, the set (Re(X(Pi0, Aio)) :X €Xi01 has compact closure.
Geom: 111 (Conical Containment) Xi, is conically contained in the sense that for all (Pie, Sio;Aio)in W i o , - Re(X(Pi0, AiJ) E
(hkis(2))
for every X belonging to Xio. There is also a group-theoretic requirement which might be invoked per gi0.
Gr For every 3E belonging to Xi,, there exists an element : W
E W(K
X; via, UiJ
such that for all (Pie, Sio;Aio)in Wio, w:(Pio I Ai,: Pi0 IAiJ IWPio, AiJ"
is the identity map.
158
so that
CHAPTER 5
WE, if it exists at all, is unique.]
So far we have said nothing about Eisenstein systems. Let us now suppose that for each io there is an Eisenstein system { E , V} belonging to each 3 in Xi,. We consider three conditions. ES: I Let 3 E xio(%?)'). Suppose that
Let wjoioE WZk(X; A,,, A'?); let w]oioE W&,(X; V,,, Vio)correspond to wjoio under the canonical bijection
E-S: I1 Let 3 E Xio(%?k). Suppose that
Let wJoioE W&,(3; A . , A,?; let wJoioE W&,(X; Wjo,wi,) correspond to wJoio J? . under the canonical bijection Then the adjoint
is given by
EISENSTEIN SYSTEMS
159
E-S: III Let X E Xio(Wk).Suppose that pt
= {pJo]is a completely equivariant system of polynomials, ie., to each j , there is assigned a polynomial pJ0e C[liJ,,] such that for all triples
(Pjh, Sjb; Ajb) (Pja, Sj8; A j d
standing in the relation
by means of scalar multiplication. Suppose that
Then
V ( 3 PI (Ai Ail : Pjo I Ajo: PioIAio: wJoio:A/o)(dHomp!o(A!o) ( To>) = d pJo(wJoi0 A!o) (V(X : P I ( A i , A i ): PjoI A j o : Pi, I Aio: wjoio: A!& To). 9
[Observe that any polynomial pt E C[li!o] which is invariant under the !" , gives rise in a natural way to a completely operations of W t k ( A i oAio) equivariant system p' = {pJo}of polynomials.] Proposition 5.3 Suppose that for each io, X i , is subject to
Geom: I Geom: I1 Geom: I11 and E-S: I.
160
CHAPTER 5
Then X(Wo) is a j n i t e set. Moreover, the normal translation in any member of X(Ws,) is real. Finally, given Wk, every equivalence class
(5
= X F k ; WO)
contains at least one principal element.
[This result is due to Langlands [p. 1831.1 We begin with two observations which essentially have nothing to do with Eisenstein systems as they are really consequences of certain general statements about automorphic forms. Suppose that
After grouping terms, we can write E K x p j 0 ( X :PIA,: P i o ( A i oTo: : A/o: km,) =
1aJo(mi)AJ.
Y’h(kmi),
h
where the A,! are distinct elements in the complexification of 6g and the &, are elements in C[a]o]@ 8,,,(6, Ujo), none of which vanish identically. Lz-(l) If for all triples ( P j o ,Sjo;Ajo)standing in the relation ( P , S ; Ail
it is true that for all h,
+ (Pjo,
WAf)E
Sjo;
AjJ
.a,Jo(qo)9
then
qo:~
E(X: P I A , :
7 ? )~E L: ~ ( Kx
M J { I }x
rM,)
for any T O E
H O ~ ( S X ( P , , , AJcus(6, ~~),
@iJ).
L’42) If E(E P I A , :P , ~ I A , T , : ~A:J ~?: ) E L ~ ( K
x M J { x~
rM,)
is nonzero, then for all triples ( P j o ,Sjo; Ajo)standing in the relation
(P, S ; Ail 3 (Pjo, S j o ; Ajoh it is true that A1 is real.
161
EISENSTEIN SYSTEMS
[Keeping in mind Assumption 5(E-V), L2-(1) follows from Langlands [p. 1043 while L2-(2)follows from Langlands [p. 1841.1 The proof of the present proposition, while not difficult,is a bit subtle. Let us break up the argument into a series of simple lemmas.
Lemma Let
a be an equivalence class in JE(Wk; %,).
Assume
xwk = Dis(X)
for eoery 3E E (5. Fix 3E E E, say X E JEi,(Wk). Suppose that
( P , S ; Ail
3 (Pi09 Sio; Ai,,).
Then there exists a neighborhood A'" o f X ( P i , , Aio)in X(Pi,, AiJt such thatfor every A!, in JV at which both the E-jiunction and all V-jiunctions are defined,
E(X: P ~ AP~~ :~ I AT": ~ ,A!,:: ?) E L ~ ( Kx ~ ~ / { xi rMi) } for any
T o E Hom(Sr(pio,A i , )
9
@io))*
Proof Suppose that ( P , S ; Ai) 3 (Pj,, Sjo; AjJ
Then w:O,,eW: 'w:,,,,,
n
(Wjoi0)-
k ( x : w J O *@to)
'(PjOIAj,: P i , I A i o ) ( d ~ ~ , ( 6 j+ J J-16jo)
Exm(wk)
intersected with X(Pi,, Ai0)+is an open subset of X(Pi,, which, in view of Geom: 111, contains X ( P i , , A,,). Let N be the neighborhood of X ( P i o ,A?) in 3E(Pi,, A,,)+ obtained by intersecting these sets over all possible triples (P,,, Sjo;Aj,) standing in the relation ( P , S ; Ail
3 (Pi,, Sjo; ' 4 j J
We explicitly note that N is in fact open, as the intersection can actually be taken over a finite set of indices. If A!, is a point in N , at which both the Efunction and all V-functions are defined, then
E K X p ~ , ( PX (: A i :PiolAio:To:A!,: kmi) =
1 a~o(mi)Al .Th(kmi), h
where, in view of the definitions and E-S: I, R e ( 4 ) E dPJ,,(6J0) for all h. Our assertion thus followsfrom the square integrability criterion cited above. JJ
162
CHAPTER 5
Let us keep for the moment to the assumptions and notation of the preceding lemma. We then claim that, of necessity, the A!o in N, at which both the E-function and all V-functions are defined, are real (and hence that every point in J(r is real). In fact, if this were not the case, then (cf. supra), for some such
E (I : PIA,: PioIAio:To:A!,,: ?) = 0 for any
T o E Ho~(SX(P,,, A,,)
9
&cuds ,
@iJ)*
Consequently, E ( 3 : P I A , : P i l ~ ~ / l?:i l A!(,: , : ?) = 0
for A/o in an open set, thus everywhere. In other words, E ( 3 : PIA,: PioIAio:?: ?: ?) = 0.
By making a suitable replacement of the data if necessary, this fact can be used to contradict the nontriviality of the Eisenstein system belonging to 3. Every point in J(r is therefore real. But then W P i o , A,,)+ =
{X(Pio, AiJ>,
which implies that X ( P i o ,A,,,) is real. Moreover, in the case at hand, 3 ( P i o ,Aio)is actually equal to the complexification of a,, i.e., 3 is principal. To summarize, if the special circumstances of the above lemma are in force, then the normal translation in any member of (E is real, and, in addition, every element of (E is principal. What is needed now is a lemma which will allow us to reduce to this particular case. ~~a Say
Let
3 E %io(%k),
an equioalence class in x ( % k ; W0)--then there exists X E (E, and q k , with
(E be
such that and such that for every X' in the equivalence class (E' of 3 in 3(%;;W0), one has 1%; = Dis(3').
Proof The proof will be by induction on n0(Wk)where, by definition, no(%k)= rank(V0)-rank(Vk)
(rank(Wk)= rank(%)).
Suppose first that n0(Wk)= 0-then, of course, %?k = Wi0 for some io and the assertion is clearly true. Assume, therefore, that n0(Wk)> 0. If %k itself does
163
EISENSTEIN SYSTEMS
not work, then there exists X, E @ such that Dis(Xo) # xy,. Since no(gk(Dis(Xo)))< the induction hypothesis implies that there exists an x(qk(Dis(XO)); WO), per Vi0say, equivalent to 3, in iX(Wk(Dis(X,J); W,), for which the various assertions of the lemma are true. But then, a fortiori, X E @, so we are done. // Corollary Let @ be an equivalence class in x ( % k ; %,)-then the normal translation in any member of Ci is real, and, moreover, @ contains at least one principal element. [It is clear that the normal translation in one member of 6 is real iff this is the case for every member of 6.Thus, there is no loss of generality in considering only an element X of @ having the properties spelled out in the preceding lemma. Taking into account what has been said earlier, we then conclude at once that the normal translation in 3 is real and that X is principal.]
Remark Let @ be an equivalenceclass in iX(Wk; q0).Let 3 be a principal element [email protected] @(Dis(X))be the equivalenceclassofXinI(Wk(Dis(X));Wo)then it is clear that all the members of Ci(Dis(X)) are themselves principal. To complete the proof of our proposition, we need only establish the following lemma. Lemma X(Vo)is a j n i t e set. Proof Fix io-then it will be enough to show that Xi, is finite. For this purpose, fix ( P i o ,Sio;Aio) in Wio. Since the normal translations are real, Geom: I1 implies that the closure oioof the set {X(Pio, Aio): 3 E xio}
is compact. Owing to Geom: I, the set
{X E x i 0 X(Pio,Aio) n wi0 # W } is finite. This set, however, is just Sioitself. // Scholium Let @ be an equivalence class in element of@, say X E xio(%k). Then
%(%?k
; go).Let X be a principal
E(X:PDis(x))Abi,(x):PioIAio:To:X ( P i 0 ,/Iio):?)
164
CHAPTER 5
[As will become apparent later, this fundamental fact (contained, of course, in the preceding discussion) is the point of departure for the construction of residual forms and their associated Eisenstein series.]
Remark We remind the reader that it has been supposed up until now that W&,(% X; %i,, %i,J is nonempty. What Langlands does to ensure this is simply to assume that Gr holds, which, however, is deceptive since the particular nature of , !w plays no role whatsoever in the proof of the proposition. To even state the next result requires a fair amount of preparation. It will be assumed that the assumptions set down in the preceding proposition are in force here. Fix %,'; fix an element ( P , S; A i ) in W k . Let (E be an equivalence class in %(%,'; go).Let X be a principal element of (E, say X E xi,(%,').Fix an element (Pie, Sio; Aio)in Vi, such that
( P , S; Ai) @ (Pi07 Si,; AiJ Suppose that %,' 3 W j 0 . Let Vj,(PIAi)be the subset of Wj,, consisting of those elements (P,,, S j o ; A,,) such that
+
( P , S; Ail
(Pj,,
Sjo; AjJ
Observe that WjO(PIAi)is not empty. Moreover, any two members of Wj0(PI A i ) are necessarily Mi-conjugate. Now it is clear that rM,\ % j o ( P
IAil
is a finite set. We shall want to choose a set of representatives for it. This, however, must be done in such a way as to reflect an additional ingredient, viz., %,'(Dis(X)). The basis for making our selections is the following simple fact.
Lemma Any member of %,'(Dis(X)) which is a dominant successor of some element in Wjo(PIA i ) is necessarily a member ofWk(Dis(X);PI Ai). [We omit the elementary verification.] Let (pDis(X)
SDis(f); A&s(X))
9
be that element in %,'(Dis(X)) such that Let
( p , s;
*
(PDis(Z), SDis(X); Abis(x))
I(Ptlis(x,
9
3
Stlis(x); A#is(x))}
sio;
Aio).
EISENSTEIN SYSTEMS
165
be a set of representatives for r M ,\ wk(Dis(x); P
I Aih
the first of which being just (PDis(T)
itself. Fix k,
EMi
9
SDis(T); Ahis(X))
with the property that k, conjugates (PDis(X), SDis(X)
; Abis(X))
(Pt;is(x)
;A#is(x))*
to
Remark Parabolics r-conjugate.
St;is(x)
corresponding to distinct indices p cannot be
Given Wjo such that %?k@ W j 0 , there are two possibilities: Wk(Dis(x)) # % j O Wk(Dis(X)) 3 W j o . Let {(Pjovor Sjovo; A j o v o ) }
be a set of representatives for
r M i \ W j & P IA i l If the first possibility obtains, then we make no special assumption. If the second possibility obtains, then we assume, as we may, that each (Pjovo, Sjovo; Ajovo)
is a dominated predecessor of some (PL;is(x) SL;is(x);A%s(x))*
This agreement furnishes a partitioning { ( P j o v o * Sjovo; A j o v o ) )
=
{(Pjovo, Sjovo; Ajovo)}p Ir
into nonempty disjoint sets, where, as the notation suggests, each member of {(pjovoy Sjovo; Ajovo)}p
is a dominated predecessor of (Phis(%)St;is(x);
166
CHAPTER 5
Fix an element (Pjo, Sjo; A j J
in {(Pjovo, Sjovo; Ajovo)}
where, when the second possibility obtains, (PDis(I)? SDis(X); Ahis(X))
3 (pjo,
'jo).
Fix kjovoE M i with the property that kjov0 conjugates ( P j o , Sjo; A j o )
to ( P j o v o , S j o v o ; Ajovo).
Remark
r-conjugate.
Parabolics Pjov0corresponding to distinct indices v,, cannot be
Let '2) E a, say '2) E JEjo(Wk)-then we form HOrn(Sg(pjo,
=
9
gcus(6, o j o ) )
@ Hom(Sg(p,,,,, A,,,o),
acus(6, ojovo))
yo
and s4)(Pjo.Ajo)
=
(8&cud6,
Ojo)
0S~(Pjoyo, yo
8 &cus(6-
AjoYo)
QjovJ
It will be convenient to view the elements in either space as column vectors. The next thing to do is to construct transformations
V : Horn -+ @. The philosophy here is entirely analogous to that encountered earlier in the study of the c-function. Let%ll,%E(f,saY '2) 1 JEjb(wk) '2)2 zj$(%k)*
Let
161
EISENSTEIN SYSTEMS
We shall then denote by
V(E: 92:91: PIAi: w;ajb: AJh) the linear transformation from Hom(SQ,(Pj,:.Aj:,), &CUS(~,
to
Ojh))
S Q ~ ~A,;~) A0, & c u s ( 6 , O j $
defined by the matrix CV(91:
PI(Ai, Ail: P j d v 8 I A j a . a :
P j d v b l A j A v d : k j d . 8 . lt3J6jd
.k,l,:~:k j h v h .AJh>I.
Observe that V(E: 0 2 : 91: PIAi: W;;,A: Ajh) is a meromorphic function of A b whose singularities lie along admissible hyperplanes and has a finite dimensional range, which, by definition, is its rank. In order to describe the functional equations which the V-functions must satisfy, it will be best to form still another, larger matrix. For this purpose, we require some additional definitions. Henceforward, we shall suppose in addition that for each i o , Xi, is subject to Gr.
Lemma Let g1,g 2E E, say
i
'9 1 E xjh(wd '92
Then: The products
E xjs(wk)*
wJ?a .b . w 10P . ( wJblo ?.)-', where
w;dio E W&k(TJ 1, x; wjh ViJ wJaioE W&k(O2 x; gj8 WiJ
{
7
9
9
describe the elements of W&k('923
01;gj6, vjjd).
[The point here is that ( w : ~ = ) ~ 1.1 Introduce now
W i k ( Ex: wio) =
jo
WLk(9,x; vjo, viol I)Ex,g(wk)
rgoe
168
CHAPTER 5
and
There is a canonical injection One may attach to each W f E WJk(@:3:W i 0 )
a unique index j,(wt). 3, be subsets of W&,(@:X: gi0).We shall then denote by Let
V(k,: Z , : P J A , :A/o) the linear transformation from to
Observe that
V(3,:3,:P I A , : Ale) is a meromorphic function of A t whose singularities lie along admissible hyperplanes and has a finite dimensional range, which, by definition, is its rank. If k, = S, = 3,then we write V(S: P ( A , :A/o)
in place of
V(k:e: PI A i : A/o). There are two cases of special importance, namely, when
k = W$,(@: 3: qi0), in which case we write V,,((E.: X: PIA,: A:,),
EISENSTEIN SYSTEMS
169
and when =
Wtk(Discr))((F(DiS(3)): 3: %?io),
in which case we write Vwk((F: X : PI AJ.
We hasten to point out that the last piece of notation is justified since the matrix in question really is a constant, as follows from the definitions and our assumptions. Let us convince ourselves that this is in fact so. A typical entry of the matrix in question has the form V(@: Xw;: X,;: P ( A i :w~*w~.(w])-': ( W ] W ~ ~ ( P ~ ~ ( , ~ )PiolAio)AJ0), IA~,(,~): where, of course, Set
w!, wi E W~,(Dis(x))((F(Dis(X)): X: Uio).
There exist indices p, and p2 such that
170
CHAPTER 5
Proposition 5.4
Suppose that for each i o , Xi, i s subject to Geom: I Geom: I1 Geom: 111, Gr,
and
E-S: I E-S: 11. Fix
(%?k,
an equivalence class @ c X((%?k; ‘KO),
and a principal element X of Ct, say X E 3i0((%?k). Then: The ranks of the matrices
{V,,(@: V,,(@:
X: PI A i : AI0) x : PI Ai)
are the same at all points N o
E W p i o , AiJ’
at which
V,,(@:
X: P I A , : AL)
is dejined.
[This result is due to Langlands [p. 1911.1
Remark It will be seen eventually (cf. Chapter 6) that this “equality of the ranks” is the basis for arriving at the functional equations satisfied by the Eisenstein series associated with residual forms. The proof of the present proposition is difficult, the argument being involved and lengthy. Set nx(Wk)= rank(Wk(Dis(X)))-rank((%?k). We shall proceed by induction on nZ(Wk). If nI(Wk)= 0, then %k = gk(Dis(X)), so that there is nothing to prove. Assume, therefore, that n f ( V k )> 0. The case when nI(%?k) = 1 requires special attention and will be treated first. Having disposed of this possibility, it will then be assumed that nX(Wk)> 1, the assertion of the proposition being supposed to hold for all integers less than ndqk). There are some generalities which it will be best to take care of before getting involved in details.
171
EISENSTEIN SYSTEMS
Technical Lemma (T-L) Let 8 , ,8, be subsets of Wtk(@:X: qi0). Assume Elc 8,. Thefollowing statements are then equivalent: (i) (ii) (iii) (iv) (v)
V(8,: 8 , :PIA,: and V(8,: PIA,: At,) have the same rank; haoe the same image; V(E2: 8 , :PIA,: A!o) and V(3,: PIA,: V ( 8 , : 8,: PIA,: A f ) and V(8,: PIA,: A!o) haoe the same rank; V ( 8 , : 8,: PIA,: Ai',) and V(8,: PIA,: At)) have the same kernel; haoe the same rank. V ( 8 , : P J A , :A!J and V(8,: P J A , :
[The verification of the equivalence of these five statements is straightforward, hence will be omitted. It should be noted that the adjoint
V(E2: 8 , :PIA,: A!o)* of V(8,: 8 , :PIA,: Ale) is given by
V ( 8 , : 8,: PIA,: -wi",!o), as can be checked without difficulty (using E-S: II).]
Remark In T-L, it is assumed that A!o ranges through an open dense subset of X(Pl0, Since rank is locally constant, in order to check T-L, it suffices to restrict AL to some nonempty open subset of X(Pio,Aio)t. Let 'I) E @, say g E xjo(%k).Given Tjo= (qoyo) in
Hom(Sw(~,~, A ~ ~&cus(6, ) ,
fljo))?
set
E('I): P I A , : Ti,: AJo:x) = E('1): PIAi: PjOvO~AjovO: 7jov0:kjovo Ajo: x),
1
9
yo
a meromorphic function of AJowhose singularities lie along admissible hyperplanes. Lemma Let 1 ' ) E (5, say 'I) E Ij&%,)-then
,
,
EKxpj6u6('I)l: PIA,: Tjb:Ag: km,)
= c
wZE@nXj@k)
w&
c
EW:~(C()Z,C(I;w J & s j b )
at (mi)kJdd'w~6Jb'h~b J M
x ( E J ~ oV(@: ~'~
where is the canonical projection.
0,:
'I),: P I A , : w J ~ ~AJA)Tj$(kmi), A:
172
CHAPTER 5
[This assertion is a direct consequence of the definitions.] For the sake of simplicity, we shall agree to write
so that
Let 3 be a subset of W$,(@: X: %,)-then g=
U gjo
there is an obvious partitioning 4
jo
into disjoint sets. This said, put Given T = (Tjo(wt))in
Hom(sx, &C",(&
set
flo)),
E(3: PIA,: T: Ajo: x) =
C 1 E(X,+: PIAi: Tj0(wt):W ' W ~ ~ ( P PioIAio)A]o: ~ ~ I A ~ ~ :x), jo zj0
a meromorphic function of A!o whose singularities lie along admissible hyperplanes. Owing to the preceding lemma, we have
EK.P~oYO(S: PIA,: T: ATo:hi) t
t
c
WjoioEWF&:
3E:Yio)jo
atJoVo(mi)k,ovo.WJoio.Ai6
x
( E t P , V ( { W ~ ~ ~8, }::P J A , :AJ0)T)(kmi), 0
where E$joI;o is the canonical projection of
8 &cus(a3
%w~o,,,(~ ~ j oj .
~
)
Ojo)
onto the vo th-factor. Let A!o E X(Pi,, Aio)t-then, for brevity, we shall say that ATo is a general point if the various
.
.
kjovo WIOio A!o
are distinct.
173
EISENSTEIN SYSTEMS
We shall now make a remark preliminary to establishing a criterion which will serve to simplify our problem. Let Uv be a nonempty open subset of X(Pio, Aio)t consisting of points at which Vwk(C?:X: PIA,: AJ0) is defined. Suppose that
E(B: PIA,: T:
?)EL’(K x Mi/{l} x
rM,)
forall A!o E Uv-then,ofnecessity,E = O.InfacttheexponentskjOvOwJoio.A!o appearing in E K ~ P J ~ ~P~J(ABi :T: AL: ?) cannot all be real (since nx(Wk)> 0), so that, varying vo and j o , the assertion follows from L2-(2)supra. Consequently, upon taking for AIo a general point in Up, the V-terms appearing in
EKxp~o,o(B: PIAi: T: A!o: ?) may be split off; thus, varying vo and j o , we then infer that V(?: 8 : PIAi: AL)T = 0 for every subset of W&,(C?:X: Wio). By continuity, this relation continues to hold at any A!o in Up.
Criterion Let U be a nonempty open subset ofX(Pio,Aio)t. Let B be a subset of W&,(C?:X: Wio). Suppose that for every wJOsE W&,(C?:X: %?io)jo
-
Ej0,
AI0 E U,
we have Re(kj0vo
WJoio
. A!J
E~p~~~,(~Jovo)*
Then rank(V,,(C?: 3: PIA,: AIo)) = rank(V(8: P J A , :A!o)) at all points
E
U at which
Vwk(C?: X: PIA,: A!J is defined. [Note that the condition on U is geometric in character.] Proof
Thanks to T-L, with
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CHAPTER 5
it will be sufficient to prove that
T EKer(V(E: W&,(@:X: Wi0):P I A , : A/o)) * T E Ker(V(W&,(@:X: Wio): P I A i : A!o)). So suppose that
V(E: W&,(@:X: Wio):P I A i : AI0)T = 0. In view of what has been said above, the fact that
V(W&,(@:X: Wio): PI/&: AI0)T = 0 will follow when it is shown that E(W&,(@:3: % i o ) : P I A , : T: A!o: ? ) € L 2 ( K x M i / { l > x
For this purpose, fix j o and
yo.
rMi).
Let wJoioE Ejo-then, of course,
V({wjoio}:W&,((F:X: WiJ: PI.4,: A!o)T = 0. On the other hand, if wJaio E W&,((F: X: %io)jo exponent of
- 8,,, then, by
hypothesis, the
v({ wjOio}: w&,(@: x : wi0):P IA,: A!~)T
E;!;;
in the constant term of E(W&(@: X: Via): P I A i : T: AJ0:?) along K x P ~ o vhas , its real part in &p~OY0(6~ovO). As j , and vo are arbitrary, L2-( 1) supra implies that
E(W&,(@:X: Wi0):PIA,: T: A!o:
?)E
L2(K x Mi/{l> x
rM,),
//
thereby completing the proof.
Let E be a subset of W&,(@:3: Wio). Form
u,
=
X(Pi0, Aio)+
n Ln n n JO
WlOiO
yo
(kjovo.
J
~ j o ~ ~ , ~ ~ ~+ ~~ ~- ~1 ~ ao ~~ o~ ~~ ~ ~j
where,foreachj,, wjOiorunsthrough thecomplementofEjoin W&,((E;:X: Wio)jo. The preceding result is then applicable to U,, provided Usis nonempty. Now this will always be the case for certain 8. For example, let %(%?k)= {w’ E w&,(@:X: %io) Wk(DiS(Xw+))# %k}. must conThen Geom: I11 guarantees that Up,yk)is nonempty (since Us(%,) tain the normal translation X(P,,, &)). Observe that it is enough to verify “equality of the ranks” on some nonempty open subset of X(Pio,AiJt, it being understood, of course, that we
175
ElSENSTElN SYSTEMS
actually consider only those points at which V,,(@: X: P I A , : AI0) is defined. [In this connection, let us bear in mind that rank is lower semicontinuous.] With this preparation, we are ready to begin the proof of our proposition. We shall treat the “rank one” case first, i.e., the case when nx(%k) = 1. Let us indicate the strategy. In this situation, E(%k)can be alternatively described as the subset of W$,(@: X: Via) made up of those wt such that X,+ is principal. Owing to what has been said above, on U=(,,) one has rank(V,,(@: X: PIA,: AI0)) = rank(V(k(Vk): P1.4,: AIo)). We would be done if we knew that there was a nonempty open subset U of Us(%,)on which this relation were true with a(%&replaced by W$,cDis(x))(@(DiS(3)> : since then v(E(%k): P 1 A i : Ai’,) is replaced by V,,(@: X: PIAi), whence rank(V,,(a: X: P I A i : /\Io))
=
rank(V,,(Q: X: PIAJ),
as desired. To be able to draw this conclusion, it would clearly be sufficient to produce a nonempty open subset U of U=(,,) with the property that for every Wjc,io E E(%k)jo-
W$,(Discx))(@(DiS(X)): X: %io)jo,
A!o E U ,
176
CHAPTER 5
is
Because X(Pio AiJ E .('%?k) 9
U is a nonempty open subset of X(Pio, sumptions and notation, suppose that wjoio
=(gk)j0
9
Keeping to the preceding as-
- Wt,(Dis(x))(a(Dis(X)):3E:%iu)j0?
EISENSTEIN SYSTEMS
177
The proofofour proposition in thecase when nr(Wk) = 1 is thereforecomplete.
178
CHAPTER 5
Remark Suppose still that nS(%‘k)= 1-then it is always the case that =(%k)
# Wt,,Di,ci))(@(Dis(XE)): x: %io).
To see that this is so, proceed by contradiction. This time, let U be the intersection of Ul(vk)with {X(Pio, AiJ
+ A!o:
Re(Ait,)E ~P~,.,.,(~t)is(X)))-
Take for $, in the Criterion supra, the empty set. Arguing as before, we find that for every wJoioE W$,(@: X : %‘io)jo,
A!o E U ,
we have Re(kjovo. WJoio . A!J E ~ P ~ ~ ~ , ( ~ J o v o ) * Consequently. V,,(@: X : PIA,: ?)
=
0.
By making a suitable replacement of the data if necessary, this fact can be used to contradict the nontriviality of the Eisenstein system belonging to X . The principle embodied here is important. Indeed, it is the basis for decomposing elements of W&,(@:X: Via) into products of “simple reflections” (cf. infra). We shall now suppose that i?x(%k)> 1 and use induction. It will be convenient to first set up some preliminary machinery. Let W‘ be another association class, Vka G-conjugacy class in V. Suppose that %k 3 W‘k with rank(%‘:) > rank(Wk). Let @‘ be an equivalence class in X(48k; go),contained in @. Let X’ be a principal element of &‘, say X’ E Xib(W:)-then nzp(g:)< n.d%k)*
Given
WTbi0 E WL,(x’,
X ;Wib, Wi0),
introduce
B(@‘: X ’ : wt.. ) = { W t ’ w;. w&io:W t ’ E W&(@’:x’; Wib)} *
1010
and
B(@’(Dis(X’)) : X’ : w&~,) = {Wt‘ . W i W&io: W t ’ E W~~(Djs(x,))(@‘(DiS(X’)): 3’:Wib)}, *
these being subsets of W&,(@:X ; Wi0).
179
ElSENSTElN SYSTEMS
There are three statements basic to our development. Let AL, be a general point in X(Pi,, A,,)+.Then Im(V(E(6’: 3’:w!”,,): E(W(Dis(X’)): 3‘:wii0): PIA,: A!,)) = Im(V(E(@’:3’:w!~,,): PIA,: A!,));
S, SII
Ker(V(H(W: 3‘:w & , ~ )P: J A , :A!,)) = Ker(V(W&,(@:X: Via): =(@’:3’:w&~,):P I A , : A!,));
S,,, Im(V(W$,(@:X: W,,): E(@‘:X’: wJoio):PIA,: A!,)) = Irn(V(W:X: %,,): E(@’(Dis(X’)):X’: w!~,,): PIA,: A!,)). The proofs of S, and S,, are similar, the main difference being that the proof of S, depends ultimately upon the induction hypothesis, while the proof of S,,does not. In order to avoid interrupting the exposition, we shall postpone them for the time being. The final statement, S,,,, is a formal consequence of S, and S,,, so we might just as well dispense with its proof now. The left-hand side clearly must contain the right-hand side, so we need only establish that the left-hand side is contained in the right-hand side. Let T belong to the domain of =(@’:X’: w & , ~ )P: I A , : A!,). V(W&,(@:X: qi0): Then T belongs to the domain of
V(E(Q?: 3’: wki0): P ( A , :A!,), so that, by S,, there exists TDisin the domain of V(E(@’:X’: w / ~ , ~E(@’(Dis(X’)): ): X’: wIbi0):PI A , : A t ) such that V(E(@’:X: w$,,~):P I A , : A!o)T = V(E((%‘:X’: w&,,,):E(C%’(Dk(X‘)): 3‘:witbi,): PI A i : Ajo)TDis. Let TDisbe the column vector with coordinates of TDis in E(G’(Dis(3’)): X’: w&,,) but zero in
E(W:3’:wTbi0) - B(@’(Dis(X’)):X’: wbio). Then T - TDisbelongs to the kernel of V(E(@’:X‘: w!~,,): P I A , : A!o), so that, by S,,,
V(W&,(@:X: %io): =(@’IX’:
w/bi0): PIAi:
AL)(T - TDis)
= 0,
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CHAPTER 5
from which it follows that v(w&,(@:I:Wio): a(@': x': wIbi0): PIAi: A!o)T = V(W&,(@:3:Wg3: E(@'(Dis(X')): X': w&~): P ( A i :A!o)TDis. Let A!o be a general point in UI('ok)at which VWk(@: X: PIA,: is defined. We shall prove that rank(V,,(@: X: P I A i : A!o)) = rank(V,,(@: X: PIAi)). Modulo SI and SI,,this will finish the proof of our proposition. Let q % ? kPr) : = {w+ E W&,(@:3 : Wi0): JEW+ is principal}. Obviously $(%k: Pr) c E(Wk). It has been seen above that rank(V,,(@: X: P I A i : A!o)) = rank(V(Wk): PlAi: A!o)). This statement can be refined to read rank(V,,(@: X: P J A i :AIo)) = rank(V(E(%',: Pr): PJA,:AI0)). One may argue as follows. In view of T-L, we need only show thaf Im(V,,(@: 3:PIA,: A!o)) = Im(V(W&,(@:X: Wio): E(Vk:Pr): PIA,: A!o)). As we already know that
Im(V,,(@: X: P I A , : AJ0)) = Im(V(W$,(@: X: Wio):E(Wk):PIAi: A!o)), it will be enough to establish that Im(V(W&,(G: X: Wio): { w t } : P I A i : A!o)) is contained in E(Wk:Pr): PIA,: A!o)) Im(V(W&,(@:X: qi0):
for all wt E E(Wk).For this purpose, fix wt E E(Wk) and set wk
= Wk(Dis(X,+)).
181
EISENSTEIN SYSTEMS
Let @’ and X’ be as before. Specify W!bi0 E
W&,(X’, X;Wib, W i 0 )
by first choosing an element W!bjO(W+)
E W,L( +
3 ’ 3
Xwt;
q i b , Wjo(wt))
and then putting
wI010 t . = Wio,o(wt) t,.
*
WjO0(Wt, ’
w+*
Note that W t E S(B:
X’: w!bio).
We now have Im(V(W&,(@:X: Wio): {wt}: PIA,: A/J) c Im(V(W&,((E:X: Wio): S(@’:X’: wJbio): PIA,: AIo)) = Im(V(W&,((E:3: W,,): S(@’(Dis(SE’)):3‘:w&,~):PIA,: A:,)) c Im(V(W&,((E:X: Wio): Ev6k: Pr): PJA,:A!o)), the equality being the thrust of Slll and the final containment being a consequence of the fact that ) s ( W k : Pr). E(@‘(Dis(X’)):X’: w & , ~c
To prove that rank(V,,(@: X: P J A , :A!o))
=
rank(Vy,(@: X: PIAi)),
it suffices, thanks to T-L, to prove that Im(V,,(@: X: P I A , : A!o)) = h(V(W&,(@:3: Wio): W&,(Dis(x))(@(DiS(X)): X: %io): PIAi: A!o)). Because (cf. supra)
Im(V,,(@: X: PJA,:Ab)) = Im(V(W&,(@:X: Wio):
=(%k:
Pr): PIAi: A!o)),
it actually suffices to prove that Im(V(W&,(@:3: Wio): {wt}: P J A , :A!o)) is contained in
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CHAPTER 5
for all wt E S(%?k: Pr). To this end, we shall argue by induction on the length I(wt) of wt. [Note: wt has length zero iff
wt E Wtk(~is(x,,(@(Dis(X)): X: wi0I.I Fix wt E E(%?k:Pr) of positive length-then
E(qk:Pr),of length I(w7) - 1, and an element
there exists an element w&ioE
w,t E wtk(Xwt, 3’;%jo(wt),
%?Bib),
X’ being the principal element of @ associated with w Wt
/ such ~ that ~ ~
~
= w! * w;o ’ Wlbi0.
Here, w,‘is a “simple reflection” in the sense that w! induces a map in w&k(wk(Dis(Xw* wk(Dis(X’))) which, upon the insertion of specific data, corresponds to a simple reflection in the usual sense (cf. the Appendix to Chapter 2). We then justify the notation by remarking that the implicitly defined simple root determines a G-conk fitting into our general setup; in particular, C5‘ and X are as jugacy class v before. It is clear that Wt E
a(@’:3‘:Wt,).
Furthermore, any member of E(@’(Dis(X’)): X’ : wti0) has length I(wt)
-
1. This said, we have
Im(V(W&k(@: 3: GZi0):{wt}: PIA,: AJo)) c Im(V(W&,(@:3: giJ: B(B: X’: wii0):PIA,: A!o)) = Im(V(W&k(@: X: gi0): E(a‘(Dis(X’)): 3’:wiio): P I A , : AI0)), by S,,,. Since
E(B(Dis(X’)): 3‘:W!bio)
c
E(%k: Pr)
and consists of elements of length I(wt) - 1, the desired assertion follows by applying the induction hypothesis to wt. There remains the task of demonstrating S, and S,,. Let {(Pi.,s:.;A;.)}
be a set of representatives for \vk(P
IAi)*
EISENSTEIN SYSTEMS
I83
Givenjk such that Wk 3 W j b , there are two possibilities:
i"u;
3 %jb Wjb.
Let us suppose that the second possibility obtains. Each (Pjbvb 9 Sjbvb; Aiovb)
is then a dominated predecessor of a r,,-conjugate of some ( P i , , S:. ; A:,),say vb). There exist, therefore, elements yjovb E rMi and niovbE Niovb with index i'(jo, such that (Pi,(jb,v b ) , Si'(j0, vb);
Ai'(j0,v b J
3 ( P J b v b , S J b v b ; AJbvb),
where (PJ[& 9
= yjbvbnhvb(Pjbvb, Sjbvb;A jbvb)n;,!b
This agreement furnishes a partitioning {(PJbvbr
SJbvb;AJbvb)) =
i'
{ ( P J b v b , SJbvb; A J b v b ) l i *
into nonempty disjoint sets, where, as the notation suggests, each member of
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CHAPTER 5
Recall that there is also an intermediate choice relative to Dis(X’), but it will play no role here. The preceding notation is then applicable in the obvious way, and therefore will be used without further comment. Let
where
Let
where
by specifying their action termwise. For this purpose, let W t ’ E W&((E‘: x’: Vib)
be given-then
It will be enough to define
{L&J!
on the components of the direct sum associated with wt and (i’, w”). Since =
“.*toio
X’,
it is clear that X,t
=
x:,..
185
EISENSTEIN SYSTEMS
Moreover, the symmetric algebras
i
s x vt(P,o"o.
-4jO"O)
ST.,+'(P;bvb. A&bJ
are isomorphic. Of course, here,
j o = jo(w+) = jb(w+') = jb
and vo
=
vb, with i ' ( j o , vb) (dHorn
=
i'. This said, our map will be taken as
QiJ;v;(.A!o)(?))
0
(Yjbvbniovb)-
'
A!J(?)h where . A!o is the appropriate image of A&. Observe that we are permitting ourselves a small abuse of the notation with respect to yibvb and nibvb,since the present yjbYb is in r and not in rM,,the present nhvb then being chosen to reflect this shift in emphasis (bear in mind that rM,= Mi n r - N ) . Let A!o be a general point in X(Pio,Aio)t-then, for each i', A/,, determines a general point A:;,i.in X'(PitIi,,Ai;li.)t and a point in X'(Pii,i,,Aibi,)t:Tacit restrictions will be placed on AIo as necessity dictates. We shall now prove S,.One has (Yjbvbnjbvb)
*
(&
Q:,bV6(*
v(E(e: 2:w&io):PI.4,: A!o): ($)H,,,,,
+ ($)a.
Write
Vw@: 3':P I A , : A t ) for
@ Vw@': X': Pibie1 Afbir:A&.). i'
One has
vw,(a': X': PIAi: A!o):
($')H~,,, + ($')a.
Form the diagram THom(A!)
1 ($')Horn
V V,,(V: 3':PI.4,: A!,)
1 To" ' ($'I@
This diagram is in fact commutative, as can be seen from Assumptions 2(V), 3(V), and 4(V). Consequently, rank(V(Z(@':3': = rank(V,@':
P I A , : A!o))
3':P I A i :A!o)).
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CHAPTER 5
Write
vwL(@':3':P I A J
for
@ VwL(@':x': Pibi'I Albi'). i'
The induction hypothesis implies that rank(V,@':
X': P I A , : A!o))
=
rank(VW,(6': 3':PIA,)).
But, for reasons analogous to those above, the last term is also equal to rank(V(E(6': X': wlbi0):E(@'(Dis(X')):X': wJbi0):PIA,: Ai0)). S, thus follows upon invoking T-L. We shall now prove S,,. One has
E(%(@':X ' :
wlbi0):
PIAi: ?: A/o: ?): ($)Horn
C'-(G/(T n P ) * Ai * N ) .
-P
Write E(W&(@':x': Wib): PIA,: ?: A/o: ?) for
1 E(W&(@':X': Wib): PIA,: ?: A!bir:?). i'
One has
E(W&'@': 3':Wi6): PIA,: ?:
?): ($')Horn
+
Cm(G/(T n P ) . A i . N ) .
Form the diagram
I
E(E(Q': X': wL,J: P [ A , : ?: A;,: ?)
($)Horn
E
THOrn(N")
($')Horn
C'(G/(T n P ) Ai . N )
E(W&(CF': X ' : Y,,,). P I A , : ?: A!,,: ?)
I
C' (G/(T n P ) . A i . N )
This diagram is in fact commutative, as can be seen from Assumptions 2(E) and 3(E). Suppose that V(E(6': X': w & ~ ) :PIA,: Ait,)T = 0. Set
T
=
THorn(A!o)T.
Then, thanks to the commutativity of diagram V, VwL(6': 3':P I A , : A!o)T = 0.
187
EISENSTEIN SYSTEMS
Let TI. be the i’-component of T-then
VwL((f’:3’:PibitI A&, : A/&)Ti,= 0. Consequently,
E(W&‘L(E’: X’: %?ib): PitlAip:Ti,:
?) = 0.
Assumption 4(E) thus implies that
E(W&‘&E’: X’: %?ib): PIA,: Ti,: A/bi,:?) = 0. Therefore, due to the commutativity of diagram E,
E(E(E‘: X‘: w&J: PIAi: T: A!o: ?)
= 0.
But then the usual reasoning leads to the conclusion that
V(W&((f: 3:Wi0): E@‘:X’: w & ~ ~PIA,: ) : A,6))T = 0, as desired.
Remark It is interesting to observe that the only place that E-S: I1 enters into the proof of the proposition is through T-L.
To help put the preceding developments into perspective, it will be a good idea to look at the canonical Eisenstein system. The following simple result, whose verification is completely elementary, provides a convenient framework for getting at certain functional equations (cf. infra); it will also play a technical role later on. Lemma 5.5 Let
be a matrix whose entries are linear transformations offinite rank. Thefollowing statements are then equivalent: (i) A and M have the same rank; (ii) There exist U and V such that B=AU C = VA D = CU = VB.
Corollary Let the notations and hypotheses be as above. Assume in addition that A 2 = A. Then, under either (i) or (ii), D = CB. [Proof: D = CU
=
VAU
=
VAAU = CB.]
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CHAPTER 5
Example (The Canonical Eisenstein System) We recall that in this situation, X is the equivariant system of admissible affine subspaces attached to 5fio via the prescription X(Pio,Aio)= complexification of 6,,.
We shall now explicate the relevant data as they appear in the present setting. @jA), Viewed as a linear transformation from 8,,,(6, Ojb) to 8c,,(& V(6: 9 2 : g1:PIA,: wJ&jb:A]b) is defined by the matrix
P(O1 : PI(Ai, Ail: PjBvBIAjAvi:P j b v b I A j b v b : k j d v A * w ] A j h .kjb&: k j b v h * A j h ) I , or still, by the matrix
I
C C c u s ( P ~ ~ vA]&vA: A Pfbvb
I AJbvb: kjAvA
*
WjAjh *
kjb:b:
kjbvb
*
AJb)I,
which, of course, is simply C ~ , ~ ( P ] ~ P]hJA]b: I A ~ ~ : ~ ] d j b Ab). :
-
There is a canonical one-to-one correspondence
W$,(@: 3: Via)
jo
W(Ajo, AJo).
Furthermore, since here Vk(Dis(X)) = V,,, W~k(Dis(x))(~(Dis(X)): X: wio) * {lie},
liobeing the identity in W(Ai',,AJo).Fix indicesjh,ji. Let = {w:,
{El 8 2
=
(w:jA)-'I
{wi",,W]&,}.
One has W$,,,i~(x),(~(DiS(3E)):3: %io)
C
El,%2
C
w$,(@:3:
%io).
Therefore rank(V,,(@: X: PI A,)) = rank(V(Z2 : El : PI A,: AJ0)). In the case at hand, V,,(6:
x : PIAJ
is a one-by-one matrix whose single entry is
EISENSTEIN SYSTEMS
189
while V(&:
81:
PIAi: AJ0)
is a two-by-two matrix with left column C , . , , ( P ~ ~ IP!olAi',: A ~ ~ : lio: ATo) {ccuS(Pj~lAJ~: P/olA!o:wj*io:A/o)
and right column
is the identity matrix, it then follows from Lemma 5.5 and its Corollary that
C,,~(PJ~ I AJd: PJhI AJh: wJdi0wtjh: Ajtb) is equal to ccus(PJd1AJ~: P!oI A!o: wJai0:~!~jaAJh) ~ C , ~ , ( P ~PJhl ~ JA A ~ J~w:~! ~:~Aj'd), A:
which is just the daggered version of Theorem 4.4. We stress, however, that these considerations do not serve to give a new proof of the functional equations for c,,, ,since the functional equations and meromorphic continuation are established simultaneously for c,,, ,the latter being assumed here. Proposition 5.6 Suppose that for each io, Xi, is subject to
Geom: I Geom: I1 Geom: 111, Gr, and
E-S: I E-S: I1 E-S: 111.
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CHAPTER 5
Let p' = {pJo} be a completely equivariant system of polynomials. Let X E %(%k; W), say X E fio(%?k)--then E(X: PIA,: PioIAio:dHomp/o(A/o)(TO): A/o: x) = P / ~ ( A / ~ ) E P( X I A: i : P i o ( A i oTo: : AJo:x) and
V(X: PI(Ai, Ail: PjoIAj,: PioIAio: WJOio: ~ / o ) ( d ~ o m P / o ( h / o ) ( ' T ; . o ) ) = P/(A/~)V(X:PI(Ai, A,): PjolAjo:PioIAio:wJoi0: AlfO)TO. [This result is due to Langlands [p. 200].] Let us first consider a special case.
Lemma Suppose that X is, in addition, principal-then the proposition is true with W kreplaced by Wk(Dis(X)). [The proof is not difficult. We shall defer it for the time being simply because some new considerations will be required.] Proof of Proposition 5.6 It is clearly enough to make the verification at the general points of X(Pio,A,,)+.Accordingly, since the two equalities are then equivalent, we need only establish one of them, say the second. For this purpose, in view of E-S: I11 and the fact that
No> = P/o,
PJo(WJ0io
it suffices to prove that the operator
d,
PJo
- pJo(wJoioA/o)
annihilates Im(V(X: PI(Ai, A i ) :PjolAjo:PioIAio:wJoio:A!o)). Replacing P, by a r - N,-conjugate if necessary and keeping in mind Assumptions 2(V) and 3(V), matters can be so arranged that the setup surrounding the previous proposition includes P i , and P j , among its data. Naturally, the notation there need not coincide with the notation here, but this is irrelevant. Furthermore, as will become apparent momentarily, there is no loss of generality in replacing the present X by a principal member of its equivalence class E, which, in agreement with our earlier notational principles, will still be denoted by the same symbol. This settled, let
pjo(Aj0) = @ Pjovo(kjov0 . AJo) yo
ElSENSTElN SYSTEMS
191
and form
If 8 be a subset of WJk(@:X: Via), put and then assign to the symbols
the obvious interpretation. Observe that v(9,: 8 2 : PIA,: /\~o)~dHomd2(l\!o) = d,g,pfr,(AI0) 0 V(EI : 8,: P J A , A!o). : The operator Vwk(@: X: PIA,: A!o) contains the initial V among its entries, so our proposition in the case at hand will be established as soon as it is shown that the operator d , P h k ( Q : x :w,")"
annihilates
-
P!o
X: P I A , : A!o)). Im(Vwk(@:
is contained The orthogonal complement of Dis(X)(Pio,AiJt in X(Pio, in X(P,,,,A,,)-. We can therefore write, in a self-explanatory notation,
+ A!o.
AIo = A!o(Dis)
Fix A!o-then,
with AJ0(Dis)varying throughout Dis(X)(P,, Aio)t, the rule pr(AJo(Dis)) = p!o(A!o)
defines a polynomial
pr which is clearly invariant under the operations of
W&k(Dis(x),(AioA,,); thus it determines a completely equivariant system pDis= {pjDd'"} of polynomials. Since the derivatives are taken perpendicular to X(Pio, Aio)-, one has 3
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CHAPTER 5
on subsets S of Wt,(Dis(a))(@(Dis(X)):X: %iO), in particular on
S = Wt,(Discr,,(@(Dis(3E)): X: Wio) itself. The preceding lemma evidently implies that the operator Dis
d@h ' d ,pi.cz))(B(Dis(X)): I:Wio)(A!o(Dis)) annihilates
- pp(A!o(Dis))
Im(VWk(@: X: PIA,)).
But then the same is true of the operator d,P?J.:,(e:r:Wio,(~ito) - P!o(No).
Since (cf. Proposition 5.4) Im(V,,(@: X : PIA,)) =
Im(V(W$,(Dis(x,,(@(Dis(X)): X: Wio):W$,(@: 3:Wio): P J A , :A!o)),
it follows that Im(d"0m PLt,((E:I:Wi0)"
- d0(Ait,))
is contained in
Ker(V(W$,(Dis(I,,(@(Dis(X)): X: Wio):W&,((E: X: Wio):PI A i : AI0)), i.e. (cf. T-L), is contained in
3: P ( A , :AJo)), Ker(VWk(@: so that the operator d,
&; k(B:x: ,,o,(A!o)
-
dO(Ai9
annihilates Im(V,,(@: X: PIA,: A!o)), as desired. This completes the proof of our proposition. // Remark In a certain sense, one may view the equalities provided by the preceding proposition as the "functional calculus" for the E-functions and the V-functions. It is possible to use them to place a priori restrictions on the range of V; we shall not stop, though, to go into further detail here.
EISENSTEIN SYSTEMS
193
At this juncture, it will be necessary to shift gears so to speak. Up until this point, analysis has played little role in our considerations, but now it will move to the forefront of the discussion.The residue taking process furnishes a method of producing a new Eisenstein system from a given one. Moreover, at least for the Eisenstein systems of interest to us, the geometric conditions Geom: I, 11, and I11 will be preserved upon taking residues; the verification is fairly direct. The same will also be true of Gr and E-S: I, 11, and I11 but, in contrast, the verification is decidedly indirect. To carry it off, a new condition, An (cf. infra), will be formulated. It will be shown that An implies Gr and E-S: I, 11, and 111 (cf. Proposition 5.8). Because An is an analytic condition arising naturally in the “philosophy of residues,” determining when it holds becomes possible in practice. Our first task will be to define a space G 0 ( ( GI P)/r; & @ o )
of functions in LZ(G/(r n P ) A , . N ) , the so-called space of “partial wave packets,” formed from the E-functions in the canonical Eisenstein system X. For this purpose, let (Pi0 3
so that
Sio; Aio) E wio(P I Ail
( P , S ; Ai) 3 (Pi09 s i o ; AiJ
Write It for the common value of the f$ Let is the partial wave packet ciated with
01”E X&(B, Bi,,)-then
asso-
). that Oo;,(P(Ai)is the lift to G of a where A: is a point in ~ , ; o ( 6 , ’ , Observe hence wave packet on K x M i / {1) x rM,; O,;o(PIAi)~LZ(Gn / ( rP ) . A i . N ) , it being understood that LZ(G/(T n P) . A i . N ) is supplied with the inner product
194
CHAPTER 5
As a space of functions on G, Li,((G I P)p-; 40,) is independent of the choice of the split component A i . In the special case when P = G, we recover Li,(G/T; 6, no).It is clear that L : , ( ( G I P ) / ~a,,; 0:) 1L:,((G
Ip)/r;6,, 0 3
unless
6, = 62 = 0;.
{o: The sum
10 LE,((G I P ) / r ; 6, 001, 6
call it
Li,W I p ) l r ; 001, is evidently a G-stable subspace of L 2(G/(rn P ) . Ai N ) . Finally, there is an exhaustive decomposition L2(G/(T n P ) * Ai * N) = %k
C 1 0 Li,((G 1 P ) / r ; 00). >/YO Qo
Remark Fix a real number Rt > IIp[,ll. By the Rt-tube we mean the tube over the ball of radius R' with center zero in 61,).Denoting by SA$Rt) the space of all holomorphic functions in the R '-tube which decay at infinity faster than the inverse of any polynomial, set XAln(6,
R') = XA:,(R')
oio;
There is a strict inclusion oio) 4
u
R'
*Aio(6,
o gcus(6, oiJ oio;R').
Let a!, belong to Oio; Rt)-then, utilizing a limit process, one can show that it is possible to associate with a!, an element Oaro(PIAi)in t2(G/(rn P ) . Ai . N) which is, in fact, the L2-limit of partial wave packets formed from functions in XA;t,(8, Oio). The following inner product formula will play a technically important role during the subsequent discussion.
Lemma 5.7 Let f be a diferentiable function in Let (Pjo
L:,W
I P)/l-; 0,).
Sj,;
A,,) E wjo
9
195
EISENSTEIN SYSTEMS
4,
where the AL ure elements in the complexiJcation of and the Y hare element in C[ajo] 0 b,,, (6, Ojo). Let YjoE XAj0(6, Ojo)-then
(1,@YJ;o(p I Ail) = 1 (rh? h
dy,(- x',)).
Proof We begin the computation with an examination of the left-hand side of the purported equality, which, to within the factor (2n)It, is equal to
or still
It will be enough to show, therefore, that
(2nyt . (Yh, d Yj)(- XI)) is equal to
196
CHAPTER 5
For this purpose, fix the index h and drop it from the notation. Write
r = C u j 8 aj i
(uj 8 Qj E CCaJoI6 R A 6 , oj,J)*
Fix the indexj and drop it from the notation. We must then prove that (24'' (u
6 a, dYJo(- At))
is equal to
Since (u 8 a, dYJo(-At)) = (0, d,*YJo( -AT)),
the right-hand side of this equality being, by definition,
we need only establish that
which, however, is a simple exercise in manipulating Fourier transforms. // For the proofs which are to follow, it will be necessary to set up some machinery tailored to the present context, but entirely analogous to that encountered in the theory of Eisenstein series associated with cusp forms (cf. Langlands [pp. 121-1221). Fix a real number R t > ~ ~ pas~above. o ~ ~Because , the IIp/oll are all the same, no matter what the choice of the index i,, we shall denote their common value by llptII. Let f t = {fJo} be a completely equivariant system of bounded holomorphic functions in the Rt-tube, i.e., to each j , there is assigned a bounded holomorphic function j J oin the Rt-tube such that for all triples (Pjb, Sib; Ajb) (Pjd, SjA; Aid)
197
ElSENSTElN SYSTEMS
standing in the relation
one has
fb(w&
j,:
AT$ = fJA(AJA)
for all
AjA). To such an f there is assigned a bounded linear transformation 3(f t, on L$,((G I P)/l-; 6, U,), characterized by the relation WjAjA
w&k(AjA,
3(ft)@,:,(PIAi) = @,jo,lo(PIAi)
(a!,
EzA:,(a,
oio;
io being any index. One has
I13(ft)II 5
IIftolIm.
If
f+' = where '
{fj;},
ff;(qo> = fjo(-Ajo),
then
3(f')* = 3(ft'); hence 3(ft) is actually a bounded normal operator. It is easy to check that 3(f t, intertwines the G-action on L$,((G I P)/r; 0,). A central part in this theory is also played by an essentially self-adjoint operator 0+, usually unbounded, characterized by the relation
Ot@,:,(PIAi) = @,?,?,,:o(PIAi)
(@Io~ z A : , , ( 8 ,Uio; Rt)),
where (
?)@!,)(No) =
No)@t(No)3
i, being any index. In terms of representation theory, the Casimir operator o
is defined on the space of differentiable vectors in L$,((G I P)/l-; 0,)and, as such, is essentially self-adjoint.One has
0'
=0-
Ct(O),
ct(o) being a certain scalar. The spectrum of 0' is contained in the interval 3 - m, (lpt11 2]. This may be seen as follows. Let R ( [ ; Elt) be the resolvent of
198
CHAPTER 5
0 at the complex number (. Now fix a complex number c with Re(c) > (R ' ) 2 . Let fjo,cbe multiplication by 1
5 -
ft
wc; 0') = w:),
the arbitrariness of R' implies our spectral assertion. This is a good place to tie up a loose end, viz., that part of Proposition 5.6 which was left open in the form of an unproved lemma.
Lemma Suppose that X is, in addition, principal-then the proposition is true with %k replaced by %k(DiS(X)). Proof In this situation, WPio, Ai0It = {X(Pio, Aio)>, the single point X ( P i o ,Aio)which, of course, is general. The two equalities in question being then equivalent, we need only establish one of them, say the first. Let pt = {pJo}be a completely equivariant system of polynomials-then we must prove that
E(X: PDis(X)IAbis(X):
dHomP!o(X(Pio,Aio))(To): X(pi0, Aio): x) = Pi',(X(Pio,Aio))E(X: PDis(X) I ALis(X): Pi0 I ~ i o T: o : X(Pi0, AiJ: piol A i o :
By hypothesis, the E-functions vanish if ord(Tn) % 0. Select R t large enough to guarantee that X ( P i o , Aio) belongs to the Rt-tube (computed per Dis(X)(Pio,Ain)'). Let f J o be an invariant bounded holomorphic function in the R '-tube with the property that UJo -
PJo)(X(Pio, A i J )
vanishes to such a high degree as to secure the relation
E(3E:'Dis(X)IAbis(X): p i o l A i o : d H o m f f o ( X ( P i o , A i o ) ) ( T o ) : x(pio, Aio): x) = E(3:PDis(X)IAbis(X): PioIAio: dHomP!o(X(Pio,Aio>)(T,,):X(Pio,Aio): X) for all To.Let f = {fJo}be the corresponding completely equivariant system of bounded holomorphic functions in the Rt-tube. We can and will assume in
EISENSTEIN SYSTEMS
is contained in the kernel of
Taken together, these statements yield the lemma.
199
200
CHAPTER 5
Choose a positive integer N such that the degree of
Taking into account the preceding computations, induction on n leads to the equality
20 1
EISENSTEIN SYSTEMS
is orthogonal to every
1 Abis(X)),
oY]o(pDis(X)
hence is zero. But if a vector is in the kernel of some power of a bounded normal operator, then the vector is in the kernel of the operator itself. In particular, IE(3: PDis(X) I which is (1).
Pi, I Aio: To:X ( P i 0 ,A d : ?) = 0,
Proofof(2) Form the difference between the two sides and compute the inner product of the result against any
1 Ahis(%)>.
oY~o(pDis(X)
Using Lemma 5.7, we get the expression
c
w;oioEW:*(Di.,I))(X:
AJO. Ai,)
(da ffo(wJoioX(pio,
(v(3:PDis(X)I(Abis(X), -v(x: pDis(X)I x
(dHom
P i o I A i o : wfoio: x ( p i o ,
Abis(X)): p j o l A j o :
(Abis(X), Abis(X)):
'io))To)
pjol Ajo: PioIAio: WJoio: x(pio, Aio))
f!o(X(pio, AiJ)(To)), dyIo( - WJoioX(Pi0,Aio))),
which, in view of our assumptions and E-S: 111, is zero, thereby proving (2). The lemma is thus established. // Suppose that for each io, JEi0 is subject to Geom: I Geom: I1 Geom: 111. Fix an R with the property that all the normal translations are in the R-tube. [In view of Geom: 11, this is clearly possible. The point, of course, is that one number, viz., R, is being used simultaneouslyfor all gk.] Introduce the set
> 0) consisting, by definition, of all points of the form IB:(Wpio, Aio))
X(Pi0, AiJ
+
(E
-NO,
where A t is in the projection of 3 ( P i o ,Aio)" onto it!o and has norm less than E. Observe that here X ( P i o ,Aio) is not necessarily real (cf. Proposition 5.3). Write bt for the common value of the dim(3(Pi,, Ai0)').
202
CHAPTER 5
Let us recall that one may associate with each 010
E xAlo(6,
oio; R)
an element
@,i0(P IAil E L:o((G Ip)/r;6 , 0 0 ) *
If p b is any complex-valued polynomial on JfA:,(d*
oio;
we understand the subspace of which are divisible by pib, i.e., the Set Ae = R 2 - E~
@!o
P!O
*
(E
> 0).
(@o!
then by
R ; P!J Oio;R ) made up of those functions
E Jf'~:,(d,
oio;
R)).
An Suppose that
Then: There exists an orthogonal projection
Q ~L:,((G : I wr;6,0,)
+
L:,((G
w r ; 6,0,)
onto a closed subspace, commuting with the 3(ft), and, for every E > 0, a family of polynomials pi = { P J ~ , (depending ~} on E ) such that
I
P10.e g(pjo,Ajo)t
#
(9 %jo(Vk))
satisfying the following conditions. Let @ o!
E x A : o ( 6 , Oio; R ; ~ 1 0 . e )
If 3E E %io(%k)has the property that
(V(X: PI(Ai,A i ) :PJoIAjo:PiolAio:wfoio:A]o)d@!o(A!o), d ~ J o ( - w J o i o ~ ~ o ) ) is holomorphic in a neighborhood of ZB!(3(Pio, Aio)).In addition: An, The differencebetween
(WC;0')QP
@@jo(P I
@,j0(P I Ail)
EISENSTEIN SYSTEMS
203
and
summed over all X E Xio(Wk)having the property that Re((X(Pi0,
AiJ,
X(Pi0, A i o ) ) ) > 1,
and over all wjoi0E W&,(X; A,, Aio),can be holomorphically continued to the region Re([) > ie. An, The difference between ( Q ~ @ s i ~ ( P l A i ) R(C; ,
and
S'
(2n)b'
nt)@,;o(f'IAi))
(V(X:P I ( A i , A i ) :PjolAjo:PioIAio:wJoio:
I B : ( X ( P , , . Ato))
summed over all X E Xio(Wk)having the property that Re((X(Pi0, AiJ, X(Pi0, A i J ) ) > 1, and over all wJoioE W&,(X; A,, Aio),can be holomorphically continued to the region Re([) > i t . Remark It is tacitly assumed that QPis independent of the indices io and
j , . QP is not necessarily unique.
Some clarifying comments are in order. Write
204
CHAPTER 5
Hence, if then
Re((X(Pi0, AiJ, X(Pi0, Aio))) >
)be,
IIWX(Pi0, Aio>)It < Consequently, the normal translations X(Pi,, AiJ figuring in An, and An, must lie in a compact region. It therefore follows from Geom: I that the sums involved are actually finite. As for the analytic conditions, observe that by general properties of the resolvent, the inner products
( R ( l ;0t ) Q @,:,,(P ~ I A i l @,j0(PI Ail) {(QP @.:o(P I Ail, NC; 0')@,j,(P I Ail) are holomorphic in the region Re(l) > A,, except, perhaps, along the segment ]A,, IIpt1I2]. Now ask: When do the integrals become singular? Under the assumption that they actually exist, any integral per an X with Re((X(Pi0, Aio), X(Pi0, Aio)>) 5 A, is automatically holomorphic for Re(5) > A,. Ofcourse, these are the integrals not taken into account by An, and An,. In practice, E will be allowed to approach infinity. Consider Fig. 1. Here the dots represent the various
u x x x x x X. o(xxxxxx
xx xx xx xx xx
1
cwcwMc*
I lptl I
(XXXX.
uxxxxxxx
XX.
XXX.
FIG.1
R2
t
X
ElSENSTEIN SYSTEMS
205
Off of the finitely many thatched segments, the integrals are holomorphic. An, and An, postulate that the difference between the inner products and the integrals can be holomorphically continued across these segments. Remark Since QP commutes with the 3(ft), QP also commutes with R ( [ ; Elt). Obviously, R ([;Ot)* = R ([; Elt).
Therefore the inner products involved are equal. It will appear, a posteriori, that the same is true of the integrals. Proposition 5.8 Suppose thatfor each io, Xi, is subject to
i
Geom: I Geom: I1 Geom: 111
and
An. Then for each io , Xi, is subject to
and
{
E-S: I E-S: I1 E-S: 111
Gr. [This result is due to Langlands b.2033.1 Surprisingly enough, the proof is not especially difficult. We shall preface the discussion with an auxiliary result (cf. Lemma (A) infra) which will serve as the technical vehicle for the demonstration. Let us begin by dispensing with a technicality. Lemma Let X E Xi0(Wk). Suppose that
Let A!o be a general point in X(Pio,A,)'-then
the set
206
CHAPTER 5
set
Needless to say, it is still true that Yjz E %A;0.] Since AI0 is, by hypothesis, a general point, the various wjoiohI0are distinct. But then it is not difficult to see that one can choose YJoin such a way that I* t 0# duw:oioy,o (WjoioA!oL wioio
which is a contradiction. //
Lemma (A) Let X E ili0(Vk). Suppose that
EISENSTEIN SYSTEMS
207
Let there be given a function A(wjoio: on w$k(x;Ajo,
AiJ
x X(Pi0, AiJ'
which is a linear trans$ormarion from Hom(S.%(P,o. Ato), gcus(6, 03) to %P,,,
A , ~ ) ~ ; , ,@ , ~gcus(4 o j o )
such that (A(Wjoi0:N J T o y Tj,) is meromorphic on X(Pio,Ai0)+and vanishes identically ifthe order of is suficiently large. Let @!o {vjo
E Jf',-~~~(fi~ %A,,
+
Toor To
o i o ; R ;P ~ O , (6, ojo;
w.
Make the following assumptions: d'J'jo( - w J ~ ~ ~ K ! ~ ) ) (A(wjoi0:AJo)d@]o(A!o)y
is holomorphic in a neighborhood ofIB!(X(Pio,Aio)) and
summed over all wJoioE W&,(X; A,, Aio),can be holomorphically continued to the region Re(() > i t .
Conclusion : If Re((X(Pi0, Aio), X(Pio, Aio)>) > A t ,
then A(?: ?) = 0. Proof We shall proceed by contradiction. Suppose that A(wJoi0: A!o)To ord(To) > N , but that for some wjoioand some Toof order N , A(wJoio: A,',)T0 # 0 on some open subset of X(Pio,Ai0)'. Fix such a To.Select an element = 0 if
Ti0 E CC6!oI @ &cus(fi,
oiJ
208
CHAPTER 5
with the property that ord(K;, - dYio(?)) > N on X(Pi,, AiJt. Given q], E C[d{,,], put
.
~~f~
= pf,, q/o
Qqi0
E HAio(J, oio;
'!)
e("9
. r.
10.
Obviously,
R ; ~ 1 0A* ,
Moreover, it is clear that ord(d@,,l0(?)- p,',,, . q!o . e(?.?) KO) > N on 3(Pio,Ai0)t. Consequently,
is equal to
EISENSTEIN SYSTEMS
tZ
+ (S
-2J-It
- Re((X(Pio, Aio), X(Pio, Aio)))
209
+ (A]o,
Integrate this expression with respect to s from Re(lo) to C-then we get 2J-1 times thedifferenceof
210
CHAPTER 5
Put EO
= (Re((X(Pi0, Aio), X(Pi0, AiJ>)
- Re(CO))"*,
a positive number which is strictly less than E. Then the limit as t 10 of the preceding difference is a.e: equal to - 2 x J - I times the characteristic funcAio)). Consequently, the integral tion of IB~o(X(Pio,
I
t
f(A!o) . e("lo*"d. q/o(A!o)1 dA/oI
~ B t $ ( W t 40)) ~,
vanishes.Duetothearbitrarinessofql0,weinferthatf = Oon IB~o(fi(Pi,,Ad), hence on all of 3E(Pio,AiO)'. Using the density statement supra, it is then a simple matter to contradict the supposition that for some wjoio,A(wjoi0:A/o)To # 0 on some open subset of X(Pi,,,143'. // We shall also need a classical result from operator theory. Spectral Lemma (S-L) Let T be a self-adjoint linear transformation on a Hilbert space V, R(C; T )its resolvent. Let { E(A)}be the resolution ofthe identity for T-then, for all real p and v with v < p,
+ { ( W L9) + (W- O)f,
9)) - m ( v ) f , 9)
+ (E(v - O)f,g)}
is equal to
lim 010
1 ~
2sJ-1
s.(fl,v,",v)
(NC;T)f,9 ) 4,
where &(p, v, u, u ) is the contour
v+J-1ut V
-
G
V-J-IU
U
p-J-rv
p - o u
We shall now tackle the proof of our proposition. Fix X0 E xi0(%k).[The subscript has been appended momentarily in order to allow X itself to be a running variable.] Pick an E > Q with the property that Re((XdPio,Aio), XdPio, AiJ>) > It has been pointed out above that there are but finitely many X E xio(%k)such that Re((X(Pi0, Aio), X(Pi0, Aio)>)>
21 1
ElSENSTElN SYSTEMS
or finitely many 9 E Xjo(Wk) such that Re((Y(Pj0, A,,), Y(Pj0, Ajo)>) > For these X and 9 we can therefore choose a polynomial qJ0E C [ ~ I /whose ~] restriction to Xo(Pio, Aio)tdoes not vanish, but whose restriction to 3(Pio,Ai0)+ (3 # 3,) vanishes to such a high degree that the composition of i b E .
V(X: P ( ( A i ,Ai): PjolAjo:PioIAio:wJoio: with dHom q!o(A/o)
is null and whose restriction to gw;o,o(Pi,,, such a high degree that the composition of
(?)w!o,o
# 3,) vanishes to
d 8 q!,,(w!,ojo~Jo)
with
V ( g : P J ( A i Ai): , PioIAio: PjoIAjo:w!ojo:AJ0) is null. This in place, let us indicate the strategy. We want to verify E-S: I, 11, and 111, as well as Gr, for 3,. Because An involves not just Xo alone, we must devise an artifice to split off To, as it were, from the rest of the pack. The properties of q1!,, in conjunction with a little manipulation, will accomplish exactly this. In An, take,
@lo
where, of course,
{
--
= ~ l ’ , , e. q!o * @o!
YJo= Pjo.E* vjo,
a/o ExAfo(d,
oio;R )
{qjoE xAjo(d, ojo;R). In the second difference(i.e., An?), first interchange the roles of io andj,, then replace [ by c, and finally conjugate the inner products. Subtract the resuft from the first difference (ie., An,) to get, after multiplying through by (2n)”,
V(3: PI(Ai, Ai): PjolAjo:PioIAio:wJoio:A\t)
212
CHAPTER 5
and over all wjoi0E W&,(3;A j o ,A,,), less the conjugate of
summed over all 9 E JEi,(‘iR,) having the property that Re((Y(Pj0, AjJ, Y(Pj0, Ajo))) > and over all w!,~,E W&,((D;A,,, Ajo).By hypothesis, the expression in toto can be holomorphically continued to the region Re(() > &. In view of our assumptions on q1,;in the first part only 3, itselfremains. Taking into account the fact that the adjoint of dB qito*(wit,joA,’J is
dHomq!o( - w!ojoA,’,-,), in view of our assumptions on q!,, in the second part only those ’2) and w t j 0 such that (Dw;?,, = 3, remain. At this point, it will be convenient to change our notation slightly and write 3 for 3,. Recalling that
(V?, *Y)= (V*?II,?), our expression reduces to the integral over IB;(X(P,,, Ai0))of (dBp,’~e(w,’oioA!o) 0 V ( 3 : P I ( A i , A i ) : PjoIA;,: PioIAi,:wjoi0: A!,)
summed over all wjoi0E W&,(x;A,, Aio) such that 3 w ~Eoiit j,, ( W k ) . The verification of E-S: I and E-S: I1 for 3 is now immediate. In the notations of Lemma (A), let us define a function A(wJoi0:A!,) according to the following
ElSENSTElN SYSTEMS
213
prescription. Let wJoioE W&,(X; A,, Aio).IfX,jo,o # xjo(%k), let A(wJoio:AI0)be
V(X: PI(Ai, Ai): PjolAjo:PioIAio:wJoio:A!o), composed on the left with d , p J ~ E ( ~ ~ o i oand A ~ oon ) the right with dHo,q~o(A~o). If XWj0,,E xjo(%?k), let A(wJoio:A!o) be the difference of
V(X: PI(Ai, A i ) : PjolAjo:PioIAio:wJoio:A!o) and V(Xnfoio:PI(&
A J : PioIAio:PjolAjo:(WJoio)-l: -wJoioXit,)*,
composed on the left with d,pJ&(w~oioA!o) and on the right with dHomq!o(A!o). Lemma (A) then implies that Since
r
A(?: ?) = 0.
d , pJZ&(~~Joio~!o) q/o(A!o) are isomorphisms for a.e. AJ0, the validity of E-S: I and E-S: I1 is thereby established. There remains, then, the task of addressing E-S: 111. For this purpose, note first that for any ft, dHom
(R(C;ot)Qp3(ft)0,~o(PlAi), %j0(PIAi)) = (R(C; 0 t ) Q @,fo(P ~ I 3(ft*)@,jo(PI Ai)). Working only with the first difference (i.e., An,), take as data
and then subtract. The terms containing the resolvent thus disappear and so, proceeding as above, after multiplying through by (27~)’~ we are left with
summed over all wJOioE W&,(X; A,, Aio),where A(wJoio:A!o) is, by definition, the difference d ~ f J o ( W J o i o AV(X: ~ o ) ~PI(Ai, Ai): PjoIAjo:PioIAio:wJoi0: - V(X: PI(Ai, A ( ) :P j o l A j o :PioIAio:wJoi0:AL)odHomf!o(A!o) composed on the right with dHomq!o(A!o).It follows from Lemma (A) that A(?: ?) = 0.
2 I4
CHAPTER 5
A simple approximation argument, which has been used explicitly before, then yields E-S: 111. We have yet to consider Gr.
Lemma Let X E 3Eio(vk)-then w t k ( X ,
X; q i o , WiJ
z 0.
Admit the lemma for the moment-then the fact that Gr holds can be seen by the following argument. Note first that Proposition 5.3 is now in force. The equivalence class in 3E(V,; go)to which X belongs thus contains a principal member, say 9 E jejo(%k). One has
wt,($, X; q j o , q i o ) # 0 iw: 9;%joy qjo) k(Dis(g))@,
#
0
9
say
{
3; q j o , WiJ E W t k ( D i s ( g ) ) ( 9 , 9 ; q j o WjJ
WJOio E $0
wtk('D,
3
Because 9 is principal, wyohas the properties postulated in Gr per 9;but then W E = (wjoio)-'WO JO w? Jot0 . has the properties postulated in Gr per X. Proof of Lemma Suppose the lemma to be false. Assuming that io = j , , take in the first difference
-
qlo aq0. For the usual reasons, all the terms not involving X drop out; in addition, the terms involving X itself also drop out, an empty sum being zero. We conclude, therefore, that = p!o,E*
(R(i;
0' ) Q p @,jo(P I Ail, @,jo(P I Ail)
or still
(R(i; n t ) Q p @ ~ ~ o ( P I A i )Q, P @ @ ~ ~Ail) (PI is holomorphic in the region Re(5) > &.Because R 2is not in the spectrum of El1,it follows from S-L that
((1 - U & ) ) Q p @ o t o ( P
is equal to lim lim
1 ~
I A i l Q P @ , ~ ~ ( PA,)) I
(R(C;n t ) Q p @,lo(P I Ail, Q P @,lo(P I Ail) 4
EISENSTEIN SYSTEMS
215
i
R d O = X, FIG.3
where (tc,ais the contour shown in Fig. 3. As the integrand is holomorphic, the integral tends to zero with 6. Consequently, E(Ie)Qp@olo(PIAi)= Q ~ @ g l ~ ( P l A i ) * This, of course, implies that
(R(5; 0 ' ) Q p @,;to(P I Ai), @vJ~(P I Ail) is holomorphic in the region Re([) > Ie for any YJo. Therefore
V(X: PI(Ai, Ai): P j o l A j o :PioIAio:wjoio: A!o)
summed over all wjoi0 E Wf,(X; Ajo,Aio), is holomorphic in the region Re([) > I , . It then follows from Lemma (A) that
V(X: P I ( A i , Ai): PjolAjo:PiolAio:wJoi0: A!o) ~ d ~ , , q ~ o ( A ~ o ) and hence
V(X: P ( ( A i ,Ai): PjolAjo:PioIAio:wJoio: A$ is zero for all wJoioE W$,(X; A,,, Aio). This forces the conclusion that E(X: P I A , : PioIAio:?: ?: ?) = 0.
216
CHAPTER 5
By making a suitable replacement of the data if necessary, this fact can be used to contradict the nontriviality of the Eisenstein system belonging to 3.In other words, the set Wik(3,X; Uio,Wi0)is nonempty, as was to be shown. //
I
To summarize: E-S: I
Geom: I Geom: I1 AND An Geom: I11
E-S: I11
Remark It is more or less obvious that the canonical Eisenstein system satisfies the conditions on the left-hand side of the implication. We shall now consider the problem of constructing certain closed subspaces of L:,((G IP)/l-; &@,) by means of the machinery introduced above. In spirit, the results obtained are entirely analogous to their counterparts in Chapter 4. They play an important role in the proof of a major theorem of Langlands (cf. Theorem 5.12 infra). We place ourselves in the setting of the formulation of An, which we assume to be in force (along, of course, with Geom: I, 11, and 111). Bearing in mind Proposition 5.3, we can then say that 3(Uk;U o )is a finite set. Furthermore, the normal translation in any member of %(%?k; go)is real. Finally, every equivalence class in X ( % ? k ; U,) contains a principal member. This being so, let X w k ; Uo) = a, be a decomposition of X ( % k ; U,) into equivalenceclasses (35,. Select a principal Set element 3,in (f,,say Xi E 3io(i)(Uk). 0,' =
# (wtk(xi
31;
Uio(i), Uio(i)))-
Write
Supposing now that io and E vary, let L:,((G IWl-; 4 00;3)
be the closed subspace of L:,((G I P)/l-; 6,O0) spanned by the ( I - E ( & ) ) Q P @ ~ ~I Ail ~ ( P (@!o E J f ' ~ f ~ (o6i o, ; R ;P ~ O .el), (Pi,),Sio;Aio)being, as always, a generic member of Uio(PIAi).Let
I W r ;&@,)
Li,((G I P ) K ; 40,;2) be the corresponding orthogonal projection.
Qt: L:,((G
+
EISENSTEIN SYSTEMS
Let Given I , let us agree to write
Then the function is holomorphic on
217
218
CHAPTER 5
Let (pE(dX@!,)equal
[This result is due to Langlands [p. 21 11.1 These two results are, of course, closely related. For the moment, let us remark only that the second is proved first. We begin with a spectral computation.
Lemma Suppose that E 9 0. Let @ ,! E ~ f A j , ( 8 90 i o ; R ;P!,,
iPj, € s A f 0 ( 8 , 0,; R). +
Then
A
ElSENSTEl N SYSTEMS
2 19
is equal to
being, as before, the contour shown in Fig. 4. By our assumptions, the difference between
(R(i; and
t)Qp@@!o(P
IAi)?@YJ,,(P I A,))
220
CHAPTER 5
summed over all 3 E xio(%?,() and over all wjoioE W.&,(X;A,, Aio), can be holomorphically continued to the region Re(C) > I,. Observe that it is a question here of all X E Xi0(Wk),as is permissible. Adding and subtracting the obvious quantity, we can therefore say that ((1 - E(U)QP@,;to(P I A i l @,j0(P I Ail)
is equal to 1 lim lim . T+ C I A , 610 (27C)
~. 1 27C-
Jac, ~ L d ( X ( P i oAio)i( . x d[
O "
i- (?, ?>
V(X: PI(A,, A i ) : PjolAjo:PioIAio:wjoio:
](A/o), d Y ~ o ( - w ~ o i o A ~ o )
summed over all 3E E Sio(%?k) and over all wJoioE W$,(X; Ajo,Aio). Using E-S: 111, bring (c - (?, ?))-' to the outside and compute 1
thereby obtaining
EISENSTEIN SYSTEMS
22 1
Observe that there is a natural map d, (sic) which assigns to each element
2122
CHAPTER 5
Point (1) is, of course, a simple consequence of E-S: 11. Points (2) and (3), however, are not so immediate; it is necessary, for a complete verification, to dispose of some rather fulsome analytic details. The key is the spectral computation supra, in conjunction with an approximation argument. Owing to a variant of the Stone-Weierstrass theorem, to be discussed later, it is possible to find an open dense set
D,c I@o(%(pio(,),Aio(i))), clonsistingof general points at which
(v,(A!o(,))?,?"I is defined, with the following property. Let E > 0 be given. Let
F,: zB!(x,(pio(,), Aio(t)))+ Hom(S,, &cus(6, OiJ) be a continuous function, with support contained in D,, whose components
E
wt,(@, : 3,: w i o ( t ) ) i o )
satisfy a symmetry condition, namely, =~w?ow~o(,)~~~(,~~!o~I~~ (Sym), Fwlo(W:(,)w!o(,)A!o(,)) for all w!o(,)
E
wt,(x, 3,;w i o ( r ) * wio(iJ 9
Such symmetry is evidently possessed by the components of the d,@g. This said, it can then be shown that for every N 2 0, F, is the uniform limit of certain d,@/o, modulo Homs of order > N , the d,. ( I ' # I ) having order :z N . By shrinking D,if necessary, one can arrange that the approximation is actually effected by certain d,@lo, being divisible by pl0,&. We shall now establish (2); (3) may be discussed similarly, hence need not he dealt with in detail. Suppose that E 9 0-then E , is defined. Obviously: E , < E. Since I - E ( I E )and QPare both projections which commute with one another, the product ( I - E(AE))Qpis also a projection. Consequently, thanks to the spectral computation supra,
'S
OlC-.
(v,(A!o(,)NI@!o, dI@!o)I dA,to(,) I
d , ( % ( P i o ( ~A ~i os( d )
for all R ; plo, Choose N 9 0 so large that (V?', ?") = 0 when the orders of ?, ?,, are greater than N . If, therefore, d,@!o + F,, then still @ o!
EsAl0(8r
(V,(Aito(,))F,F,) I d N 0 ( l )I . 9
ElSENSTElN SYSTEMS
223
-
2;!4
CHAPTER 5
be the corresponding orthogonal projection. The Qx are expansive since, as will be seen below, c1 > e2 Im(QS)
so that Qx = lim
Qk
t+m
= WQ3,
(strong operator topology).
On the other hand, fix i, and allow E to vary-then by -Go((GIP)/r;6,@0;W i o
we understand the closed subspace of G o ( ( GI P ) / T ; 6900 ;3)
spanned by the (1 - E ( ~ ) ) Q P @ ~ ~I,Ail ,(P
(@!o
Ex
~ j ~ (o 6i o ,; R ; P!O,
A)*
Let G , ( ( G IP)/r; a,@,; 3;E l i 0
be the closed subspace of J%O((G I P)/r;6,@0;W i o
obtained by freezing both io and E. It is clear that
lJLio((GI P)/r; a,@,;
JE; E l i 0
E*O
is dense in G o ( ( GI P)/r; 6,oo;
W i O .
In order to provide an explicit realization of the space G 0 ( ( GI P)/r; 6,@0;JEIi,,
as well as its truncation Lio((GIP)/r; 6300;JE;
Eli0 9
il. will be necessary to introduce another class of Hilbert spaces.
Let
y i o ( ( G I P)/r;6,@0;
Wio
be the Hilbert space made up of those collections F = (F,) of measurable functions F,: lBL(JEI(Pio(i), Ai0(1)))+ Hom(S1, 6 c u d 6 ,
oio))?
EISENSTEIN SYSTEMS
225
whose components are subject to (Sym),, with
the inner product then being the prescription
Let E B 0-then we can repeat the precedingdefinition to form a Hilbert space Yio((GIP)/r;6 7 8 0 ; 3;E)io,
where now the integration is carried out only over the IB,t,(Xi(Pio(i) Aio(iJ). 9
It is clear that is dense in
E
lJ Y i o ( ( G IPYr; 6 9 0 0 ;3;E l i 0 *O
~ i o ( ( G I p ) / r ;6 3 0 0 ; s ) i o *
Given @!o E
x A l o ( 6 , o i o ; R),
let dx@!o
=
(d,@!o).
Then, on the basis of the approximation argument indicated earlier, the divisible by p!o,e, are dense in restrictions di@!o, with 2iO((GIp)/r; 6900;3;E ) i o .
Consider the diagram
226
CHAPTER 5
The spectral computation supra implies that IIQZ@,fo(PIAi)II
=
IIdi@!oII.
Consequently, the dotted arrow may be filled in to establish an isometric isomorphism Lio((G I P)/r; 6 , 0 0;JE;
E l i 0 --t
y $ o ( ( G I P)/r; 6 9 0 0 ;
E)io
9
the space on the right then providing an explicit realization of the space on the left. Thanks to the expansiveness of our projections, these considerations then lead, in the obvious way, to an isometric isomorphism L$o((G I P)/r; 6 , 0 0 ;W i o
+
y$o((G I P)/r; 6,flo; W i O -
Lemma Let @!o E
x A f o ( 6 , Oio; R),
"hen the restriction dfr@i',belongs to p i o ( ( G I P)/r; 6900;JE; [Note: This is not a priori evident since p,',,, need not divide @J0.]
Proof Let f stand for the projection of O,io(P I A i ) onto L$,,((G IP)/r; 6 7 0 0 ;
Eli0
and call F the corresponding element of
Yio((GIP)/r;6 9 0 0 ; JE;
s o that We shall prove that Let
llfll
&)io,
= IlFll.
F = a:@!,, (a.e.). P/o,&(NoE JQo(8r o*o; R ; do,,).
Then we have 0 = (QZ @pro. .ero(P I Ail, f - f ) = (Qi @pio, .m,t,(P IA i l @,r0(P I A,)) - (QB Opto,.&.,t,(PI = ( Q i @ p ~ o , . ~ ~ o ( ~ ~O~ i, ), ,t o ( ~ ~~ (di~i',,c6/0, i)) F), t:hat is
'S
O = COIi . I
~Vl(~ito(l))dlP!o, &&it, dl @;o - Fl) I"1)
l d , ( X ~ ( P i ~ho(nd) ~t~.
I
f)
I
EISENSTEIN SYSTEMS
227
for every pit,.e &o!
EzAlo(d, oio;
R ;pit,, e>.
Given Ti0
E H W S i3
&cuds, oio>),
it then follows from the approximation argument used before that (Vt(A!o(i))Tio,
di@!o
- FJ = 0 (a*e.Aito(i)).
Since functions which agree a.e. are identified, we are done. // Observe that the preceding lemma implies that the QE are expansive. Let @ o!
EXAfo(d, oio;
R)*
Then we have IIQx@~~o(PIAi)I12
2
IIQE@~~o(Pl~i)l12
Because the integrands are nonnegative, we can let E approach infinity to get
2;!8
CHAPTER 5
is integrable (hence holomorphic) on
IBt,(K(Pio(i) Aio(i)))* 7
The inner product (Qx @,,to(P I A i l %l0(P I Ail)
can be evaluated by the limit
I AiL @vl0(f'I4)), lim(Qi @alo(P
E+W
the latter being, however, exactly 1
(Wf.
1
? 3.J-,B:w*o(,)*@lod, 'yjo)I ( v l ( N o ( l )9 ~ l
Ai0(,)))
"1,
I
*
Proposition 5.9(V) is therefore established. It remains to consider Proposition 5.9(E). Let
@loE
Oio; R).
That
E,(d,@!o:A!o,,,: x) is holomorphic on IBL(K(pio(i), Aio(i)))
ciin be seen by employing an argument similar to that used in the proof of Proposition 5.1, thus need not be considered further. Our only responsibility, then, is the inversion formula, viz., Q x @ a ~ o ( PA(i ) = 1.i.m. qE(dX@/o). E-W
We begin with a preliminary observation. Let
F E Y i 0 ( ( GI P ) / r ;6,oo ; W i o . Assume two things: For each index
I,
(i) F, is continuous, x) is continuous. (ii) E,(F,:
Naturally, the symbol in (ii) is to be assigned the obvious interpretation. Let f be the element of G 0 ( ( GIP ) / r ;6,oo;
corresponding to F, so that
llfll
=
IlFll.
3)iO
229
ElSENSTElN SYSTEMS
We then claim that
for a.e. x. Observe that the left-hand side is square integrable, while the righthand side is slowly increasing. Proceeding in the usual way, form the difference A and, using Lemma 5.7 (or rather an obvious variant thereof), integrate it against any partial wave packet, the result being, as a short computation reveals, zero. In order to conclude that A = 0, one need only verify that a * A = 0 for every LY E IF(G),which can be done by imitating the argument of Lemma 4.1 1. [Note: Tacitly, Assumption Rep($ has been invoked here.] Hence the claim. The inversion formula is now immediate: Just take F = dx@loand let E approach infinity. But the discussion actually gives somewhat more, namely, G,((GI
w r ;4
=
00;
c 0 Go(@I w r ;
6900;
@I),
I
where
G , ( ( G I n/r;4 0 0 ; @ I ) is the closed subspace of
G,((G I spanned by the lim &+m
S
w r ;4 0 0 ; J%d, @lo No(,) -4IdNo(,,I
9
f B J ( & ( P i o ( t ) *Aio(a1))
where, of course, io is allowed to vary. This is best seen by taking this time
F:
{F,
= d,@,
F,. = 0
(1’
# r).
Observe that the decomposition does not involve the projection QPin a direct manner. The proof of Proposition 5.9(E) is now complete. Remark There are a couple of subtleties implicit in the foregoing which should be commented on. First of all, it is perfectly possible that some of the terms figuring in
@lo
(Vl(N,(I))dl
7
dl
y;,)
may have singularities on IBL(Wpio(i) Aio(i)))9
230
CHAPTER 5
For a numerical example, cf. Langlands [p. 313). The point, therefore, is that alter a suitable combination, the possible singularities are removed. Second of all, there is no reason to expect that
w,:Go(,):x)
will be well behaved on
IBt,(Xi(Pio(i) Aioci)))? 9
even, e.g., if F, is differentiable with compact support. Before continuing, we had best fill in the details of the approximation argument left open above. Stone-Weierstrass Theorem Let X be a compact Hausdor- space. Let B be a Jinite-dimensional commutative *-algebra containing C which is generated by C arid the self-adjoint nilpotent elements. Let 8 be a subset of C ( X ,a).Assume 8 is a *-subalgebra of C ( X , a)containing the constants with the property that for all points p, q E X , p # q,
{f(PI : f ( 4 ) = 0,f E 81. Then: 8 is unformly dense in C ( X , a). =
[The proof is not difficult.In order to avoid interrupting the exposition, we shall defer it to the Appendix of this chapter.]
W0),say 3 E Xio(%k). Consider, in the usual Let X be a member of z(%k; notation, H O ~ ( S . XA~, ~P) ',&cus(d, ~, OiJ). Denote by A ~ ~ ) Ho~/N(SX(P~,,
-
OiJ)
the quotient of HO~(S.X(P,~, A(,) &cus(d, 9
OiJ)
b y the elements of order > N. Needless to say,
HO~/N(SX(PI,, Ccudd, OiJ) 9
is finite dimensional. Lemma Let
E
> 0 be given-then the set of restrictions to IBt(X(Pi0 AiJ)
23 1
ElSENSTElN SYSTEMS
IB,t(x(Pio AiJ) 9
of the
d@!,
(@it, E J f A l , ( R ) )
is uniformly dense in C(IBZ(X(Pi0 AirJ), Hom/N(Sx(p,,,A,,) C))* 9
9
To begin with, note that Hom/N(SX(P,o,A , o )
9
c>
is a finite-dimensional commutative *-algebra containing C. Here, of course, the *-operation is the obvious one, viz.,
T * ( u ) = T(u*).
Moreover, HOm/N(S3E(P,,,Aio)7C) is generated by C and the self-adjoint nilpotent elements. This being so, put
a = Hom/N(SX(P,o,A,o)' c)*
Let
x = IBJ(X(Pi,, Aio))u ( 0 0 ) .
Then X is a compact Hausdorff space. Let 8 be the direct sum of the constants and those continuous functions f:X-+%
such that: ( 0 f ( m )= 0; (ii) f I IB:(X(P,,, Aio))= do!, (some @loE HA,*,(R)). It is a straightforward exercise to verify that 5 is a *-subalgebra of C ( X , a) with the properties stipulated in the Stone-Weierstrass theorem. Therefore 8 is uniformly dense in C ( X , a),and this, as is easily seen, implies our assertion. //
232
CHAPTER 5
We can now supply the details for the approximation argument. For each 3 E Xio(%',,), choose a polynomial p i which does not vanish on X(Pio,A,,)+ but which does vanish on all r l ( P i o ,Ai0)+,X' E Xio(%J (rl # X), to order N + 1. Fix an index 1. Given
3E
n JEio(%'J,
let Y(p1)be the set of all No(,)
E ZB!
(xt(Pio(t),
Aio(,)))
with the property that
+
P1(w/oio(~)wiO,(t)~!o~~0 ~)
for every w,foio(,) in W&,(X, X,; Aio, Aio(,J.Call D,the subset of
nx pi)
consisting of the general points at which
(V,(N0(,)),K T) is defined. It is then clear that D,is an open dense subset of ZBL(X,(Pio(,), Aioc,,)). Let Fi lB!(Xt(Pio(,),Aio(i)))
-,H W S ,
3
gcus(8,
OiJ)
be a continuous function, with support contained in D,,whose components are subject to (Sym),. The components of F, indexed by the W/oio(r)E W&k(X,
X,; vio
3
%'io(rJ
may be planted on IB,t(X(Pio AiJ) 9
according to the rule
FXNJ
=
Fw:o,oc,,((Wibio(i)
wiO,(tJ-
'No).
Thanks to (Sym),, the definition of FIE is independent of the choice of witoio(,). Since SPt(FIE) = {No : Piz 01, the preceding lemma implies that there exist (@!&
E z A f o ( 8 , oio; R)
such that F: is the uniform limit of the d ( p 1 . > N . This being the case, set =
1x P1
*
(W0)x.
modulo Horn's of order
ElSENSTElN SYSTEMS
233
It then follows from our definitions that F, is the uniform limit of the d,@,!,, modulo Hom’s of order > N, the d,, (I’ # z) having order > N. The approximation argument is therefore established. Let X be a member of %(%?k; W0), say X E xio(%?k).By definition, IBL(X(Pi0, AiJ)
= E
IJ IBf(X(Pi0, A i o ) ) * >O
As these sets are ultimately the ones of basic importance, we shall take, at this point, a closer look at the singularities of the E-functions and V-functions along them. In so doing, we keep to the preceding assumptions and notations. Proposition 5.10 Let X E xio(%?k)- Let A!,, be ageneral point in IBL(X(P,,,, A,”)) lying on no X’(Pio,Aio)t (X’ # 3, X ’ equioalent to X)-then V(X: PI(& A i ) : PjoIAjo:PioIAi,:wJoio:?) is holomorphic at A!o.
[Since this result plays but a secondary role in what follows, we shall merely indicate the line of proof (which is technically a bit messy but not really very difficult). Let ( X I be the equivalence class of X,XIa principal element in (3,. According to Proposition 5.9(V), the function ( v l ( ~ ! o ~ l@it,, ) ) ~ ldl VJo) is holomorphic on IBt,(Xi(Pi,,(i),A i o ( i J )
for all
i
E xAlo(d, oio; R )
YJoE x a j o ( d , ojo;R ) ,
The main thing now to realize is that if the assertion were false, then by choosing and Vj,, judiciously, one can pick off the V-term in the sum defining (vl(~!o(l))d, dl YJ0l which, in so doing, leads to a contradiction.] Proposition 5.11 Let X E X i 0 ( q k ) . Let H , be a singular hyperplane of
E(X: PIA,: PioIAio:T o :A!o: x) which meets IBL(X(P,,, A,,))-then (Hs
- X(Pi0, AiJ)
+ Ai
3
Dis(X)(Pio, AiJ.
[This result is due to Langlands [p. 2173.1
234
CHAPTER 5
An equivalent statement is this: H , - X(Pio,Aio) contains the projection of Dis(X)(Pio,Aio) onto li!o.Necessarily, then,
Consider, in particular, the special case when X is principal-then the dimension of the projection of Dis(X)(Pio,AiJ onto &I,, is the same as that of X(Pi,, Ai0)+.On the other hand, H , is a hyperplane in X(Pio,Aio)t; thus it has dimension one less than that of X(Pio,Ai0)+.Conclusion: If X is principal, then
E(X: P I A i : PioIAio:To:ATo: x) is liolomorphic on IBL(X(Pio,Aio)).This, in fact, is the first step in the proof of the proposition. Lemma Let X E Xio(qk).Suppose that X is principal-then E ( X : PIAi: PioIAi,: To:A!o: x)
is holornorphic on IBt,(X(Pio,Aio)).
Proof It will be convenient to change the notation a little and write XI for X, io(i) for i o . In this way, we are in a position to make free use of the machinery developed above for Proposition 5.9(E and V). Proceeding by contradiction, let H , be a singular hyperplane of
Aio,,))). Let A1 be a real unit normal to H , in which meets IBL(Xl(Pjo(l), X,(Pio(,),Aio,,,)t-then, on H , , the Laurent expansion gives E(Xi: PIAi: p i o ( i ) I A i o ( i ) : m
To(l):ATo(,)+ [Al:
X>
235
EISENSTEIN SYSTEMS
with the following property: For all wib(i)E Wt,(xi,
xi; Aio(i)r
AiOct))
such that the restriction of w:(,)wito(,)to H , is not the identity, w:(t)w!o,t)A!l# A & -
Otherwise said, AL is in general position with respect to H , . In addition, we can and will assume that AX lies on no other singular hyperplane of E(xi: PIAi: Pio(i)IAio(i): To(i): A,!,,,):
XI*
Finally, it may be supposed that
E - N ( T ~ (Ah: ~ ) :X )
# 0.
This said, let B be a small ball contained in
Hs n lBL(xi(pi,,(t),Aio(i))) and centered at AA. B will be allowed to shrink in due course. Let 4 be a nonnegative continuous function on
Hs n l B L ( ~ t ( p i o Aioti))) (i~~ with support contained in B such that the measure of the set ofx E G for which - N(
/$(A!o(i))E
I
: A!~(i):
&I)
I#
dA!~(i)
+
is positive. Let be a nonnegative continuous function on the line with support contained in ]1/2, 1[ such that Jii2 3 = 1-then, putting $mcV)
we determine a sequence {$,} that
=
m* N
~Y),
of nonnegative continuous functions $, such
{;m&,:l;y%
1/mC
The points
No(l) +fiYA1, where
N0(i) E Hs n lBL(xi(Pio(i),AiociJ)
236
CHAPTER 5
defines a sequence { I,} of continuous compactly supported functions Jn:
IEt,(xi(pio(i)yAio(i)))
+
R.
Extend To(,) to an element T i o ( i )E
Horn(% gcus(6, 9
@io(i)))
by requiring that all the components of Tio(l) be zero save the one corresponding to w & ~ , ,which we take as To(l) itself. With this preparation, let {F,} be the sequence of functions defined by specifying that the components of F, are zero except for the one associated with I, in which case we put
FXA&i)) =
1
,fm((wL(t)w!o(i))-
'A!o(i))
*
( ~ i ' o ( r*) x i o ( a J -
W?O(l,
Claim: If m B 0, and E is sufficiently small, then F m E z i o ( ( G I p)/r;6,oo ;J E ) i o ( i ) .
Indeed, as is easily seen, under these circumstances, the integral
'.I
(Vl(N0(l))F~9 F 3 IdNo(,)I
~ & W P t o ( t ) Ato(,))) .
is equal to the integral of fm(A~oc,,)2 times (V(3i: PI (Ai, Ai): pio(i)I Aio(i): Pio(r)I A i o ( i ) : w&i): A!o(i))TT;.o(i)>To(,))
taken over
E x 11/2m, l / m [ . J - l ~ l .
Owing to Assumption 4(V), that part of the integrand coming from the Vfunction is actually constant; in fact, it is just
I p i o ( i ) I Aio(i): p i o ( i ) I Aio(i): w & ~ ) X : , ( P i o ( i ) , Aio(i))) applied to To(1)and paired with To(l).Denote this number by Cv and let pDis(X,) (A&s(E,)>
C=
sB 42.It is then clear that the norm of F, is finite, being given by 1
rl/m
EISENSTEIN SYSTEMS
237
or still
Hence the claim. We are now in a position to derive a contradiction. First note that lim IIF,II
= 0.
m+m
On the other hand, let f, be the element of G o ( ( G I p)/r;6 3 0 0 ;JE)io(,)
corresponding to F,,,, so that Ilfmll
= IIFrnll.
If m 9 0 and B is sufficiently small, then the set B x 11/2m, l/mc. f
i ~ l
is contained in the domain of holomorphy of
E ( 3 , : PIAi: p i o ( t ) I A i o ( t ) : To(,):A/o(i): XI. We can therefore say that for a.e. x, fm(x), multiplied by (27r)’+,is equal to
or still
where
L
c, =
Let m approach infinity in this expression-then all terms but the first drop out. Consequently, 1 lim fm(x) = ~. m-tm (274’’
. J”
I9
~ ( N ~ ( , ) ) E - N ( T ~ ( ~A/~(,): ):
I~A/~(,)I.
238
CHAPTER 5
By construction, the function on the right is nonzero on a set of positive measure, thus contradicting the earlier conclusion that lim llfmll = 0. m-rw
This completes the proof of the lemma. // The remainder of the proof of our proposition depends in an essential way on Proposition 5.4. We can now, of course, suppose that X is not principal. It will then be convenient to change our notation: Write 'I) for X , j , for io, and let @ be the equivalence class of 'I),X a principal member of @, say X E Xio(Wk). Placing ourselves in the setting of Proposition 5.4, we can and will assume that ( P j o ,Sjo;Ajo)is part of the data, as there. Extend To,in the obvious way, to an element I c u d S , ojo))* Tjo E Hom(SQ(pjo,ajo),
Lemma ZfHs is a singular hyperplane of
E ( 9 : P I A , : PjoIAjo:To:AJo:x) which meets ZBL(V(Pjn,A J ) , then H , is a singular hyperplane of'
v(w$k(Dis(x))(@( Dis(X)): X : %io):
{ wJoi0}: P I Ai : w,P( wjoi0)-'Ajo)Tjo.
Proof It amounts to showing that if V(W$,(Dism)(@(DiS(X)):3: %io): {Wjoio}:PI Ai: ?)Tjo is defined at N o
then
E IB',(X(Pio,
Ai,-,)),
E ( 9 : PIA,: Pjo)Ajo:qo:?: x)
is defined at w J o i ~ w ~ AglB',(9(pjo, !o AjO)).
This said, choose a fixed right inverse t: Im(V,,((F:
X: PIA,)) -+ Dom(V,,(@: X: PIA,))
to VWk(@: X: PIA,)). Because Proposition 5.4 implies that
V(W~,(,iscx,,(@(Dis(X)): X: Wi0):{ w ~ ~PI~ A~, :} ?), : when defined, takes values in Im(V,,(@: 3:PIA,)),
239
EISENSTEIN SYSTEMS
it makes sense to form the following meromorphic function zjo of E X(Pio,A,,)+:
A!,
r(V(Wtk(DiScr))(a(Dis(X)): 3: %?lo): { wj0io}: P IAi: A!o)Tjo).
Claim: There is an equality of meromorphic functions:
E(g:P I A , : PjoIAjo:To:w ~ o i o w ~ Ax)~ o : = E(Wt,(Dis(x,)(a(DiS(X)): 3: gi0): PI Ai: zj0(A!,):A!o: x).
Admit the claim for the moment. Owing to the preceding lemma, the righthand side of this relation is defined at A!, E IBt,(JE(Pi,,A,))
whenever this is the case for tj,;as this is so only if V(Wtk(Dis(r))(a(DiS(X)): X: %?i,):
{ W j o i o } : P ( A i : ?)Tio
is defined at A!,
IBt,(X(Pio, AiJ),
the lemma will follow. To prove our claim, we can take A!, to be a general point at which everything is defined. Set T;~(A!~)= rjo(AT0)0 ( -Tjo),
an element of
@ Hom(Sxwt(P,,l,+).A j o I w t ) ) ' 8cus(6,
'joOvt))),
3 = Wt,(Discx,,(a(DiS(X)):x: %io)
U (W;,i,}.
E
where Observe that
E(%:PIA,: T>~(A[~): A!,: x) is thedifference between the right- and left-hand sidesofthepurported equality. Bearing in mind the expression for the constant term of E(%:. . .) along K x PJovo, in order to show that E($: . . .) = 0, it suffices, thanks to Theorem 3.6, to verify that V(Wt,(@: X: Wi0): %: PIA,: A/,)$,(A!,)
=0
or still (cf. Proposition 5.4), to verify that V(Wtk(Discx,,((F(DiS(X)): X: via): %: P I A i : A!,)t>,(l\~,)
=
0.
240
CHAPTER 5
But the latter is equal to the sum of V,,((€: 3: PIAi)tjo(A,6)
and -V(Wtk(Dis(x))(Q(DiS(X)): X: %to): {Wjoio}: P J A i :h/o)Tjo.
Since, by construction,
V,,(@: X: PIAi)tjo(A/o) is equal to V(Wt,(Dis(x),(Q(DiS(x)): 3 :gio): {WJoio}: PIAi: A!o)Tjo,
we are done. //
Lemma I f H , is a singular hyperplane of V(W&k,Dis(x,,(Q(Dis(X)): X: Wi0): {wJoio}: PI A i : wi",(wfio)-'AJo)Tj0 which meets IB',(g(Pjo, Ajo)),then H,is a singular hyperplane of
V ( g : P I ( A i , Ai): PjolAjo:PjolAjo:wyo:Ajo)To. Proof By inspecting the terms which make up v(Wt,(Dis(x),(~(Dis(x)): X : %io): { wJoi0}: 'I 1 Ai: A!o)Tjo, one can say that there exists an index I and an element w!o(i)jo such that H, is a singular hyperplane of V(r): PI(Ai, Ail: PiO(i)IAio(i):P j o I A j o : w!o(i)jo:AJJqO. Needless to say,
xi = 'Dw!,,, , ,,,~E @(Dis(X)), hence is principal, thereby justifying our notation. Let us agree to write qoinside
'.
M ~ J ~ for ~ ~ (w!o,l~jo)-' (~) Place
Hom(Si, &cus(6, OjJ) by stipulating that all its components be zero except the one corresponding to M J ) ~ ~ ~ which ,~), we take to be qoitself. Let E Hom(Sx,(pto(,), A , ~ ( , )9 )gcus(6,
Place To(1) inside Hom(Si, g c u s ( 6 , Oio(i)))
Oio(i)))-
EISENSTEIN SYSTEMS
24 1
by stipulating that all its components be zero except the one corresponding to w:,(~,, which we take to be q,,(l) itself. Now specialize inequality (3) of page 221 to this particular situation and derive that I(V(g: p I ( ~ iAil: , pio(t)IAio(i):P j o IAjo: Wito(,)jo:Wjtoio(i,W,P(i)Ait,ci))Tjo, T o ( t J 1 2 5 I(V(g: PI(Ai, Ai): P j O I A j o : P j o I A j o : wyo: wjnio(i)w,P(i)Ar!o(i))Tjo, TjJI x I(V(xi: P I ( A i , Ai): pin(i)IAio(i):pio(i)IAio(i):w,P(i):‘!o(iJTo(i), To(iJI*
It should, perhaps, be recalled that Aitoci) E lBL(x,(Pio(t), Aio(tJ)*
The second factor on the right-hand side of this inequality was encountered a short while ago; it is a constant (cf. Assumption 4(V)). Our assertion is therefore manifest. // Proof of Proposition 5.11 Keep to the notational changes agreed to earlier. Let H , be a singular hyperplane of
E ( 9 : P I A , : P j o l A j o :To:Ajo:x) which meets l B L ( g ( P j o ,Ajo))-then it follows from the two lemmas above that H sis a singular hyperplane of
V ( g : P I ( A i , Ai): PjoIAjo:PjoIAjo:wy0: Aj0)IT;.,. In view of Assumption 4(V), this function depends only on the projection of Ajo onto the complexification of Dis(g)(Pjo, A,,)+. Consequently, H,, and therefore H, - Y ( P j o ,Ajo), is a union of cosets of the projection of Dis(Vj)(Pj0, Ajo)onto i5jc,.So, if we can prove that Y ( P j o ,Ajo)E H,,then we will be done. Referring back to Proposition 5.10, we see that Hs
lBL(g(Pj0 A j o ) ) 9
must be contained in the union of the various “w-hyperplanes” H w and the various
g(pjo,AjoIt n g’(Pjo9 Ajo)t. The Baire category theorem implies that
Hs r‘\ lBL(g(Pjo3 AjJ) is actually contained in one of the members of this union. Since the real dimension of
Hs n lB!o(g(pjoT AjJ) is evidently bt - 1, there are two possibilities: Either
Hs n zBL(9(f‘jo, A j o ) )
=
Hw
n lBL(g(Pjo3 AjJ)
242
CHAPTER 5
for some H,, or else Hs n IBL(’2)(Pjo,Ajo))
=
YI’(PjO3
AjJt n ~ ~ L ( ~ (AjJ) P j 0 ~
for some ‘2)’(Pjo,Ajo)t. In either case, it is easily shown that Y(Pj0,A j J E H s ,
as desired. // The theorem toward which the efforts of the present chapter have been directed can now be stated. Theorem 5.12 Given V o ,write =
IJ W i o . io
For any integer b, 0 I b I rank(%:,),
one may then attach to each Via, in a unique way, a collection Xi, of equivariant systems of admissible afJine subspaces of dimension b and an Eisenstein system bdonging to each element oJ’Xio,subject to
I
Geom: I Geom: I1 Geom: I11
and
An, with the following property: For all ( P , S ; A i ) in W k , the space
G o ( ( GI
w-; a,@:,)
can be written as the orthogonal direct sum of the G 0 ( ( GI P ) / r ;d,@o ;
the sum extending over those collections whose dimension lies in the range
rank(%) Ib I rank(%:,). [This result is due to Langlands [p. 2221.1
Let us agree to some notation. Recall that %(go) =
JJ io
Xio.
243
ElSENSTElN SYSTEMS
If it is a question of a collection of dimension b and if this is to be emphasized, then we write JE(V0; b ) = JEio(b)io
Similar principles will be applied whenever it is convenient, e.g., assigning to the symbol L:,((G I wr; 6, oo;3,b) the obvious meaning, our theorem asserts that is equal to
L:,((G I
c
rank(%) 5 b 5 rank(Yo)
w-; 6,00)
o L:,W I wr; 6,00;
Consider, in particular, the special case when (G, G ; { I})-then
%?k
JE,b).
= {C},
(P,S ; Ai)=
G o ( G / r ; 6900)
admits the orthogonal decomposition
1
0 5 3 5 rank(Yo)
0 L:,(GIr; 6, 0,;JE, b),
a fact which, as will be seen in due course, leads to the orthogonal decomposition hinted at near the end of Chapter 4. The uniqueness is relatively easy and will be dispensed with momentarily. The existence, on the other hand, is more difficult. Its proof is inductive in character. To allow induction, it will be necessary to formulate still another set of assumptions and conditions. In this connection, the residue taking process will play an important role. Remark To avoid any possibility of confusion, we stress that %?k ana ( P , S ; A i ) are “running variables”. The collections Xi,and their associated Eisenstein systems are independent of them. Suppose that for each b we have collections 3?(V0;b), 3”(Vo; b) and associated Eisenstein systems {E’, V’}, { E ” , V”} with the properties stated in our theorem. The proof of uniqueness then begins with the following observation. Since the spaces
w-;
raws
L:,((G I d,oo; JE’, i Li O (I wr; ( ~ 6, oo; r,rank(%)) both give the discrete spectrum of 0’ in
L:,W I P ) F ; 6,
00l
244
CHAPTER 5
they must be equal. This said, let ExAlo(S,
@/o
oio;
R)*
Then (cf. Proposition 5.9(E)) the projection of O,;o(P1 A i ) onto the space spanned by the eigenfunctions of 0 is given by either
or
c‘ E’(3E’:P ( A i :Pio(Aio:d@/o(X’(Pio, Aio)):X’(Pio, Aio): ?)
c“E”(3E”: P(Ai:P i o ( A i od@/o(X”(Pio, : Aio)):X”(Pio,AiJ:
?).
Here c‘ or Z’’ stands for a sum over those 3’or 3” in
{
Xi0(Wk; rank(%)) Xy0(Wk;rank(%)).
In the case at hand,
i
X’(Pi0, Aio) = {X’(Pio, o (hi X”(Pi0,Aio) = {X”(Pio,Aio)) o (ai
+ + J-rAi),
so the normal translation determines the affine subspace. Using a simple approximation argument, it then follows readily that Xio(%k;
rank(%‘))= xy0(%‘k; rank(%‘))
for all i o . Furthermore (cf. Assumptions 4(E) and 4(V)), the E‘- and E functions, as well as the V - and V-functions, are the same. Now let 0 Ib I rank(V0). By varying %‘k over those G-conjugacy classes of rank = b, we conclude from the preceding discussion that the collections X’(%‘,,; b), Z”(W0; b) have the same principal elements and that the associated Eisenstein systems coincide. In order to draw similar conclusions in general, we need some preparation. Lemma Let X(%‘o) =
xi0
io
he given. Suppose that for each io, X i , is subject to Geom: I Geom: I1 Geom: I11 and
An.
ElSENSTElN SYSTEMS
245
Taking Wk = ( G } , let be an equivalence class in X(Wo), Fix a principal element X, E 6,.Then: The 3rloio,where
X E @o(Dis(Xo))
{
wjoio
having the property that
E W,,,(X;
W j o 7 WiJ
V(3: Gl({l), {I}): P j o l A j o :PiolAio:wjoi0:Aio) Z 0, describe g o . Proof That such elements belong to @” is a consequence of E-S: I. Proceeding by contradiction, let 9 be an element of@,, say 9 E X,, such that whenever
9 = finJoio
for some
3 E @o(Dis(Xo))
{
Wjoio
V(X: GI((l), (1)):
E W(,j(X; Vjo3
WiJ,
PjolAjo:PioIAio:wjoio:Aio) = 0.
In view of E-S: 11, this would mean that V(9: GI({l), (1)):
PioIAio:PjolAjo:wiojo:Ajo)= 0
provided ~ w , o l o E &,(Dis(X,)). Now fix X E @,(Dis(X,)), say 3E E Xi,. Write 9 = XwJoi0(some wjoio).We can then assert that
V(W,,,,i,cx))(@(DiS(fi)): 3: WiJ: {wjoio) : G I { 1) : Aio) = 0 or still, thanks to Proposition 5.4, that V(W(&: 3 : Wio): {wjoi0}:G l { l } : Aio) = 0.
This relation, when unraveled, implies that V(9: G J ( ( l ) , (1)):
...) = 0,
from which it follows, in the usual way, that the Eisenstein system belonging to 9 is trivial, which is an impossibility. // Let X0 be a principal element in JE’(Wo; b) or x”(Wo; b). Let Wo be the equivalence class of X, in X’(W0; b), @: the equivalenceclass of Xo in JE”(Wo; b). Claim: Wo = Wh. To see this, note that at least Wo(Dis(Io)) = Wi(Dis(3Eo)). But then our claim follows from the description of @; and Wi provided by the preceding lemma. Consequently, X’(Wo; b) = JE”(W0; b). Denote their common value by X(W,; b).
246
CHAPTER 5
It remains to show that the Eisenstein systems {E’,V’}, {E”,V”} belonging to the elements of JE(W:,; b) coincide. For this purpose, let (E be an equivalence class in JE(Wo; b), X a principal element of @, say X E Zio-then it will be enough to show that ViG)((E:X: G l { l } : Aio) = V;b,(E: X: G I { l } : Aio). Let
i
9 2
=
W,,(,is(31))(@(Dis(X)): 3: WiJ
= W,G)(Ci:X: Wio) - W,,,,iscx,,(@(Dis(3E)): X: Wi,).
We then have
i
V’(9, u 9,:9,: G I { l } : Aio) = V”(9, u 9,: E l : G I { l } : Aio) V‘(E1:EI U 9 2 : G l { l } : A i o ) = V”(B,:=I U 9 ~ : G l { l } : A i 0 ) .
Indeed, the second relation is a formal consequence of the first (cf. E-S: II), while the first is a consequence of what we already know to be true. Using 9, and 9,, write, in the obvious way,
On the basis of what has been said so far, A‘ = A”, B’ = B ,
C’ = C”.
Taking into account Lemma 5.5 (applicable because of Proposition 5.4), write B = A”U” B’ = A’u‘ C” = V”A” C’ = V’A’ D“ = C”u” = V“B“ D’ = C‘u‘= V’B’ Then
I
{
D’= C’u’ = V“A“U’ = V“B“ = D”
Therefore ViG)(@:X: GI(1): Aio) = V[Gl(@: X: GI{l}: AiJ, desired. Having taken care of uniqueness, let us pass to the problem of existence. It will be convenient to start off by introducing three types of naturally defined sets, viz., “cylinders,” “cones,” and “real or imaginary balls.”
iis
247
EISENSTEIN SYSTEMS
Let X be an equivariant system of admissible affine subspaces attached to Vio.Given zl, e2 > 0, let C(X(Pio,Aio):61: ~
be the set of all points of the form X ( P i o , Aio)
2 )
+ A:o + - A t ,
where A,!o, A;o
with
E X(Pio,Aio)-
{ wo;;c
n
ai0
61
< E2.
Such a set will sometimes be referred to as a cylinder (of radii and E ~ ) . Given E > 0 and an open set w on the sphere of radius E in X(Pio,Aio) n ai0, let V(X(Pi0,Aio):E : w ) be the set of all points of the form t X ( P i o ,Ado)
where 0 < t < 1,
+ (1 - t)Aio, Aioeu.
Such a set will sometimes be referred to as a cone (of radius X ( P i o ,Aio)and base w). Given A0
and E > 0, let
e X ( P i 0 , AiJ
+ X(Pio,Aio)'
E
with vertex
n aio,
i I:{ 1
RBe(X(Pi0, Aio): zBe(X(pi0, AiJ:
be the set of all points of the form
&Aio,
where
Aio E
X(Pi0, Aio)- n 4,
with IIAio)I< E. Such a set will sometimes be referred to as a real or imaginary ball (of radius E and center Ao). When A. is just the normal translation, we
248
CHAPTER 5
suppress it from the notation. Daggered versions of these three types of sets will be used without comment. We shall now define what we mean by an amalgamation. This is a composite concept, tailored for induction arguments. Let Xi, be a collection of equivariant systems of admissibleaffine subspaces of dimension b attached to Via. It will be assumed that there is an Eisenstein system belonging to each element X of X i o . Let I:, and I:o be two disjoint index sets, Iio = I:, u IL. We shall suppose that there is associated with each index i E Iio an equivariant system XI of admissible affine subspaces of dimension b - 1 attached to Via. It will be assumed that there is an Eisenstein system belonging to each X, (i E Iio).In addition, for each X E Zi0,we suppose given a nonincreasing function EX:
R + + R+,
and for each index i E Iio, we suppose given a nonincreasing function
R + + R+. Finally, for each 3 E Xi, and E > 0, we suppose given a nonempty open subset W P i o IA i o : E )
of X(Pi0, AiJ
+ Wf'io,
AiJ
- n hi0
such that 81
> ~2
KdPiolAio:&~)c
and for each index i E Iio and subset
E
h(PiOlAi0: ~ 2 1 9
> 0, we suppose given a nonempty open
UPio I Aio : E )
of XdPio, AiJ
+ %(Pi,,
Ai0)-
n hi0
such that El
> E2
* P p i o l A i o :E l )
c lgPiolAio:E Z ) .
In both cases, we assume that v ~ ( x P i o x - l I x A i o x -El): = x . Vx(PioIAio: E) ~ ( x P i o x - f l x A i o x - E' :) = x . &(PioIAio:E )
for all x E G for which x P i o x - is again r-cuspidal.
249
EISENSTEIN SYSTEMS
The specification for each io of data of the preceding type is said to be an amalgamation of dimension b if all of the following assumptions and conditions are met. To begin with, we shall suppose that itio, as well as the collection determined by Iio,is subject to Geom: I Geom: I1 Geom: 111. Fix an R with the property that all the normal translations are in the R-tube. [In view of Geom: 11, this is clearly possible. The point, of course, is that one number, viz., R, is being used simultaneously for all Wk.]We impose the blanket hypothesis that the various cylinders, cones, balls (real and imaginary), or whatever that appear below are all contained in the R-tube. The functions E~ and E, play a technical role vis a vis these sets (cf. infra). [Actually, enters does not. We work with both only for notational consistency.] in but There are some additional geometric requirements which must also be fulfilled. Am-Geom: A For every X E itio, For every i E Iio, Am-Geom: B Let
E
> 0. For every X E itio, U P i o I Aio : E )
is a convex cone of radius E ~ ( E )(with vertex X(Pio,Aio)). For every i’ E Iio,
I
I/;*(PioAio: E )
is a convex cone of radius
(with vertex XI,(Pio,Aio)).For every i” E Ira, I/;..(Pi0I Ai,: E )
is convex and
Re( V,.(PioI Aio:E ) )
is contained in the interior of the convex hull of and - ~ p , o ( ( ~ i * * ( PAio) i o - n 4J1)* Apart from these geometric assumptions, we shall impose a series of conditions on the location of the singularities of the E-functions and the V-functions. -~
P i o ( ~ i J
3
250
CHAPTER 5
Am-Sing: I For every X @ X i , and each A,
E
VE(PiolAio: E),
IBd WPio AiJ : A d meets no singular hyperplane of 3
V(X: GI({l}, (1)): ?I?: PioIAio:?: For every i E Iio and each Ai E v ( P i oI Aio:E), IBe(Xi(Pi0 A i o ) 9
?).
4)
meets no singular hyperplane of V(Xi: GI({l}, 11)):
?I?:PioIAio:?:
?).
Am-Sing: I1 For every X E X i 0 , any singular hyperplane of
V(X: Gl({l},
(1)):
?I?:Piol.4io: ?: ?)
mhich meets C(X(Pi0 Aio): E X ( & ) : E ) 7
also meets For every i’ E I;,, any singular hyperplane of which meets also meets IBm(Xi4Pio AiJ)* 9
Am-Sing: 111 For every i’ E Iio, any singular hyperplane H, of
E(3,v: PIA,: PioIAio:?: ?: ?) which meets IBt,(Xi,(Pio,AiJ)
has the property that (Hs - Xi,(Pio,AiJ)
+ 6i
3
Dis(JEv)(Pi0,Ai,J
Remark Am-Sing: I and Am-Sing: I1 have been formulated in the special case: WRk= {G}. But then, in view of Assumption 4(V), this serves to cover all the cases. Let us also bear in mind Propositions 5.1 and 5.2. Am-Sing: 111is, of
25 1
EISENSTEIN SYSTEMS
course, suggested by Proposition 5.11. It will appear shortly that this condition is automatic for the elements of Xi,; hence no assumption in this direction is necessary (thus accounting for the apparent asymmetry in its formulation). Finally, there is an analytic condition which must be met. Am-An
Suppose that
Then: There exists an orthogonal projection
Q ~L:,W : I wr ; 6,00)
-+
L:,((G
I v r ; d,oo)
onto a closed subspace, commuting with the 3(ft),such that for all
r0
+
(6, oio;R )
vjoE2Aio %AJ,(~,
oj,;
R)
and all E > 0 the following hold. Am-An,
The difference between (R(C;0 t ) Q @O!,(P ~ I Ail, @YJ~(P I Ail)
and
the first expression being summed over all X E iTi0(Wk) having the property that Re((X(Pi0,
A i o ) , X ( P i 0 , AiJ>)
and over all wJoioE W&,(X; A,,, Aio),where AX E
I
vS(f'i0 Ai,:
EL
>
-&'
252
CHAPTER 5
the second expression being summed over all i E Iio(Wk)having the property that Re((Xi(Pi0, AiJ,
Xi(PiO3 A i O ) > )
> -c2
and over all wJoioE W&,(Xi; A j o ,Aio),where Ai E
V,(PioI Aio: E ) ,
can be holomorphically continued to the region Am-An2 The difference between
and
the first expression being summed over all X E Xio(Wk) having the property that t Re((X(Pi0, Aio), X(Pi0, AiJ>) > -c2 and over all wJoioE W&,(X; A,, Aio),where AX E VXPiO I Aio: E),
the second expression being summed over all i E Iio(Wk)having the property that Re((Xi(Pi0, Ai,J, Xi(Pi0, Aio))) > and over all wJoioE W&,(Xi; A j o ,Aio),where
4 E UPio IAio: E), can be holomorphically continued to the region
Wi) > 1,.
-8’
EISENSTEIN SYSTEMS
253
Remark It is tacitly assumed that QPis independent of the indices io and
j,. QP is necessarily unique.
The integrals figuring in Am-An, and Am-An2 are evidently holomorphic in the region Re(() > R2. [Needless to say, the sums involved are finite.] Elementary considerations imply that the possible singularities of the integrals in the strip R2 2 Re(c) > & lie in certain parabolic sectors, with vertex the inner product of the conal point and focus the inner product of the normal translation (see Fig. 5.) In the figure the circles represent the various (A!& L A ! , A!>,
while the dots represent the various (X(Pi0, A i o h X(Pi0, Aio)) (XdPio, Aio), Xi(Pio, Aio)), Y
FIG.5
254
CHAPTER 5
Off of the indicated parabolic sectors, the integrals are holomorphic to the right of the line Re([) = A,. It should also be noted that the integrals involving the X or XI such that
i
Re((X(Pio, AiJ, X(Pi,, AiJ>) I-E' Re() -E'
are automatically holomorphic for Re([) > AE. We shall denote an amalgamation by the symbol ( ( x i o ~ l ~ o ~ lErXo, ,~
i ,
6 ,{ Q P } ) ) .
Example (The Canonical Amalgamation) Put b = rank(gi,) + 1. We then set X i , = @,I;, = 0,ly0 = I@}. Assign to the unique index i E li, the equivariant system Xi of admissible affine subspaces attached to Vi0via the prescription XI(Pio,Aio) = complexification of hi0
and take for the Eisenstein system belonging to Xi the canonical one. This said, let be any nonincreasing function. Moreover, for all E > 0, let &(PiolAio:E ) be the points in Fplo(&io) lying in the R-tube. Finally, let QP be the identity map. It is then clear that the data
Iio, constitutes an amalgamation. ((Xi09
IYo, E X ,
~
i
VX, r 6
9
{Qp}))
In what follows, we shall frequently find it convenient to write
A(X: wToio:A6: c)
in place of
V(X: P I ( A i , A i ) : PjolAjo:PioIAio:wjoio:Ai',)
Similar abbreviations with 3: replaced by some other symbol, e.g. Xi or fi, will be used without comment. Lemma Let ((xi09
EX, ~ i VX, ,
VI, {Qpl))
be an amalgamation of dimension b-then, for each i , , Xi, satisjes An. Proof Naturally, the projection QP appearing in Am-An will be the projection Q P appearing in An. The main point, therefore, is to produce the
255
EISENSTEIN SYSTEMS
requisite family of polynomials. So let E > 0 be given-then we can certainly find a family of polynomials p! = {pJo,,}(depending on E ) such that pJo,&IWPjo,A j J + + 0
but such that
('2) E xjo(wk))
~ X ~=( 0, ~io, V(3i: PI(Ai, A{):f'joIAjo: Pio(Aio:wJoi0:?)0 d ~ o m ~ ~ o , E A,,)+ where Re((Xi(Pio, Aio)r Xi(Pi0, Ai,)>) > - E 2 (i E Iio(wk))* In addition, it may be supposed that V(3: PI(& AJ: Pjopljo:PioIAio:wJoio:?) 0 d"ompio.&, t where is holomorphic on
Re((X(Pi0,
Ai,J
X(Pio, Aio)>) > - c 2
IBd(X(Pi0 AiJ) 3
Now let
A ! ~ ( oio; S , R; {yloE s A f o ( d r ojo; R). @ o!
Then, by construction,
( 3 E Xio(wA)*
E~
A
(V(X: PI(Ai, Ai): PjolAjo:PioIAio:wJoio:Ale) d@!o(A/o), dYjo(- w ~ o i o ~ ~ o ) ) is holomorphic in a neighborhood of IB!(3(Pio, Aio))for all 3E E xiO(%k)with Re((X(Pi0, AiJ, X(Pi0, AiJ>) > -cZ, hence a fortiori for all 3 E Xio(wk)with Re((X(Pi0, Aio), X(Pio, Aio)>) > R2 - ez. Furthermore, in Am-An, and Am-An, ,only the terms involving t h e 3 E Xio(Wk) having the property that Re((X(Pi0, AiJ, X(Pi0, AiO)>) > - c 2 remain. This said, let us verify An, and An,. We need only deal explicitly with An,, the discussion of An, being entirely analogous. Claim: If then the difference
s
Re() > -c2,
lB~(.%(Pio, Aio): A s )
A(X: wJoio:A!o: 4') IdA!ol
256
CHAPTER 5
can be holomorphically continued to the region Re([) > A,. Admit the claim. For An,, it is a question of showing that the difference between
(NC;')QP
%t0(P I Ail, @vJ~(P IAil)
and
summed over all 3E E Xio(Wk)having the property that Re((X(Pi0, AiJ X(Pi0, AiJ>) > A, and over all wJoioE W&,(X; Ajo, Aio),can be holomorphically continued to the region Re([) > A,. The strategy therefore is obvious: First, add and subtract the sum over X and wJoio of the integrals
I
A(X: wJfoio:A!o:
I B t (%PI,, Aio): Ax)
where
Re((X(Pi,,
Air,),
[) IdAfoI,
X(Pi0, A i o ) > ) >
-E';
second, add and subtract the sum over X and wjoioof the integrals
s
A(X: wJoio:A!o: c) IdAfoI,
IBd(X(Pi0. a d )
where Ae
2 Re((X(Pi0, Aio), X(Pio, A J > ) >
-E'*
Bear in mind that the latter integrals are automatically holomorphic in the region Re(<) > A,. An, then follows upon quoting Am-An, and the claim. It remains to consider the claim. Note that
- X(Pi0, AiJ is real. Let be the set of all points in the projection of 3E(Pio,Aio)- onto ?i!o which are orthogonal to AI
-
X(Pi0, Ai,J
and have norm less than E. If 0 I c I 1, then
A(X: wfoio:A!o:
[)
EISENSTEIN SYSTEMS
257
Y
FIG.6
and, by Fubini’s theorem, its integral over this region can be written in the form -
~
I
I -Ax ( P~i l , ,Aio)lt
x A(X: wjoi0:X(Pio,Aio)
j’ J
+
D!.
I
W: A to)
+ z(A&- X(Pio, Aio)):i)dz IdA!oI,
the inner contour being as shown in Fig. 6. Call this integral I(c: c)-then our claim amounts to the assertion that
I(1: C)
-
I(0:[)
can be holomorphically continued to the region Re(c) > 1,. Suppose that Re([) > R2-then Cauchy’s theorem enables us to write I(1: [) - I(0: i)
as
258
CHAPTER 5 Y
Our claim will be settled when it is shown that the real part of this expression is II, on C(AI0). Write
Then Re((X(Pi0, Aio), X(Pi0, AiJ>) - IINoII’ + Re(zZ)IIA&- X(Pi0, AiJII’ = Re((X(Pi0, A i o L X(Pi0, AiJ>) - E~ + x211A&- X(Pi0, AiJII’ 5 (Re(X(Pi0, AiJ), Re(X(f‘io, Ai,-,))> - c2 + IIAL - X(Pi0, AiJI12 = IIRe(Al)II’ - E’ IRZ - c2 = I , . Hence the claim. The proof of the lemma is thus complete. //
Of course, not all the assumptions defining an amalgamation were used in the preceding argument. The point is that now for each io, JEio is subject to Geom: I Geom: I1 Geom: 111 and
An.
Therefore this part of the data may be fitted into the picture developed earlier. The following lemma is the main result on amalgamations.
Main Lemma Let ((xio,
KO,EX, &I,
VX, V,,
{Qp)))
259
EISENSTEIN SYSTEMS
be an amalgamation of dimension b-then there exists an amalgamation ReS{((Xio,
&o,
I;o,
EX, EI,
VX, 6 , {QP}))}
of dimension b - 1 such that
RNQP) = QP - Qx. [The objects of Res{(. . .)) are denoted by placing the symbol Res in front of the corresponding objects in (. . .).I Grant the lemma for the time being-then one has
XI. Im(Qp) = Im(Res(Qp)) 0 J%,((G I W r ;400; Proof of Theorem 5.12 For integral n 2 0,define by recursion a sequence {Amal(n)}of amalgamations according to the prescription
i
Amal(0) = canonical amalgamation Amal(n) = Res{Amal(n - 1)).
Let us agree to write
. . . , {QP’})).
Amal(n) = ((Xi:’, Observe that
dim(Amal(n)) = rank(%,)
+ 1 - n,
so the amalgamation is trivial when n > rank(%,) 0 I b I rank(W0),
set JE(%o;
b) =
IJ Xio(b), io
where Xi001 =
xi,(rank(Yo)+
For each io, Xi,@) is subject to Geom: 1 Geom: I1 Geom: I11 and An.
1-3).
+ 1. For any integer b,
260
CHAPTER 5
Now let (P, S; A i ) in W kbe given. Claim:
Here, of course, is the closed subspace of
L:,((G I Pyr; 6, oo; W ) G , ( ( G I PYr; 4 00)
associated with Amal(r). This can be proved by induction on n. If, to begin with, n = 0, then QP) is the identity map and the sum is empty. In general, thanks to the lemma supra, we have Im(Q',"))= Im(Q',""+'))0 L$,((GIP)/I'; a,@,; X)O(nJ, thus pushing the induction forward. If b < rank(%,),
i.e., if n > rank(q0) - rank(%,)
+ 1,
then the sums figuring in Am-An are empty, so, as the inner product per R ( ( ; Rt)QP) is entire, Q',") = 0. It therefore follows that G , ( ( G IP)/r; S, 0 0 )
is equal to
c
+1
c
o L:,W I wr; 6, oo;x, b),
0 5 r 5 rank(%,) -fank(Yk)
or still, is equal to
rank(%*) 5 b 5 rank(W0)
0L~,((G I wr;6, oo; XP
the space L$,((G I W r ;4 Oo; WC0) being trivial. The theorem is thereby established. // Remark The space L;,(G/r; 8,oo; w o )
was constructed near the end of Chapter 4. In the present context, it corresponds to L+,(G/r;6, oo;3,
EISENSTEIN SYSTEMS
26 1
i.e., the data per Amal(1). The methods employed are completely different. The initial construction relied on estimates, while the current construction depends on the spectral theory of 0. We must now deal with the lemma. Given the amalgamation , K, {QP))), ((xi09 Iio, I&, EX, ~ i VX,
it is then a question of manufacturing Res(Xio),. . . , Res(Q,) so as to satisfy the various conditions. The key rests with a potentially confounding farrago of integral manipulations. We have, of course, already specified Res(Q,): R ~ ~ Q=P QP ) - Qr. This being so, let
i
R) YJoE .YfAjo(6, (!Ijo; R).
@/o
Then
EzAto(b,
oio;
( R ( l ;0t, R~S(QP)@,J~(P I Ail, @YJ~(PI A,))
) Q @,lo(P ~ IA i l @YJ~(P IAil) - (R(C;0t)Qx @,;t,(P I Aih @YJ~(P I Ail)*
= (R(C; 0t
Suppose that Re(() > R2-then, using Proposition 5.9(V), we can write
( R ( l ;0' ) Q x @@lo(PI A i l @YJ~(PI Ail) as
The integrals taken from E to infinity are evidently holomorphic in the region Re(() > 1,.Consequently, the difference between ( R ( ( ;0t ) Q @,lo(P ~ I Ai), @,f0(P I Ail) and
can be holomorphically continued to the region Re(() > A,. For the sake of brevity, let us agree to write
A(3 : Wjoio(r): w!oio(t) : A!o(j) : i)
262
CHAPTER 5
in place of
applied to
and paired with Let
stand for
summed over all i E Iio(%,Jhaving the property that and over all wfoioE W&,(X,: A,, A,,), where
We can therefore say that the difference between
263
EISENSTEIN SYSTEMS
and
=-
A,. Note that the can be holomorphically continued to the region Re([) extra integrals thrown into the first expression are automatically holomorphic, thus cause no difficulty. Analogous considerations apply when [ is replaced by 4. Consider
{A(Ai: X I . . .) - A(X:. . .)}. By hypothesis, A(A4: X: . . .) is holomorphic throughout the region of integration. On the other hand, A(X: . . .) may well have singularities somewhere in the region of integration. [In this connection, bear in mind that it is the sum of the A(X: . . .) which is holomorphic throughout the region of integration; cf. Proposition 5.9(V).] It will be necessary to move the integrals to another region where the summation signs can be pulled outside the integral sign. This requires some preparation. Because %(go) is a finite set, we can form EX(&)
=
minx.X(Wo) Ex(&),
and then assume without loss of generality that KAPio I Aio : E ) has radius E ~ ( E ) ,so that, henceforward, we work only with C(X(Pio,Aio): E X ( & ) :
E).
Let X E Xiopthen, by definition, Fr will be that subset of Cpio(g,ai0) consisting of those with the property that X(Dis(X)(Pio,Aio) = 0. This said, suppose that $ is a hyperplane in X such that $(Pi,, Aio) meets I B m (X(Pi0 AiJ)* If 9
E(X: PIA,: PioIAio:To:A!o: x)
264
CHAPTER 5
Both RBe(X(Pi0, AiJ: X) RBe(WPi0 A i J : f‘~) 9
admit a finite decomposition into a disjoint union of open convex sets which, heuristically, may be thought of as “chambers.” Of course, distinct chambers per RB,(X(Pio,Aio):3)may lie in the same chamber per RB,(X(Pio,A d : FS). In particular, therefore, we can write WAC:Q
RBs,(e)(WPio,Ai,J: X) = C
the set on the right having cardinality c(X), say, a number depending only on the equivalence class of X in X ( { G } ;W o ) (=X(Wo)). Fix general points A,(&: X) E WC(.e: I). Take now two general points A:o,
E
RBEx(c)(X(Pio,AiJ:
which lie in the same component of RBEX(e)(X(Pio,AiJ: f‘d
.I"ilP (cv
ssal
592
266
CHAPTER 5
the difference can be holomorphically continued to the region Re([) > A,. The same conclusion holds if we multiply through by l/m/ and sum over 1. In contradistinction to the first expression, the summands in the second expression are themselves holomorphic throughout the region of integration; thus, for it, the summation signs can be pulled outside the integral sign. Therefore, making an obvious change of variable, we can say that
is equal to
x
A(X: wJoio:A!o: [) ldA!ol.
For each w,foio(l),a given chamber in RB,,(&)(X(Pio Aio) 3)
contains one and only one of the points w ~ o i o ( l ) w ~ ( X,) l ) ~ c ~ ~(1: 5 c 5 CI).
S o , adding and subtracting the integrals per /Ic(&: X) and taking into account what was said in the preceding paragraph, we conclude that the difference bet ween
and
x
A(X: wjoio:
c) ldA,fol
cttn be holomorphically continued to the region Re([) > I , . Let us recapitulate what has been established up to this point: The difference between (R(4'; 0'1 Res(Q~)@,f~(f'I Ail, @YJ~(P I A,))
EISENSTEIN SYSTEMS
267
and
can be holomorphically continued to the region Re(c) > A&.Analogous considerations apply when is replaced by [. It will now be necessary to work with
<
in order to pick up residues. Write
the set on the right having cardinality c(F,), say, a number depending on 3 (and not just its equivalence class in X({G}; Wo) ( =X(Wo))). A given chamber WC(&:F,) contains at least one and perhaps several chambers of RBs,(e)(X(Pio, AiJ: 31, say C , of them. Fix general points A,(E:F,) Rewrite
in the form 1 I-x C(3)
c c
1 s c s c ( x ) wfo,o
E
WC(&:FX).
268
CHAPTER 5
ex
The notation is slightly deceptive since the first usage of the symbol refers to a sum over @, n Zi0(Vk)whereas the second usage of the symbol Ex refers to a sum over &o(%?k).One should also bear in mind that c(X) = c, for all X E (5,. This explained, employ the principle spelled out earlier to conclude that the difference between
and
[J
I B , ~ ( ~ ( P ,A,,): , , AX)
-s
1
A(3:wjoio:A!,: 5) ldA!ol
I B ~ ( S ( P , , A,,): ,
ME:
F ~ ) )
can be holomorphically continued to the region Re([) > I,. For purposes which will become clear shortly, we shall now introduce a third index set I;, whose elements i l are ordered triples (3,c, fix where 3 E Xi,, 1 Ic I c(F,), and fi is a distinguished hyperplane in 3 with the property that A, and A,(&: F,) are on opposite sides of $(pi0 AiJ n RBc,(,)(X(Pio Ai0))3
9
It can be supposed that 1; is disjoint from I;, and Iy0. There is a certain set of data which must be attached to the elements of 1;. To begin with, associate with (3,c, 8)the Eisenstein system
belonging to fi, {E, V} being the Eisenstein system belonging to 3. [Delete the index if the corresponding Eisenstein system is trivial.] The real unit normal implicit in this construction is chosen so as to point out from S(Pi0, AiJ n RBc,(e)(X(Pio, Aio))
toward A,. Next, assign to (3,c, fi) the “epsilon function” determined by 3 itself, namely E~ (cf. supra). It remains to specify the ambient nonempty open “ I/-subset ” of H(Pi0, Aio)
+ &(Pi,, AiJ“
naio.
269
ElSENSTElN SYSTEMS
Note that H(Pi,, A,,,) = X ( P i , , Aio).Select a polygonal path
10, 11
~ c :
+
RBzr(e)(X(Pio,Aio))
starting at A,(E:F,) and ending at A, which crosses the “ w-hyperplanes” and the “A-hyperplanes” in a semiregular manner. This can be done in such a way that each hyperplane is crossed orthogonally, no hyperplane is crossed twice, and no hyperplane is crossed at all unless Ac(&:F,) and A, are on opposite sides of it. The connected component of the point where yc crosses $(Pi,, AiJ in ($(pi09 Aio)
-
u
$'(Pie, AiJ) n RBzx(e,(X(Pio, Aio)),
8‘+5
8’ being a “w-hyperplane” or a “A-hyperplane,” will be taken as the “ V subset” corresponding to (3,c, 9).Denote this set by the symbol Va(Pi0I Ai,
when no confusion is possible. To secure the shrinking property, it can be assumed, with no loss of generality, that > c2 VXPioI Ai,:
~ 1 = )
I/x(PioIAio: ~ 2 n ) RBc,(e,JX(Pio, 40)).
Suppose given X E Xi,. Fix c, 1 Ic Ic(F,). Let us assume that xyk is 3-admissible, i.e., that X E Xi0(%J. Consider those indices (X, c, $) such that xy, is $-admissible. Write for a sum over the set of fi singled out by this requirement. Since X and c are fixed, the notation is permissible.
c5
Lemma Let As E V5(PioIA,,: &)-then the difference between
A(X: wjoio:A/o Aio): A&: F x ) )
and
A($: wJoio:A/,: () IdA/, can be holomorphically continued to the region Re(() > &. [Note: In the first expression, the sum over wJoiois relative to the elements of W&,(X; A,,, A,,), while in the second the sum over wjoio is relative to the elements of W&,($;A ~ , A,,).] , Proof It should be pointed out that the integrand appearing in the integral relative to 9 is holomorphic throughout the region of integration, so that, for the usual reasons, there is determined a function of (,holomorphic in and A2,s lying on that Re(l) > R 2 . This being the case, choose points part of yc which intersects $(Pi,, A,,) orthogonally. We shall assume that lies in the same half-space as A,(&: F,) and that A2,5 lies in the same
270
CHAPTER 5
half-space as A,. Modulo a function holomorphic in the region Re(() > A,, one can then write
[s ;[s
-I -s
I B . ~ ( x ( P , ,A . , ~ )A: ~ )
as
1 1
A@: wjoio:A!o: i)I dA,toI
I B ~ ~ x ( PA , ~ ).A'(&: : FX))
IBd(X(Pt0.Atoh Az9 8 )
A(3: wJoio:A!o: () ldA~ol.
IBd(W'n0, Atoh A I . 6)
Fixing fi, let A0,$ be the point at which ye crosses b(Pio, Aio).It will be enough to show that wJot0
less
[s
IB~(x(P,,A , , ~ )A*, : 6)
271.
-s
c1
wfoto
1
A ( 3 : wJoio:A!o: i)ldA!ol
IB.~(x(P,,A . ,~):
8)
A(fi: wjoi0:A/o: () IdA!ol
IBt(b(Pio,
Ao, 6)
can be holomorphically continued to the region Re(() > &. Let Al be a real unit normal to !ij(P,,, Ai0)pointing in the direction of A2,5. Determine real constants cl, and c2, by writing
i
+ Cl.bA1 + C2.bAl
< 0) > 0). Fubini's theorem, in conjunction with the residual definitions, then implies that the difference is equal to h . 5 = A0.5 A2.5 = A0.5
x A(X: wjoio:A!,
(C,,$
+ z A 1 : () dz ldAJOl,
the inner contour being as shown in Fig. 8. Here, of course, the summation over wJoiois relative to the elements of W&,(3;Aj,, A+. Under the assumption that Re(() > R2,use Cauchy's theorem to shift the integral over
C(C,,~: c2,&:0: A!o) to the contour shown in Fig. 9. To finish up, we need only show that Re((A/,
+ z A f , AIo f z A 1 ) ) I A,
on the contour. As this kind of calculation has been written out several times before, it can be omitted. //
27 1
ENENSTEIN SYSTEMS
FIG.8
I__
f
+ FIG.9
Remark There is a potential minor difficulty in the preceding computation which we deliberately ignored, viz., in order to apply Fubini's theorem legitimately, it is necessary that the contours stay at least a fixed distance from the singularity. The way out is to put, if need be, a bump of predetermined radius r into the horizontal line segments at the point where they cross the y-axis. This does not affect the argument in any way. Now where are we? We can certainly say that the difference between
( R ( i ; 0') Res(Q~)@~l~(f' I Ai),@vJ~(P I Ail)
272
CHAPTER 5
and 1
ob’.
:co 1
1
cc c
wf,,
X
+ can be holomorphically continued to the region Re(() > &. Analogous considerations apply when [ is replaced by 4. Recall that Iio= 11, u 1s. Change the notation so that this time Iio = I:, u 1r0u 1;;. Suppose we throw away those X such that Re((X(Pio, AiJ X(Pi0, AiO))) I Naturally, nothing is lost in doing this. The preceding lemma then implies that the difference between and
(Nl; 0’)Res(Qp)%io(P I A i l @ Y J ~ (I Ail) P
(2n)bt l -
SIB:(XdPi0,
A(&: wjoi0:A/o: [) ldA!ol,
Ato): AI)
summed over all i E Iio(Wk)having the property that Re((Xi(Pi0, Aio), Xi(Pi,, AiO))) > - c 2 and over all wjoioE W&,(Xi; A j o ,Aio), where E UPio
I Aio: E),
can be holomorphically continued to the region Re([) > &. Analogous considerations apply when is replaced by 4. We are, at last, in a position to take the first step in the construction of the residual amalgamation, viz., the definition of Res(Sio).Tentatively, Re@,) will be the set of all equivariant systems of admissible a f i e subspaces of dimension b - 1 attached to Wi0 for which there exists an index i E I, such that X = Xi. Observe that here it is a question of a finite set of i. We then attach an Eisenstein system {E, V} to each such X according to the prescription
jE(X:. . .) = P ( X : . . .) =
1E(Xi: . . .)
c V ( X , : ..
i:X1=X
i:fr=X
.).
273
EISENSTEIN SYSTEMS
Delete X if this prescription yields a trivial Eisenstein system. [Note: Distinct indices i may very well yield distinct Eisenstein systems even though they give the same X,.] There are, of course, certain conditions which the elements of Res(X,,) must be shown to satisfy. The fact that
i
Geom: I Geom: I1 Geom: I11
are in force is clear. The verification of Am-Geom: A for Res(X,,), i.e., that for every X E Res(Xi,), -Re(X(Pio, Aio)) E dP,o((X(Pio,A i J " n Q'), is simple enough. Indeed, the only possible issue would be if X were equal t o some element coming from 1: but equal to no element coming from I:,, or I:.:,. Suppose, for definiteness, that there exists an index (Xo, co, 90)such that X = Go. Write Jso(Pio,
Aio)
= W P i o , AiJ n (XtAPiO, AiJ
+ ~er(&)),
where 1, E Fxo. Recall that Ho(Pio,A,) = Xo(P,,, Aio). One must show therefore that -Re(Xo(Pi,, AiJ) E dPi,((GO(pio, Aio)" n aio)'). Since it is already the case that -Re(Xo(Pi,, Aio)) E dP,,((WPio, A i J " n hJ'), there exist, by definition, elements I Z E E ~ , , (aio), ~ , with xlXo(Pi,,A,,)" = 0, and nonnegative constants c1 such that
- Re(XO(Pi0 Aio)) = C ~1 1. 9
Because Sjo(Pio,A,,)" is contained in Xo(Pio,A,,)", this fact implies our assertion. Am-Geom: B, Am-Sing: I, and Am-Sing: I1 (Am-Sing: 111 is not applicable) are predicated upon the existence of E~ and Vx(PiolAio:E), which, accordingly, must be defined before anything can be said. What will be done is this: Construct E~ so as to render Am-Sing: I1 true and then construct Vx(PiolAio:E ) so as to render Am-Geom: B and Am-Sing: I true.
274
CHAPTER 5
Let 6,be the set of all hyperplanes missing IBm (3(.. .)) which are singular hyperplanes of some V(3,: . . .), 3' = X . The collection 6,(Pi0IAio)of such in 3E(Pio,Aio)is locally finite, thus their union is closed. Let &x(PiolAi0):
IBm(3(Pi", AiJ)
+
R+
be the corresponding distance function. Put =
t min{min dQI(P,oIA,o)IIB,(3(Pio,Aio)X R - IIRe(X(Pi0, AiJ>II, 11.
Then it is clear that any singular hyperplane of V ( 3 : . . .) which meets C(3(Pi0,Aio):Ex(&):
E)
also meets IBm(X(Pi0, AiO)), and this is the content of Am-Sing :I1 for 3. We must now modify t, to produce and then define V,(Pio I Aio:E). For this purpose, let
E,
bn,(Pioy AiJ be an enumeration of the singular hyperplanes of V ( 3 : . . .) which meet B l ( P i o 9 AiiJ
* * *
9
C(X(Pi0,Aio): 1 : E). As E increases, more hyperplanes may appear. We can and will assume that the enumeration then satisfies an obvious consistency condition. Write
Bi(Pi0, Ai,J = {Aio E X(Pio,Ato): Ji(Aio) = ci}. Put
,in{
I W c i ) - Re(Ji(X(Pi0, Aio))) I . IIiiII
M c i ) # Re(Ji(X(Pi,, Ai,))) (1 Ii In,)
11 .
Needless to say, Am-Sing: I1 is still valid for 3. Take then for Vz(PiolAio:E ) any component of RB,,(,)(WPio AiJ: 3). With this choice, Am-Geom: B for 3 is, of course, immediate. To discuss Am-Sing: I for 3, let 1
AXE VdPio I Aio: E )
275
EISENSTEIN SYSTEMS
and suppose that IBc(X(Pi0 AiJ : AX) meets Bi(Pi0,A,) for some i-then, of necessity, 3
1 Therefore
the first expression being summed over all X E Res(Xio(Vk)) having the property that Re((X(Pi0,
AiJ
X(Pi0, A d > ) > - c 2
and over all wjoioE W&,(X;A j o ,Aio),where E
h(Pi0I Aio:E),
the second expression being summed over all i E Iio(Vk)having the property that Re((Xi(Pi0, and over all w;,]~,] E W',,(X,;
AioX
Xi(Pior AiJ>) >
- E ~
A,,,, Ai,,), where E
&(pi0I Aio: E ) ,
can be holomorphically continued to the region Re((') > 1,.Analogous considerations apply when (' is replaced by 4. [Note: Conventionally, we take A,, = As for all i such that Xi = 3.1
276
CHAPTER 5
Observe that the first expression in the preceding difference is exactly what we want for Am-An in the residual picture. Let us now move on and define Res(Ifo).Tentatively, Res(IIo)will be the set of all ordered pairs
(i. 81, where i E 11, u 1; and 9 is a singular hyperplane of V(&: . . .) meeting ZH,(3Ei(. . .)) with the property that @(Pi,, AiJ n RBca(Xi(Pi0, AiJ) separates A, and AXl. Associate with (i, 8)the Eisenstein system {Resb(E), Resb(V)}belonging to sj, {E, V} being the Eisenstein system belonging to Xi. [Delete the index if the corresponding Eisenstein system is trivial.] The real unit normal implicit in this construction is chosen so as to point out from $(Pi0
9
Aio) n RBca(Xi(Pi0 AiJ)
toward Ai. It is not difficult to concoct E,. and i$(PiolAio:E ) in such a way that the requisite conditions are met. The details are similar to but simpler than those presented above, and hence can be omitted. Performing a familiar manipulation, we can then say that the difference between (R(C;
'1
R~~(QP)@cJ~(P I Ail, @,j0(PI Ail)
and n
n
the first expression being summed over all X E Res(Xio(Wk))having the property that Re() > - e 2 and over all wJoioE W&,(X; A,, Aio),where AX E G(Pi,-,
IAio: E),
the second expression being summed over all i' E Res(Iio(Wk))having the property that Re() > - c 2
277
ElSENSTElN SYSTEMS
and over all wJoioE W&,(Xit;A j o , A d , where A,. E c.(PioI Aio: E ) ,
the third expression being summed over all i" E If0(Wk) having the property that Re( -E' and over all wJoioE W&(X,,,; A,,
Aio),where
A,.. E &,,(Pi0I Aio:E),
can be holomorphically continued to the region Re(c) > 1,. Analogous considerations apply when [ is replaced by 1. Observe that the first and second expressions in the preceding difference are exactly what we want for Am-An in the residual picture. We have yet to come to grips with the most difficult part of the proof, viz., the determination of Res(Ifo).The reader may have wondered to what extent some of our conditions have actually been needed. Of course Am-Sing: I, 11, and 111, as well as Am-An, have all seen service. On the other hand, the role thus far of the geometric conditions Am-Geom: A and B is less clear. In fact, they have scarcely been used at all. But this is the place where their presence will finally be felt. Let us begin by stating a simple geometric criterion. Let fi be an admissible hyperplane in X, say
Lemma
$(Pio, Aio) = {AioE x(Pi0,Aio) : &AiJ = c>.
Let A0 E X(Pio,Aio)
+ X(Pio,A,,)"
n &io*
Then fi(Pi0, AiJ n IB,(X(Pio,Aio):Ao) #
0
?fs (i) (ii)
4 E
~
*
-0X(Pi0, Aio)) = ~ e ( c X(X(Pi0, AiO))), 1 1 0 p2 I ~ m ( c - %X(Pio AiJ)) I*
II1I W P i o AiJ9
9
Corollary Suppose that A,,
are such that
A2
E X(Pi0, Aio)
+ x(Pio,A,,)"
$(Pi,, Aio)n IB,(X(Pio, Aio): Ai) = 0
n 61,
(i = 1,2).
If E .
IIXIWPio, A i J * I O P 2 I I ~ (-C%X(Pio, Aio)))I,
278
CHAPTER 5
then there exists t E 30, 1[ with the property that $(Pi,, AiJ n IB&(X(Pi,,Aio):t A ,
8
i
X(A1
+ (1 - t)Az) # 0'
X(Pio,Ato))- Re(c - X(X(Pi0, Ad)) - X(Pio,A,,)) - Re(c - X(X(Pi0, A,,))) -
have opposite signs.
Fix an index i" E I;,-then sets
attached to i" are two nonempty open convex
i
&(Pi, IAio:E ) Vx,#,(Pio I A,,: E ) .
Bear in mind that Vxl,,(Pio~Aio: E ) is actually a cone, defined earlier in the E ) be the set of all singular hyperplanes $3 of Res(X,,) picture. Let So(?', V(XI.,:.. .) such that $3(Pi,, Ai,) n IB&XdPi", A,,):
for some 12, E Xlt,(Pio,A,,)
Z0
+ XI.,(Pio,A,,)" n tii,.
Since { & ~ o-
X,(Pi,, A,,)): ~o
E xv(~i,,A,,)
fills out R,it is clear that $3 E So(?', E ) iff E.
+ ~ i , , ( ~~i ,i,, ) -n
IIJ-IWPio* Aio)"IIop 2 I I ~ ( C J-(Xi,,(Pio, AjJ))I,
it being assumed, of course, that b(Pi0, AiJ = {Aio E &**(Pi,,A,,) : J-(A,,J = c}.
The union
8o(i") =
u So(i",4
&>O
,
is the set of singular hyperplanes of V(X,..: . . .). This said, fix $3 E So(?',E ) and take points
i
A1
E &,,(Pi,IAi,: E )
A? E Vxl,,(PioIAi,:E).
Owing to Am-Sing: I,
$3(Pio,Aio) n IB&(Xv(Pio,A,,): A,)
=
0
(i = 1, 2).
v
or, there exist A, {A2
E
€
t+(PioIAi0:E ) V~ 1 (Pi0 I A io : E )
and t €10, 1[ with the property that
$(Pi,, Aio) n IBz(XdPi0, Aio):
+ (1 - t)A2) f 0.
Let @(i”, E ) be the set of all fi in &,(i’‘, E ) possessing either one of the two preceding equivalent properties. Suppose that > &,-then
i$,(Pio~Aio: E l ) c I/;..(PioIAi0: E,) vxl~~(Pio 1 A;,: ~ 1 c) vxl,,(f‘io I : E Z ) , so it is clear that
$(if’,
E,)
c $(i”,
E ~ ) .
280
CHAPTER 5
Put
$(i")
=
u $(i",
&
E).
>O
We are now in a position to define Res(Iro).Tentatively, Res(I;,) will be the set of all ordered pairs
V', $1, where i" E Ir0 and $ E $(i"). Associate with (irr,$) the Eisenstein system {Res,(E), ResB(V)}belonging to !+j, {E, V} being the Eisenstein system belonging to &. [Delete the index if the corresponding Eisenstein system is trivial.] The real unit normal implicit in this construction is chosen so as to point out from (120 : $(Pi0 AiJ n IBc(XdPi0 A iJ : Ao) # 0 } 3
7
toward V,4PioIAio: E),
it being assumed, of course, that 8 E $(i", construction of the sets Yi,,, ,,(Pi0
E).
Our first objective will be the
I Aio : 6).
Unfortunately, this is a rather involved and complicated business. Lemma
Let $ be an admissible hyperplane in X, say
@(Pi,, AiJ = {Aio W P i o ,AiJ : 4 A i J = C I . Then there exists a unique point
A d $ ) ~ W p i oAio)" , n 6io with the following property: For all E X(Pi0, AiO)
+ WPiO, AiO)" n 6io
such that A.
+ flAs2($)
$(Pi09 AiJ n IBe(X(Pi0, AiJ: A,)
+ 0,
is that element of
$(Pi09 Aio) n IBe(WPi0, AiJ: Ao) closest to A,.
[Let Bw(Pi0,Aio)stand for the A0 E X(Pi0, AiJ
+ X(Pi0, Aio)" n 6io
28 1
EISENSTEIN SYSTEMS
such that $3(Pio, AiJ n IBco(X(Pi0, A i J :
One then has
i
H(Pi,, AiJ = HR(Pi,, Air,) + H(Pi,, AiJ + $(Pi,, AiJ aio as can be checked without difficulty.]
Let &(i”,
E)
=
+ 0.
fia(PiO3 Air,)
+~ M B ) ,
be the set of all fin,$ belonging to %(if’,E). Put &(if’) =
u &(i”,
E).
&>O
Let 2l(i”, E ) be the set of all !& n !iji !, & and .f& belonging to $jsr(i”,E). The elements % of “(i”, E ) are thus affine subspaces of codimension 2. Split 2l(i”,E ) into two parts by the definition
ia1(i”,
E ) = {aE %(i”, 8 ) : Xl,,(Pi,, Ai,) E %(Pi,, Aio)} 212(i”,E ) = {% E a(?’, E ) : XI..(Pi,,A,,) # %(Pi,, Aio)}.
Put
/
i
a l ( i N )=
u Nl(i”, u a2(i”,
E)
e>O
a2(i”)=
E).
&>O
It should be noted that both Wl(i”,E ) and 212(i”,E ) are finite, that in addition %,(ill)is also finite, and that finally a#’) is at most countable. By our assumptions, each l+(Pi0 1 Aio: E )
has a compact closure. Furthermore, the collection of such is nested with respect to E. Therefore, the intersection &>O
W P i o I Aio
E)
is nonempty, containing, say, A$. Let W2(i”, E)* be the set of all in a2(i”,E ) which meet the line segment joining X,..(Pi,, Aio)to A:,. Let RB(i”,E ) and RB(i”,E)* be real balls centered at XI.,(Pio,Aio)and A:,, respectively, such that the only E &(in, E ) which meet the convex hull of RB(i”,E ) and RB(i”,E)* are in N2(i”,E)*. One can certainly arrange matters so that E,
> E2
*
i
RB(i”,E ~ c) RB(i”, t 2 ) RB(i”,E ~ ) * c RB(i”,t 2 ) * .
28 2
CHAPTER 5
Let ‘illE 212(i”,&)*-then, by definition, Xi4pio AiJ 3
4 ‘WPio, AiJ
Therefore the pair (Xi*,(pio 7
%(pi03 Aio))
determines in a unique way a hyperplane in Xi4piay AiJ
+ %4Pi,,
AiJ“ n hi0
containing both. Since ‘9I meets the line segment joining XIt.(Pio,Aio)to A:,, such a hyperplane necessarily passes through A;,. Varying ‘illthen gives a finite collection of hyperplanes which, in view of what has just been said, slices RB(i”,E)* into chambers. A generic chamber %‘(ir’,E)* is said to persist if
%‘(i”,E)* n i$(PioIAi0:E ) # for all E‘ 2
0
E.
Lemma Persistent chambers exist. Proof Let {gP(i”, E ) * } be the set of chambers in RB(i”, E ) * . If, to the contrary, no chamber persists, then for each p we can choose an.c, 2 E such that
%,,(ifr, c)* n c..(PioIAio:c,,)
=
0.
Put c0 = max
E,,
P
Since I/i.,(Pi,JAi,:E O ) c c*,(PiolAio: E,,)
for all p,
%Ji”, E)* n l+(PiolAio:cO) = 0, from which it follows that RB(i“,E)* n c..(PioIAio:eO) = $3,
the union of the %‘,,(if’, &)* being dense in RB(i”, E ) * . Consequently, A?, 4 &,*(PioI Ai,: EO),
which is a contradiction. // Suppose that cl > &,-then it is easy to see that every persistent chamber of RB(i”,E ~ ) * contains a persistent chamber of RB(i“, As E increases, the number of chambers may also increase, but, of course, only in jumps. Since it
283
EISENSTEIN SYSTEMS
is a question of an at most countable sequence of selections, one can choose a persistent chamber %'(i'', E)* in RB(i", E)* such that c1
> t 2 3 %'(i", E ~ ) * c %'(i",
E~)*.
Put l+(Pio I Aio:E)* = %'(i'',
n l+(PioIAio: E).
E)*
Although vt,(PioI Aio:E)* is a nonempty open convex subset of Xi*,(Pi0, Aio)
+ xi,*(Pio,AiJ"
n
such that E1
> E l = L$(PioIAio:E l ) *
c l$*(PiolAio: Q)*,
it is still not the candidate used for producing
I
q i , , , @)(Pi0 Aio : E).
Further reductions are necessary. Recall that b,**(PioI A i o :
E)
was taken to be any component of RBcXI,.(E)(xi4Pio AiJ : xi,,). 9
We shall now make this choice more definite by requiring that meet the interior of the convex hull of
284
CHAPTER 5
then XdPiO, AiJ + t(Aio - Xi4Pi0, A i J ) E ~ ~ i ~ * ( p i o InA(iXoi )4 p i o , Aio) + &,(Pio, AiJ“
hioh
for t E 10, I]. Our contention is now immediate. Consider ‘Bl(i”,E ) u 912(i”,E ) * . Given % in this union, there are then two possibilities :
i
A? 4 %(Pi0 AiJ A:, E %(Pi,, Aio). 7
If the first eventuality obtains, then the pair (A?*, %(pi03 A i J )
determines in a unique way a hyperplane in X d P i o , Aio)
+ &(f‘io,
Ai0)- n hi0
containing both. If the second eventuality obtains, then this is no longer true: One simply has A:, E %(Pie, Aio),the latter being of codimension 2. Varying E then gives an at most countable collection of affine subspaces. The Baire category theorem implies that there exists a point At,,, which lies in none of these subspaces, but which is in &,,,(pi0I Aio: 1) n ~ ~ ~ ~ I AiJ , ( p i o
Let % ’ E %,(i”)-then, by definition,
At,,!# %(Pi, AiJ 9
Therefore the pair (A&,* %(Pi, AiJ) 9
9
determines in a unique way a hyperplane in Xi”(Pi0, Aio) + &(Pio, A,,)“ n 6io containing both. Varying %, this procedure then yields finitely many hyperplanes which may or may not meet i+(Pio I Ai,: &)*. Consider only those that do. The collection of such slices h,,(Pi01 Aio:E)* into chambers. Fix one and, without changing the notation, label it by the symbol l$(Pi0 I Ai,: E)* as well. We can and will assume that El
> E2
-
~ : . . ( P i , i A iEl)* o : c J$*(PiolAio:&2)*.
For t > 0, let us agree to write A&(t) = Xi*,(Pi,,Ai,)
+ t(A&
-
Xl*,(Pio,Aio)).
285
EISENSTEIN SYSTEMS
Lemma Given Ai.. E i+(Pi0 I Aio:E ) * ,
there exists a T > 0 such that if0 < t < T , then the line segment joining A&(c) to A,. misses every 2I E a(?’,E).
Proof Start the verification by choosing T small enough to secure the relation (0 < t < T),
A$,..(t)E RB(i”,E )
Let 2I E Nl(i”,&)-then, since the pair (A$,**(t), %(Pi0 AiJ) 9
determines the same hyperplane in Xi4pio9
Ai,J
+ xi,,(Pio,Aio)“
n hi,
as the pair
(AI,*. W P i o , AiJL and the latter misses l$,(PiolAia:E)*, it must also be the case that ‘9I itself fails to cut the line segment joining &,(t) to Alp..Recalling next that RB(i”, E ) and RB(i”, E)* were constructed in such a way that the only 2I E Nz(i”,E ) which meet the convex hull of RB(i”,E ) and RB(i”, E)* are in N12(i”, E)*, we see that if 2I E !&(if’, E ) - Nz(i”,E ) * , then again 2I fails to cut the line segment joining A,*,..(t) to Ai... Finally, we must consider what happens when 2I E 2112(i”,E ) * . Let us remind ourselves that 3
i$,(Pi0I Aio:E)* c %(i”, E)*.
In particular: The line segment joining Xi..(Pio,Aio)to A,. misses every element of N2(i”,E)*. Because N12(i”, E)* is a finite union of closed sets, there exists a neighborhood N of Xi@ia., Aio)in RB(i”, E ) with the property that for all Aio E N,the line segment joining Aioto A,. misses every element of ‘U,(i”, E)*. Adjusting T, if necessary, to force the relation A$,.,(t)E Jlr
(0 < t < T )
then finishes the proof. // Remark The preceding argument yields an obvious uniformity, namely, that the assertion is true as it stands when A,. is allowed to vary in a compact subset of V,.,(PioI Aio:E ) * .
Let !ijE &i”, &)-then $,(Pi,, Aio)separates l$(Pi0 IAio: E )
and
&,,,(Pi,IAio: E).
286
CHAPTER 5
Form fig(Pi0,
AiJ -
u (bd,
O(E)
the symbol on the right standing for the union over the complement of !&(Pio, Aio)in [$,,(i”, E ) ] ~(obvious notation). Given Ai,, E F,,(PioI Aio:E ) * ,
choose T as in the lemma supra. There is no loss of generality in supposing that matters have been so arranged that tE
10, T[ * A&,,(t)E V,,,,(PioI Aio:E ) n RB(i”, E).
This said, fix t €10, r[-then segment
the crossing point in !+&(Pi,, Aio) of the line
+ (1 - s)Ai,,
(s E [0, 11)
sAg,..(t)
determines a component of &(Pi09
AiJ -
u (8d.
O(E)
Ostensibly, the component thus singled out depends on the choice of Ai.., T, and 1. But, in fact, this is not the case. Let, say, A,’.., t,,t , , and t 2 ,t 2 be given. The line segment L , , joining A,?.to A:, is compact; hence one can select a 2‘ with the uniformity mentioned above and used below. Denoting by seg(. . .) the line segment joining ..., the union of {seg(AxIw(t), A:,) n SR(Pio, Aio): r/2 I t I t l > {seg(Ax&/2), AiJ n B d P i o , A i o ) : Aio E L I ,21 (seg(A,,..(t), A:,) n 8,(Pio, Aio): r/2 It It 2 } is a polygonal path connecting the respective crossing points in &(Pi03
AiJ
-
u (lid.
Let q i, , , BJPio
I Aio: 8 )
be the intersection of the R-tube with that component of 4SdPi0, A i o )
-
u
($01)
o(e)
distinguished by the preceding data. It is clear that
I
Ci,,,B)(Pio A i o :
EISENSTEIN SYSTEMS
28 7
is a nonempty open convex subset of !&(Pio, Aio)such that
such that There is no need to specify an “epsilon function” since it plays no role in the double prime picture. One must now verify that the axioms are met by Res(I:’,) and associated data. Geom : I and Geom : I1 are immediate, as is Am-Geom : B (which requires no argument at all in the present case). Geom: I11 and Am-Geom: A are interconnected. Let us recall that they assert, respectively, that
Put 6, = 5(Pi,l,Ai0). n hi,,-then 6 , = n ( A = 0), A in some subset of Xp,o(g,Ai0). [Note: Needless to say, 6, is not necessarily distinguished.] Our
288
CHAPTER 5
constructions imply that there exists a point Aio E 5j(Pio,Aio)with the property that - Re(Ai,) belongs to the interior of the convex hull of and
~P,,(6io)
&P,,(a$).
By definition, Q!JP,,(ak)
is contained in 6.; If we project onto 6$, then -Re(Ai,) goes ,to - Re(H(P,,, A,,)), so we can say that - Re(H(Pio,A,,)) belongs to the interior of the convex hull of pro,(~P,o(6io))
and
.aP,o,
Prog(Wp,,(lli0))standing for the projection of Wp,,(6io) onto 62. Suppose we could show (i) Prog(Wp,,(6io))is contained in &p,,,(6k), (ii) &p,o(6~) is contained in &pai,J6&(,)). The first point would then imply that - Re(H(Pi,, A,,)) belongs to the interior of the convex hull of &P,,<6$>
and
.aP,o
i.e., to &,,,(6$) itself, thus giving Am-Geom: A. The second point, in conjunction with what has just been established, then leads to Geom: 111.This said, let us proceed to the verification of (i) and (ii). For (i), observe that Pro8(Wp,o(6,,)) (cf. Lemma 6 is contained in WP,,(6$), which, in turn, is contained in &,,(a$) and its Corollary in the Appendix to Chapter 2). For (ii), let A1, A,, . . . be the simple roots in XP,,(g, aio); let A', A2, . . . be the corresponding dual roots (so that (Ai, A j ) = Gi)-then each element of & p , o ( 6 ~can ) be written as a nonnegative linear combination of the Ai., the coefficient of Ai actually being zero iff A' E 6,. Since the span of the A', A' E ,6, is Dis(fi)(P,,, Aio),and since is the orthogonal complement of Dis(fi)(Pio, Aio) in 6io, it therefore follows that &p,,(a;)
c &P~,,.,,,(6&~)),
as desired. Of the singularity conditions, only Am-Sing: I is applicable. Let AV E Yv, g)(PioI Aio: 8 )
(fiE Hi",E))*
The claim is that IB.AJS(Pio, AiJ: Av)
EISENSTEIN SYSTEMS
289
We are now in a position to make the final push, viz., the verification of Am-An in the residual setting. In order to reorient ourselves, let us recall that we know so far that the difference between
( R ( l ;0') Res(Qp)@,:,(P I Ail, @YJ,(PI Ail) and
290
CHAPTER 5
the second expression being summed over all i’ E Res(I~,(‘ek))having the property that Re((Xi*(Pio,AiJ, Xi,(PiO, AiJ>> > - e 2 and over all wJoioE W$,(X,.; A,,, Aio), where Air E
K,(PioI Ai,:
E),
the third expression being summed over all i” E I:,(%?&having the property that Re((Xi“(Pior AiJ Xi,,(Pio,Ai,J>> > - g 2 and over all wjoioE W$,(X,..;A j o ,Aio), where A,,, E &,,(Pi,I Aio: E),
can be holomorphically continued to the region Re(() > 1,. Analogous considerations apply when ( is replaced by f . What we must do, therefore, is manipulate the double prime expression in such a way as to bring it into residual form. Keep in mind that
i
Alp,E &(Pi,1 Aio: E ) Ax,. E V’,,,(Pio I Ai,: E )
are at our disposal. Fix
A,,, E L+(Pio I Ai,: 2 ~ ) *n V,(Pio I Aio).
Let J1’ be a small open neighborhood around A,,, with compact closure P. Select a r > 0 such that if 0 c t c T, then the line segment joining A$,,,(t)to any point in JV misses every % E %(i”, E). There is no loss of generality in supposing that t €10, r[
-
A$,.(t) E Vxl.(Pi,IAi,:E ) n RB(i”, E).
Fixing t €10,T[, set A$,.(r) = Ax,. and let y,., be a polygonal path from Ax,- to A,.. lying in the convex hull of Ax,..and J1’ obtained from the line segment joining Ax,,, to Aip,,altered so as to cross each Bq(Pio,Aio)(9 E Hi’’,E ) ) orthogonally. In passing, note that distinct 5 may very well yield the same $3%. Ultimately, it will be a question of summing over the (i”, $3), xyk being !+admissible, where Re((H(Pi0, AiJ
H(Pio,
Ai,J>) > -c2*
A short calculation implies that for such Sj, one necessarily has
Re((X,..(P,,, Aio), XI..(Pi,, A d ) ) > - c 2 - R 2 . We can, of course, throw in the extra integrals arising from these Xi.. without changing the situation in any essential way.
29 1
ElSENSTElN SYSTEMS
Fix, then, an index i” and assume that the normal translation of the corresponding XI.. has the property that Re((X,,.(P,,, Aio),X,..(Pio, A i o ) ) ) >
-E’
-
R2.
For each 9, coming from an Jj E &i”, E), choose points A1,&%and A’, bs lying on that part of y,.. which intersects !jj,(Pio, Aio)orthogonally. It will be supposed that A,,8pl lies in the same half-space as AX,”and that A2,swlies in the same half-space as Ai-. Call A0,& the point at which y,..crosses @,(Pi,, Aio). On the basis of by now familiar principles, one can then write, modulo a function holomorphic in the region Re([) > I & ,
[s [s
I B b ( ~ i ‘ V r oAt,): 9 Ai*‘)
as 6%
1
-s
I ~ ( X I ” ( P , OA,w l : A2.sy1)
A(&,: wJoio:A!,: [) ldA/,1
IBf(Xi”(J‘t0. 40): AII,,)
-1
I
A(X,..:wJoio:
[) IdA!,l.
IBf(Xi”(pi0. Aio): A I , 6%)
Here the summation is taken over the 43, determined by the J3 E $(i”, E). We explicitly observe that while yi.. may actually meet an !&(Pi,, Aio),where 8 is in the complement ofb(i”, E ) in&(i”), this presents, in reality, no problem at all. Generically, let fil, . . . , &, be the elements of $(if’, E ) such that ($,), = 9,. Naturally, xy, is X,..-adrnissible. However, this may or may not be the case for the 8,.[Note that xyk is $,-admissible iff xrgk is $,,-admissible, 1 I p, v I n.] If xy, is not $,-admissible, then J3, is not a singular hyperplane of V(X,..:. . .) (cf. Assumption 4(V)), so the usual considerations imply that the associated “!&-term” supra is holomorphic in the region Re([) > I , . As for what remains, we have the following lemma.
Lemma
and 2.Ic.
Suppose that xwk is 8,-admissible-then the diflerence between
icJ
r = 1 wfoio
1 ~ f ( 8 , ( ~~ 1i,,) :, ~
o6% , +J?
A(!&,: wJoio: AJ0:[) IdA!,l AW(~,))
can be holomorphically continued to the region Re([) > I , .
[Note: In the first expression, the sum over wJoiois relative to the elements of W&,(X,..; Aj,, Aio), while in the second, the sum over wjoi0is relative to the elements of w&,($,; A,,, A,,).]
292
CHAPTER 5
Let us defer the proof of this result for the moment. The definitions show that
+
~0.5s
f
l
~
~
Ep ~ i) ,b,,)(Pio ,, IAi,:
E)*
Since working with one point in yi”,Br)(Pio I Aio:E ) is the same as working with I Aio:E), it is now an easy matter to fulfill our expectaany point in Yi,,,Br)(Pio tions. No restriction has been placed thus far on Re((Hp(Pi0, AiJ
Hp(Pi0,
Aio)))*
Of course, we are only interested in the $j for , which Re((Hp(Pi0, AiJ, Hp(Pi0, Ai,))) >
- E ~ *
Throw away those not meeting this requirement. Add in the contributions from the $ in the complement of Hi”,E ) in @(i”), where xrg, is $-admissible, with Re((H(Pi0, AiJ H(Pi0, As))) > -c2, justified because Re((?, ?)) of the integrand is < I , . Consolidating our notations, we can then conclude that
less
A($: wjoio: the first expression being summed over all i“ E
C) IdA/,l, having the property that
Re((X,,.(Pio, Aio),Xi,,(Pio,Aio))) > - 2 - R 2 and over all wjoio E Wik(Xi”;A,,, Aio),the second expression being summed over all (i”, $5) E Res(Iyo(Wk))having the property that Re((H(Pi0,
Aioh
H(Pi,,
Aio)))
>
-t2
and over all wJoioE W&,($;A,, Aio),where 41*,,6 E ) 4I”.b)(PioIAi,: EL
can be holomorphically continued to the region Re(C) > A,. Analogous considerations apply when C is replaced by g.
293
EISENSTEIN SYSTEMS
Proof of Lemma Let AL be a real unit normal to $%(Pi,,Aio)pointing in the direction of A2,591.Determine real constants c1,8rand c2,891by writing
I
A,.gs = A O , & A2AI = AO,Br
+ C1,BaA* + C2*&AL
Fubini's theorem then implies that Wfoio
[I
IBL(Xi'.(Pio.Aio):A2.691)
is equal to
-I
IB~(Xi"(PiosAi0): Ai,fm)
x A(Xi.: wJoi0:A!o
(CI,& (c2.591
I
< 0)
'0).
A(Xi2,: wJoio:
5) I dAJ0I
+ zA1: 5) d z I dA!oI,
the inner contour being as shown in Fig. 10. Here, of course, the summation over wJoio is relative to the elements of W&,(X,.,;A j o ,AiJ. Needless to say, BdPiO, AiJ = HR(Pio, Aio) + BdPio, Aio)", where, for any p, SjdPio, AiJ" = $p(Pio, AiJ- n 610, so that the symbol rst!($dPio,
b,aS)
may be assigned the obvious interpretation. Let Am($,)
=
c, .
(C, =
FIG.10
ll~%($u)ll).
294
CHAPTER 5
Then
x A(&: wJoio: AJ0
+ zA1: 5) dz ldA!ol
becomes
x A(&: wJoio: A!o
+ zA1: C) dz ldA!ol,
the inner contour being shown representatively in Fig. 11. A small positively oriented circle Ci,, is drawn around a given C, if
J-1
IC,I < JE2
- - ~ _ _ _
-
IIA!o -
Y
FIG.I I
4,$a1129
EISENSTEIN SYSTEMS
295
but no circle is drawn if The contour is undefined if there exists equality for some p (which is possible, though, only on a set of A/o-measurezero). The contributions arising from the horizontal lines yield functions holomorphic in the region Re(5) > A,, so that they may be set aside. We are thus left with the contributions coming from the circles. Let IB,t(BdPio AiJ: &. 8% : p ) 7
denote the subset of
IBb(BR(Pi0, AiJ: Ao,8%) for which EZ
- llNo - Ai,&J’ > IC,I2.
What remains can then be put in the form
296
CHAPTER 5
This is permissible, provided one knows that
+
+
flA&j,,) Re((Ab,,, whenever
Ab,,,
+ -A&(&)
+
( A i o E $ j p ( P i o , AiJ“ n aiJ - ICpI’ But the expression in question is equal to
11A/01122
E’
llRe(Ab,b,)112 - (Im(Ab,,,) or still
+ C, . + A!o, Im(Ab,b,) + C ,
+
IIRe(Ab,bsl)IIZ - (Im(Xr(Pio, Aio)) C, * A 1
A!
+ A!o>
+ AI0, Im(X,,,(Pio,Ato))+ C , .A! + A!o;
or still IIRe(Ab,8,)I12 - IIWXdPio, AiJ)II’ - ICpI’ - IIALII’, which is 1,. Set Res(Iio)= Res(I:,) u Res(I:l,)-then, between
in summary, the difference
(R(C; 0’)R~NQP)@,~~(P I Ail, @,j0(P IAil)
and
the first expression being summed over all X E Res(Xi0(Wk)) having the property that Re((X(Pi0, A,,), X(Pi0, Air,)>) >
-8’
291
EISENSTEIN SYSTEMS
and over all wJoioE W&,(X;Am, Aio),where Ax E h(f'i0
IAio : E),
the second expression being summed over all i E Res(Ii0(gk))having the property that Re((Xi(f'i0,
Aio),
XdPiO, AiJ>) >
-E'
and over all w;oioE W&,(X,; A,, Aio),where Ai E Y ( f ' i 0 I Aio: E), can be holomorphically continued to the region Re([) > I,. Analogous considerations apply when [ is replaced by 1. The construction of the residual amalgamation is therefore complete.
APPENDIX We shall establish here the following result. Stone-Weierstrass Theorem Let X be a compact Hausdor-space.Let % be a finite-dimensionalcommutative *-algebra containing C which is generated by C and the selfadjoint nilpotent elements. Let 5 be a subset of C ( X , a).Assume 8 is u wubulgebra of C ( X , %) containing the constants with the property that .for all points p , q E X , p # q, = { f ( P ) : f (4) = 0,f E 5).
Then: 3 is uniformly dense in C ( X , a).
Proof The proof proceeds by induction on dim(%). If dim(%) = 1, then our assertion follows from the usual statement of the Stone-Weierstrass theorem. Suppose, therefore, that dim(%) > 1. Fix a nonzero, noninvertible, self-adjoint element a. E %. Such exist by our hypotheses. Consider %aothen %ao is, via the identification'iNao zz %/Ann(ao),a commutative *-algebra satisfying the same general conditions as % itself except, of course, that now dim(21ao)
-= dim(%).
Similar remarks apply to the quotient %/%ao.Abstractly, there is a direct sum decomposition of vector spaces over C: %
2
%/%ao 0 %ao.
Let P be any projection of % onto %Ia,-then Ker(P)
2
%/%ao.
298
CHAPTER 5
Fix a norm on 9l so that llall = Il(1- P)4I
+ IlPaII
(UEW.
Place on 21/21ao the norm determined by restriction to Ker(P). There are canonical maps
i
C(X, C(X,
+
+
C ( X , waao>
w,%ao)
carrying 8 to a *-subalgebra satisfying the same general conditions as 3 itself. This being so, fix E > 0. Let f E C ( X , a) be given. By the induction hypothesis, there exists g E 5 such that Il(1- P)f(X) - (1 - P)g(x)II < E / 2
(XEX).
By the induction hypothesis again, there exists h E 3 such that (x E XI.
IIfYf(x) - g(x)) - a0 hWll < &/2
Hence
IIfW - g(x)
- aoh(x)ll
Since 5 contains the constants, g
(x E X ) .
+ aoh E 5,thus finishing the proof.
//
Remark It is clear that the assumptions on % or 8 are subject to some relaxation, but one cannot push things too far. For instance, simple examples show that the theorem is false if every self-adjoint element in 2l is invertible or if 5 does not contain the constants. The first point may be illustrated by taking for X the closed unit disk in C, for ‘9l the quotient C[t]/(l + t Z )(with (a bt)* = si bt), and for 5 the elements
+
where
+
{”
(4+ J--ruz> =
u1
fz = uz
+f l u z +
+ (4 + f l v z ) t , are holomorphic on X.
The second point may be illustrated by taking for X the closed unit interval in R, for & the quotient C[t]/(t2)(with ( a + bt)* = si + 6t), and for 8 the elements
f + f’t
(fE Crn(COI11)).
Eisenstein Systems
In this chapter, we shall give an account of the Langlands theory of Eisenstein systems.This is difficult mathematics, compounded additionally by the apparent lack ofa central theme. For the purposes of the present work, the theory has two objectives (closely related to one another).
( I ) Define the spaces L$,(G/T; 6, O o ;W) and obtain the orthogonal decomposition
L:,(G/r; 6 , o o ) =
1 0 L$,(G/r; 6,@,; W).
Y>,WO
(2) Develop the theory of Eisenstein series and c-functions for residual forms. Eisenstein systems provide an inductive vehicle for arriving at these goals. The reader will simply have to take this on faith: The ends justify the means. Langlands’s discussion of these matters is virtually impossible to read. [He himself has admitted that it is almost impenetrable.] Furthermore, points (1) and (2) are not developed at all systematically, even though they are absolutely essential for the trace formula. It is for this reason that we have decided to give a complete and detailed account of the theory. Our version differs in many respects from that of Langlands; but there is really nothing to be gained by making a detailed comparison here. Modulo what we consider to be elementary or obvious, our proofs, we hope, will be found accessible. 122
EISENSTEIN SYSTEMS
123
Nevertheless, the reader is warned in advance that there are many definitions to be absorbed, notation to be kept in mind, and intricate constructions to be contemplated. With perseverance, however, these difficulties should move to the background. The main result is Theorem 5.12. Its statement can be understood after digesting the definition of Eisenstein system and some of the accompanying axioms:This turns out to be the source of (1). Proposition 5.4 is of crucial importance for (2). Before defining the notion of Eisenstein system, it will be necessary to set up some notation and state a few simple facts. Let (P, S) be a r-cuspidal split parabolic subgroup of G with split component A . Let X be an affine subspace of the complexification of &then we shall say that X is admissible if X can be represented as an intersection of hyperplanes of the form 1= c(1 E C,(g, 6)). Fix an admissible affine subspace X. If
x = n (1= c),
then X admits a unique decomposition X=X"@X,
where
3-
=n(x=o)
and X is a vector which is orthogonal to X-,the normal translation in 3. We shallwriteX1fortheorthogonalcomplementofX-. ByS,,wethenunderstand the symmetric algebra over XI.There is a unique conjugate linear isomorphism *: s, {u
-+
s,
H u*
of S, with itself such that A* = -12 if A belongs to XI. Let V be a finite-dimensional complex Hilbert space. Form
{S;oidm&,
V).
There is a natural pairing S, 0 V x Hom(Sx, V ) + C
which is linear in the first variable and conjugate linear in the second variable, characterized by the condition (u 0 u, T ) = (0,T(u*)).
124
CHAPTER 5
Form
There is a natural pairing
S, Q
v x Hom(S,,
V ) -,C
which is linear in both variables, characterized by the condition (U Q
5, T) = (a, T(u)).
Lemma Suppose that A is a linear function on S , Q V-then there is a TAin Hom(S,, V ) such that
NQ)=
(QI
‘GI
(Q E S, 0 V ) .
Lemma @is) Suppose that A is a linearfunction on S, 0 V-rhen there is a TA in Hom(S,, V ) such that
N Q ) = ( Q , ‘M
(Q E S, 0
v).
One can identify Hom(S,, V ) with the space of formal power series over X I with coefficients in V. In this interpretation, we can speak of the order of an element T: ord(T) is the order of the term of lowest degree which actually occurs in the power series expansion of T. This being so, a linear transformation from Hom(S,, V ) to another vector space is said to be of degree n iff it annihilates all the terms of order >n but does not annihilate every term of order =n.
Lemma Suppose that A is a linearfunction on Hom(S,, V)-then A is of jinite degree i8there is a QA in S , Q V such that ( T EHom(S,, V ) ) .
NT) =
Lemma (bis) Suppose that A is a linearfunction on Hom(S,, V)-then A is ofjinite degree Iffthere is a Q A in S , @ such that A(T) = ( Q A , T )
( T E Hom(S,, V ) ) .
Remark Suppose given
( P ,S‘; A ‘ ) (p“,S”;A ” )
with associated data
125
EISENSTEIN SYSTEMS
Let
V: Hom(S,,, V') -+ S,,, 0 V" be a linear transformation of finite degree-then there exists a unique linear transformation
V*: Hom(S,.., V " ) + S,. 0 V' of finite degree such that (VT', T " ) = (V*T", T ' )
for all
{
T' E Hom(S,, , V ' ) T" E Horn(&, V").
We shall, accordingly, refer to V* as the adjoint of V. Let 'I) be an admissible affine subspace which is contained in X, so that 'I)='I)"@Y with 'I)" contained in X ". Call S,complement of 3' in gl-then S,
the symmetricalgebra over theorthogonal =
Sa-, Q S,.
If U O E S,-E
T E Hom(S,, V ) ,
then there is determined an element uo v T E Hom(S,, V )
characterized by the condition that One has
(uo v T)(u) = T(u0 0 u )
(u E S,).
(Q, uo v T ) = (4 0 Q, T ) (Q, uo v T ) = ( u o Q Q, T )
(Q E S, Q
i
v)
(QESX Q
v).
The discussion thus far has centered on (6, X, V ) . We shall now inject another ingredient into our considerations. Let x be a linear subspace of 6then x is said to be X-admissible if x is contained in X". Fix an X-admissible subspace x. Obviously: X' c x'.
126
CHAPTER 5
Lemma There is a canonical injection u Ha, morphic diflerential operators on x1 such that d,f(Ao) =
df
(A0
S , into the algebra of holo-
of
+ tA)
lt=o
for all A in X I . Fix a point A. in x1 and an open neighborhood N o of A,. Let @:N0+V
be a holomorphic function. By d@(Ao)we mean that element of Hom(S,, V ) specified by the rule d@(Ao)(u)= a,@(Ao)
( U E Sx).
In the formal power series picture, d@(Ao)is obtained by expanding
+ A))
= @(AO
@A0
about the origin. Observe that d@ may be thought of as a map d o : N o + Hom(S,, V ) with the property that for every u E S,, d@(?: u): No
+
V
is holomorphic on N o . Let F: N o + Hom(S,, V ) be a map with the property that for every u E S,, F(?: u ) : No + V is holomorphic on N o .As before, let '1) be an admissit.,: affine subspace v, ..ich is contained in 3. Assume x is 9-admissible. We shall then denote by do F(Ao) that element of Hom(SB,,V ) given by the prescription doF(uo @ u) = duoF(Ao:u).
Remark It is easy to check that dO(dWA0) = d@(Ao)* The notation, however, is deceptive. On the left, d@ is viewed as an element of Horn(&, V ) while, on the right, d@ is viewed as an element of Hom(SB,,V ) . In the sequel, we shall let the context dictate the appropriate interpretation.
127
EISENSTEIN SYSTEMS
Let Vl and V, be finite-dimensional complex Hilbert spaces-then Hom(Vl, V,) is again a finite-dimensional complex Hilbert space. Let
f : N o -,Hom( V,, V,) be a holomorphic function. One then has that df(A0)E Hom(S,, Hom(V1, V,)),
I.e., df(A0) E
0 Vl,
V2),
I.e., df(A0) E Horn( Vl,
3
V2N.
The composition of any element in Hom(S,, V,)with df(Ao)provides us with an element of Horn(&
3
9
V2)h
or still, with an element of Hom(S, 0 S,, Vd. The Hopf algebra map
s,
-+
s, 0 s,
then induces a morphism of restriction Hom(S, @ S,, V,)
-, Hom(S,,
V,).
There is, therefore, a natural map Hom(S,, Vl) -, Hom(S,, V2), thereby determining an element dHom
f(A01 E Hom(Hom(S,
7
Vl)? Horn(&
7
V2)).
Let us now go back to the beginning of this discussion. Starting from the fact that df(A0) E Hom(S,, Hom(V1, V2N7
I.e., df(A0)
E
0
Vl,
V,),
we obtain an element in Horn(& 0 S, 0 Vl, S, .@ V2)
128
CHAPTER 5
by tensoring df (Ao) on the left with the identity map on S,. Tensoring the Hopf algebra map
s, s, 6 s, +
on the right with the identity map on V, yields an element in Combine these two maps under the composition on thereby determining an element In general, any has an adjoint D* E Hom(Hom(S,, V2),Hom(S,, V,)) characterized by the condition
( N u 0 VJ, Lemma Let
T2) = (u 6 V1, D*T2).
-,Hom(l/,, V2) g: N o-+ Hom(V2, V,)
be holomorphic functions. Suppose that Then
[Needless to say, it is necessary to impose an obvious condition on No.] There is one final point in this circle of ideas which should be mentioned.
Lemma
There is a canonical injection u Hpu of S , into the algebra of polynomial functions on i1 such that PA
=A
for all A in . ' 3
We shall now take up the definition of Eisenstein system. The definition, in toto, is rather lengthy, so we shall break it up into a series of assumptions, developing the necessary preliminaries as we go along.
129
EISENSTEIN SYSTEMS
Let % be an association class of r-cuspidal split parabolic subgroups of G, gia G-conjugacy class in W. Let (Pl, s,;A , ) ( P 2 , s2;A , )
{
be members of qi.We then define an element W2IA2:
P1IAl)E
WA2,
,411
as follows. Select x in G with the property that x(P,,
s,:A,)x-'
=
( P , , s,; A,).
Put W 2 l A 2 :
PlIA,) = Int(x)IA,,
a definition independent of the choice of x. There are certain elementary properties inherent in this construction, e.g., transitivity. Less elementary but still easy are the conditions of descent.
Lemma Let %], %, be association classes of r-cuspidal split parabolic subgroups oj' G, W i t ,W i 2 G-conjugacy classes in %, , %,. Let
i
fP2,s;; A ; ) (pi, s;; A ; )
s;;A ; ) E W i P
(Ply
( P ; , s;; A ; )
with
i
E Wi2
s;;A ; ) 3 v 2 7 s;; A ; )
(Pl,
(Pi, s;; A ; ) 3 (Pi,s;; A';).
Then
I ( P y A ' ; : P;IA;)IA;= I(P;lA;:P,IA;). Lemma Let g1,%, be association classes of r-cuspidal split parabolic subgroups of G, gil,Vi2G-conjugacy classes in V1, %',. Let
with
Then
I(Pi I A'; : P,I A ; ) I 'A2
=
I("P, I " A , : ' P , 1 'A,).
130
CHAPTER 5
Both of these facts will be used without comment in what follows. Let V be an association class of r-cuspidal split parabolic subgroups of G, V i and V j G-conjugacy classes in V. By the symbol
W(wj,wi) we understand the set of equivalence classes in
where
i
W ( A J ,A;) wyi E W(AY, A;)
Wji E
are declared equivalent iff
w'!. = I(P'!I A'!: P'. I A'.) 0 w'.. 0 I(p;I A ; : PI' I A;). J1 J J J J Jl
Observe that if
then wji n W ( A j ,A,) is a singleton; call it wji(PjIAj; P,IA,) (or wji). Let V, V, be association classes of r-cuspidal split parabolic subgroups of G, V k ,Vio,and VjoG-conjugacy classes in V, Vo. Suppose that
This being so, let
131
EISENSTEIN SYSTEMS
be the subset of W ( A j o ,Aio)comprised of those wJoiosuch that I ( P j I A j :PiIAi).
WJoioIAi =
Let then wtk(@j0 9 @iO)
be the subset of W(Kj0,qi,,)formed of those wJoiowhich factor through the triangle
- _- . _._
W ( q j 0 ,y i o )
- - _-. *-*+
W(Aj0, AiJ
t
WJk(Ajo9
for ail choices on the right.
Remark The data (Pi7 S i ;
Ail
( P j , S j ; Aj)
P (Pio,S i o ; AiJ
+ (Pjo,Sjo; AjJ
determines, in the usual way, triples
Because M i and M j may be distinct groups, one cannot, in general, ask whether (P!07St’,)and (PJo,SJo)are associate. It is for this reason that one is forced to introduce the set W t , ( A j o ,Aio). If, however,
i
(Pi0 Sio ; AiJ (Pjo, Sjo; AjJ 9
admit a common dominant in V k ,say (P, S ; A ) , then W&,(Aj0,Aio)can be identified with W(AJo7A!o), a set, though, which may be empty (since the relation of association is not necessarily preserved under the daggering procedure). Let “be an association class of r-cuspidal split parabolic subgroups of G , V i a G-conjugacy class in W. By an equivariant system X of admissible affine subspaces attached to W i we mean a map X which assigns to each
(P,s; A ) E qi
132
CHAPTER 5
a nonempty admissible affine subspace X(P,A ) of the complexification of 6 subject to the following compatibility condition: For all
{
(P1,sl; A1)egiT (P29S2; A 2 1 ~ 9 7 i 7
w 2 ,
A2) = W
2 I A 2 :
PlIAl)X(Pl, A l l .
An X-admissible subspace x of X is an equivariant system of linear subspaces attached to V i such that for each
( P , s;A ) € 9 7 i ,
x(P, A ) is X(P,A)-admissible. The terms
{
“dimension of X” “dimension of 1’’
are to be used in the obvious way. I f q j is another G-conjugacy class in V, then one can associate with each element wji E W(Vj, gi)an equivariant system X .I, of admissible affine subspaces attached to V j by requiring that X,,,(Pj, A j ) = - w j i ( P j [ A j :PiIAi)X(P,, A‘).
Fix an equivariant system 3 of admissible affine subspaces together with an X-admissible subspace I. Let
( P , s;A ) E % i ,
The conventions and notation set down above are then applicable here. In particular, we can write X(P, A )
=
X(P,A)“ 0 X ( P , A )
and consider sX(P, A ) .
Let 6 be a K-type and 0 an M-type-then 6,,,(6, 0) is a finite-dimensional Hilbert space. Specializing the earlier discussion, we can say that there is a natural pairing that is linear in the first variable and conjugate linear in the second variable, characterized by the condition (u
0 0,T ) =
( 0 7
T(u*)),
and a natural pairing
s,,,4) 0
0)’ x Hom(S,(P, A ) , 6,,,(4 0))
~clIs(63
+
c
EISENSTEIN SYSTEMS
133
which is linear in both variables, characterized by the condition (u
0 6,T ) = (6,T ( u ) ) .
Since a polynomial function on a subspace of a may always be regarded as a function on G, the presence ofthe X(P, A)-admissible subspace x(P, A ) implies that there is a map
-
S,(P,A,
0 ~,",(6,0) Fnc(G) +
sending u 0 @ to p , @. We hardly need point out that this map depends on
W, A).
We shall now change our notation slightly and freeze the data. Fix an association class W0 of r-cuspidal split parabolic subgroups of G. Let W i , be a G-conjugacy class in W 0 , X an equivariant system of admissible affine subspaces attached to %.,; Suppose that %? WiO-then the set of all (P, s; Ai) in %? for which there exists a (PiorSio;Aio)in Wi0 such that
+
(Py S ; Ail
+ (Pi,, Sio; AiJ
is itself a union of G-conjugacy classes in %?; call it %(io). Let W kbe a G-conOne may then define an equijugacy class in % which is contained in %?(io). variant system x y k of linear subspaces attached to Wi0 as follows. Let (Pio, Sio;Aio) be a member of Vi,-then there exists a unique element ( P , S ; A i ) in %?k such that (P, S ; Ail (Pior S i o ; Aio).
+
We write X(Pio,AiC,)+for the orthogonal projection of X(Pio,Aio) onto the complexification of a/,. This said, let xvk(Pio,AiJ = 6i.
It is easy to check that x y k does in fact have the required properties. Write
%Go; X) for those G-conjugacy classes V kin %(io) for which xy, is X-admissible. Of course, V(io; X) could be empty. Let Wj0 be another G-conjugacy class in V 0 . Suppose that there exists gkin %?(io;3)such that %?k 3 WjO-then we can form Wtk(Wj,, Via), which, as we recall, is a subset of W(qj,, gi,).Each wJfoioin WLk(Wj0,Via) thus takes X to another equivariant system XW~,,,.If we agree that two elements of W?,(Wj,, Wi0)are to be regarded as equivalent when their action on X coincides, then we obtain a set of equivalence classes
WLk(X; gjo gio). 9
Let
134
CHAPTER 5
Denote by
wtk(X;A j o , Aio) the set of distinct linear transformations from 3E(Pio,Aio) into the complexification ofEijoobtainedbyrestrictingtheelementsof W$,(Aj0,Aio)toX(Pio,Aio)then there is a canonical bijection
wtk(X;qjo,gio)-,Wtk(3E;A j o , AiJ We write ~ ( p j ,A, ~ ~ ) , J=~ ~-~WJoio Wpio
9
~
i
~
<
)WJoio E
Wtk(X ; A j o
9
Remark Suppose that ( P , S; Ai)
>r (Pio,S i o ; AiJ
The daggering procedure then determines a split parabolic subgroup (P,’,, Si’,) of Mi with split component A!o which, of course, is r,,-cuspidal. There are orthogonal decompositions
{hie
aio =
@ a,
=
tiit, 0 E i i .
X(Pio,Aio)tis an admissible affine subspace of the complexification of Ei!<) with normal translation X(Pio, Aio).Conventionally, an admissible hyperplane in X(Pi,, Aio)t is an intersection of the form ( I t E Cp?o(mi,a!,).
x(P,,, A ~ , ) +n ( I t = c )
Consider, then, those association classes %? such that (i) (ii) %‘(io;
+ X) # 0. q i o y
Fix a K-type 6 and an orbit type O0.The assignment to all
i
(P. S ; A i ) E W(io; X) (Pio,S i o ; Ai,J E q i o
standing in the relation
( P , S; A i l of a complex-valued function
+
(Pi09
s i o ; AiJ
E(X: P I A , : P i o ) A i oTo: :
x)
on Hom(Sx~pi,,Ai,),
Qio))
x X(Pi0, AiJ+ x G
135
EISENSTEIN SYSTEMS
'E(3: P ( A i :PioIAio:c,T,b + c,T,Zo:A!o: x) = c,E(X: P I A , : PioIAi,: Tib: A!o: X) c , E ( X : PIA,: PioIAio:T i : x)
+
E(3: P I A i : P i o ) A i oT : o :AJo:X) =0 if ord(T,,) 9 0 (uniformly in (A!o, x))
E(X: PIA,: P i o ( A i oTo: : A!o: xya,n) = E(X: P I A i : P i o I A i o :To:A!o:x)
HO~(SX,P A ,, ~~) ,a c u s ( h , 3
(ya,nE(rn P ) . A , . N )
OiJ)
to SX(P,(,.4,,)>~;o,o 0 a c u s ( h , OjJ such that
(V(3: P I ( A j , Ai):PjolAjo:PioIAio:wJoi0:A!o)To, Ti,) if
i
=
0
ord(To) 9 0 ord(qo) 9 0
is said to be an Eisenstein system belonging to infra are met in a nontrivial way.
X provided the assumptions
136
CHAPTER 5
Assumption 1 (E) (Meromorphicity) Suppose that (PI S; Ai) 3 (Pi,, Sio; AiJ We shall then assume that
E ( 3 : P J A , :P i o ) A i , :To:A!,: x) is a meromorphic function of A!, in X(P,,, admissible hyperplanes. Let
whose singularities lie along
D(X:E : P I A , : PioIAi,) be the set of all A,toin X(Pi,, A,,)+ which admit a neighborhood N:, with the property that
E ( X : P ( A , :PioJAio: To:?: x)
Ti,, and all x. We shall then assume that E(X: P J A , :P i o ( A i o To: : A/,: x)
is holomorphic on M!, for all
is a differentiable function of (A!,, x) on D(X: E : PJA,:PiolAio)x G.
Let G M ,be a Siege1 domain associated with a r,,-percuspidal subgroup P M ,of M i . We shall then assume that
parabolic
IE(X: PIA,: PioIAio:To:A!,: hi)[
I C . II Toll* zpM,(mi)l
((kmi) E K
x G.M,),
C and r being uniform on cornpacta in D(X: E : P I A i : Pi, I Ai,).
Assumption 1 (V) (Meromorphicity) Suppose that
{
( P , S; Ai) 3 (Pi,, S i o ; AiJ ( P , S; A j ) 3 (Pj,, Sjo; AjJ We shall then assume that
(V(X: P J ( A j ,Ai): P j o J A j oP: i o J A i , wIoio: : A!,)T0, Tj,) is a meromorphic function of A!, in X(P,,, A,,)t whose singularities lie along admissible hyperplanes. Let D(X: V: P J A , :PiolAio) be the set of all A!, in X(Pi,, which admit a neighborhood N!, with the property that (V(3E: P I ( A j , A,): PjoIAj,: PioIAi,: wJoio:?)To, Tj,) is holomorphic on N!, for all wJoioand all To,Ti,.
137
EISENSTEIN SYSTEMS
Assumption 2 (E) (Equivariance) Suppose that ( P , S; Ail
3
(Pi09
Sio; AiJ
Let ni, E Nio-then we shall assume that
E(X: PI ni, Aini, ' : Pi, I n,, A i , ni, ' : n,, To:n,, . A!,: x) = E ( X : P I A , : PioIAi,:T o :A!,:
x).
Assumption 2 (V) (Equivariance) Suppose that
i s;
(P, Ail 3 (Pi,, si,; 4,) ( P , S; A j ) 3 (Pj,, S j o ; Ajo)*
Let ni,
E Ni,,
njoE Nj,-then
', '
we shall assume that
'
V(X : PI (njo~ ~ n ni, j ,~ ~ n'):i p, j , I njoA ~n,;, : p i , I ni, Ai,n,; 1.. njo~J,i,ni, : ni, * A!,)ni, * To = n j o .(V(X: P I ( A j , A i ) :P j o l A j o :PioIAi,: wjoi0: A!,)T0).
138
CHAPTER 5
their common value being denoted by the symbol yPy-'. This said, we shall then assume that V ( X : ypy- I ( y j A j y j l , y i Aiy;
1):
yj
P , y;
l
IyjAjoyj 1 : y,P,,yr:
I YiA,,y;
:
YjwfOioy; : y i . Ait,)(dHornRytr(Aito)(To)) 0 7; * (~,R~,(w~,~,A,!,)(V P I( (XA: j , Ai): Pjol-4jo:Pi,( A,,: wJ,~,:AI,)IT;.,)).
= yj
Assumption 4 (E) (Transitivity) Suppose that ( P , S ; A,) with
+ (f",s';A;,) 3 (Pio,Sio; A,,)
X(P,,,A,,)"
3
a;,
3
hi.
hi,
=
h,td @ a;.
hi,
hit, @ 6, = 'hi, @ h i ,
There are orthogonal decompositions aio = aro t'
a;,
a,, = a!, @ a, a;, = 'a,. @ a,
so that
{
a;.
=
X(Pio,AiJ' = 'X(Pio, Ai,) O X(Pio,Aio)t'v
where, of course, %(Pi,, A,,) is simply the complexification of 'hi,. Let 9 ( X : E : PIA,: P'IA;,:P i , ( A i o )
be the set of all points A!o in 3 ( P i , , A,,)+ which admit acompact neighborhood N!,in X(P,,,A,,)+ such that (a) The projection A"!: onto X(P,,,A,,)" is contained in D ( X : E : P' I A;.: Pi, IA,,); (b) The series
1
Y E rnP/rnP'
~f,(~y)"*"-""'.E(3: P ' I A : . : PioIAi,:To:A/;: x?)
is uniformly convergent for A/, = 'A,, of G (any IT;.,). One can prove that
+ A&'in A"!, and x in a compact subset
9 ( X : E : PIA,: P'IAf.:PiolAio)
is a nonempty open connected subset of X(Pi,, A,,)' whose projection onto X(Pi,, is exactly D(X:E : P'IA;.:Pi,IAio). This said, we shall then assume that E ( X : P ( A , :Pi,(Aio:IT;.,: A/,: x) =
C
U~,(xy)"Ai'-'~~''.E(X: P'IA;,:PioIAi,:To:A!:
y~rnPlrnP'
for all A!,E~(X: E : PIA,: P'IA;,:PiolAio).
XY)
139
EISENSTEIN SYSTEMS
Assumption 4 (V)
I
(Transitivity) Suppose that
( P , s; Ai) 3 (Pj,, sj,;A;.) 3 (Pi,, si,; Ai,) ( P , s; A,) 3 (Pi,, sir;A;.) (P,,, s,,; A,,)
with
*
X(P,,,A,,)"
3
6;s 3 6,.
There is a canonical injection W&'L(X;A,,, AiJ
4
wtk(X;A j o , AiJ
This said, we shall then assume that
V(X: P I ( A j , Ai): P j o ~ A j oPioIAi,: : wJ:~,: A!,) = V(X: P'l(Ai,, A;,):PjolAjo:PioIAi,: wJdi0:A/:) for all WJiioE
W&'& A,,, Ai,)
if PI. = Pi. = P', say, but that
V(X: PI(A,, Ai): P j o ~ A j oPio)Ai,: : wJii0:A/,)
=0
for all WJii0
E
W&'& A,,, Ai,)
if Pi, and Pi. are not r-conjugate.
Assumption 5 (E-V)
(Negligibility) Suppose that
i
( p ,S ; Ail
3 (Pi,, Si,;
(f', S; Aj) 3 (Pj,,
Sjo;
AiJ A,,)*
We shall then assume that
E K x p j , ( X :PIA,: PioIAio:To:A!,: km,) WfOiO€
x
c
Wt.,(X;A,,.Aio)
aJo(mj)wfo~O"fo(V(X:
PI(A,, A i ) :PjoIAj,:P i o ( A i , :wloio:A&)&,)(kmj).
On the other hand, suppose that (P, S; A) is a dominant successor of some other r-cuspidal split parabolic subgroup of G which is not in W0,say
( P , s; A ) 3 (P',s';A').
We shall then assume that
E K x , p ( X :P I A : PioIAi,:To:AL: km).., 0.
140
CHAPTER 5
[An empty sum will, by convention, be set equal to zero. In this connection, observe that (P]o,S!o)and (PI,, S:,) could very well lie in distinct association classes when projected into P / A , N = P / A , N ; but this happens iff Wt,,(X; A,,, Aio) is empty.]
There is one additional assumption which must be made. Its role in the following developments is somewhat surreptitious, which is why we have chosen to isolate it. Assumption (Rep) There are two parts. (i)
Let k E K-then we shall assume that E ( 3 : PIA,: PioIAio:To:A!,: k x ) = E ( X : PIA,: Pio(Aio:L, To:A!o: x). 0
(ii) Let u E I,"(G) or C:(G;
a), so that
IndLio.Ai,.Ni,((oio A!Jcus) (a) E Hom(bcus(a, o i o ) , Scus(a, oio))* We shall then assume that
L,,,(a)E(X: PIA,: P i o ) A i , :To:A!o: ?)(x) = E ( X : PIAi: Pi01 Aio: &om Ind~,O.A~O.N~O((OiO, A!o)cus)(a)(To):A/o: x). [It is a question here, therefore, of E-functions; there is no direct counterpart for V-functions.]
Remark To say that an Eisenstein system belonging to X is nontrivial means exactly this: There are triples
I
( P , S ; Ail (Pi09 s i o ;
standing in the relation for which
Aio)
(PI S ; Ai) 3 (Pi09 S i o ; Aio) E ( 3 : P ( A , :Pio(Aio:?: ?: ?) # 0.
As will become apparent, it is necessary to make such an assumption in order to draw certain geometric and group-theoretic conclusions.In this connection, observe that if Wk 3 Wio,sWkbeing &admissible, then for some (P, S ; A,) in %?k and some (Pie, Sio; Aio) in W,,, with
it is true that
(PIS ; Ail 3 (Pi09 S i o ; AiJ E(X: P J A i :PioIAio:?: ?: ?) # 0.
141
EISENSTEIN SYSTEMS
In fact, if this were not the case, then for every (Pie, Sio;Aio)in W i o ,one would have
E(X:GI{l}: PioIAio: ?: ?: ?)
=0
(Cf. 4(E));
thus
V(X: GI({1}, {I}): Pjolz4jo:PioIAio:?: ?) = 0
(Cf.
5(E-V)),
so that
V(X: P ' J ( A ; ,A;,): , PjolAjo:PioIAio:?:?)
=0
(cf. 4(V)),
implying E(X: P'IA;,:PioIAio:?: ?: ?) = 0
(Cf. 5(E-V)),
which contradicts the nontriviality supposition. It should also be noted that the vanishing or nonvanishing of
E(X:PIA,: P i o ( A i o?:: ?: ?) is a function of the r-conjugacy class of P in Wk (cf. 3(E)). The assumptions which define an Eisenstein system are, of course, abstracted from those encountered in the study of Eisenstein series associated with cusp forms (cf. Chapter 4).The ambient group here is K x M i (or M i ) instead of just G itself. One needs this degree of generality for the inductive arguments which are to follow, although, ultimately, the final results will be phrased in terms of G alone. Remark Let {E, V} be an Eisenstein system belonging to X-then our assumptions imply that the E-functions, viewed on K x M i (or Mi), are (or rM,). automorphic forms per { 1} x rM, Example (The Canonical Eisenstein System) Let 3 be the equivariant system of admissible affine subspaces attached to Wio via the prescription X(Pio,Aio)= complexification of ai0.
In this case, then, W(io; X ) = W(io). Moreover, Sx(pio, A,o) = scalars, so that the map
ToH To(1) sets up an isomorphism H~m(Sxcp,~, Ado)
This said, suppose that
3
&cus(a,
oio))
+
&cus(a,
@iol
142
CHAPTER 5
Put
E(X:PIA,: PioIAio:To:A!o: x ) =
1
y E rn P / r nP i ,
uio(xy)(~~o-pfO) * T0(l)(xy).
This function has been encountered earlier. In fact, referring to the discussion prefacing the statement of Lemma 4.1, we see that E(X: PIA,: PioIAio:To:A!o: km,) ,
=
1
Y E r M l / r M , n p?,
- To(l)(kmiy),
uf,(mi)("?o-Pto)
the Eisenstein series
or still
O(kmi) = ( C ~ , , ~ ( PP!oIA!o: ~ ~ ~ wJoio: A ~ ~AJoio) : To(l))(kmi).
That these data do in fact constitute an Eisenstein system is a consequence of what has been said in Chapter 4.
EISENSTEIN SYSTEMS
143
It will become apparent in the end that all the Eisenstein systems of interest to us will arise from the canonical Eisenstein system by means of a residue taking process. Even though this procedure will not be of any immediate applicability, it seems appropriate to discuss it now since it serves to provide one with a deeper understanding of Eisenstein systems in general. Suppose, for the moment, that X is an affine subspace of C". Let f be a meromorphic function on X whose singularities lie along hyperplanes. Let fi be a hyperplane in X defined over R (which, though, need not be a singular hyperplane off). One can then define a meromorphic function resdf 1 on Sj as follows. Pick a real unit normal zl to $5. Set 1
P
6 being a small contour around the origin in the complex plane, so chosen that .f is holomorphic at z (zl. It is clear that resB(f) is in fact a meromorphic function on Sj whose singularities, moreover, lie along those hyperplanes obtained by intersecting 6with the singular hyperplanes off different from fi.
+
Remark The definition of residue depends upon the choice of a real unit normal. Changing the normal, however, has the effect of only changing the sign of the residue. As before, let X be an equivariant system of admissible affine subspaces attached to Xi,.By a hyperplane fi in X we understand an equivariant system of admissible affine subspaces attached to Wio with the following property: Forall(Pin,Si,;Ai,,)E %'in,fi(Pi,,Ain)isanadmissible hyperplaneinX(Pio,Ain). Fix a hyperplane Sj in X. Suppose given an Eisenstein system ( E , V} belonging to X. We shall now define an Eisenstein system { %e5,&E), %ee,(V)} belonging to Sj. This, as should be expected, requires some preparation. Fix an association class W, dominating Wio, such that W(io; fi) # 125then, of course, W(io : 3) # (21. Again, generically, vk stands for a G-conjugacy Sio;Aio)(or(P,,, Sjo; A,)) is a typical member of class in %?(io;B),while (Pi,, Wio (or W j o )admitting ( P , S; Ai) (or ( P , S; A,)) in %?k as its dominant successor. Suppose, therefore, that
and
144
CHAPTER 5
We must then construct functions %esB(E)(X: PIA,: PioIAi,:To:A!,: x) %es6(V)(X: P I ( A j , A , ) : P,,IAjo: PioIAi,:wJoi0: A!,)
i
that satisfy the various assumptions supra. Of course, the corresponding data in terms of Sj alone would be written
i
E ( 8 : P ( A , :PioIAi,:To:A!,: x) V(Sj: PI(Ai, A j ) : PjolAjo:PioIAi,:wJoio:A!,).
According to our notational principles, we can write 'B(PiosAi0) - 'B(Pco.Ai,)-X(Pi,,Aio) @ SX(Pio,Aio)' Pick a real unit normal AL to Sj(Pio,A,,) in X(P,,,A,,)-then A 1 is a real unit normal to @(Pi,, in 3 ( P i , , A d t . Let f be a meromorphic function on X(Pi0,AiO)' whose singularities lie along hyperplanes-then, on $(Pi,, the Laurent expansion gives
f(A!o + CAI) =
m
C an(A!JC", n= -N
the a, being meromorphic functions on Sj(Pio, Ai,)+ whose singularities lie along hyperplanes. This said, put
The definition is suggested by the following considerations. Let F be a holomorphic function on $(Pi,, A,,)'. Write
F(A,',
+ CAD = n
a l
=O
AAA!Jr",
the A, being holomorphic functions on $(Pi,, resb(Pi,. Aio)t N- 1
=
(fF)
1 u-,-
n=O
(A/o)
I(A!,)* AXA!,)
Then
145
EISENSTEIN SYSTEMS
More generally, let V be any complete LF-space (e.g., the symmetric algebra over a finite-dimensional complex vector space in its canonical topology). Let J' be a weakly meromorphic function on X(Pi,, Ai,,)' with values in V whose singularities lie along hyperplanes-then, on $(Pi,, the Laurent expansion gives m
the a,, being weakly meromorphic functions on &(Pi,, Aio)t with values in V whose singularities lie along hyperplanes. Explicitly,
This said, put
I
ReS&(Pro,Aio)t(f
A!,
N-1 H
1
1 -.n!
n=O
1:
8
S&(P~o,A~,)-~(P~o,A~,)
(Al)'' 6 u-,,- l(A/,).
We shall now define %ee,(E)(X: PIA,: PioIAio:Ti,: A/,: x).
For this purpose, note that E(X: PIA,: PioIAi,:?:
x)
may be regarded as an element of the dual of Hom(Sx(~,,,,A , ~ )& c u d s , 9
oiJX
so that, being of finite degree, it is represented by an element Q(X: P I A i : P i o ( A i , :A/o: x)
in
146
CHAPTER 5
It is clear that Q(X: P ) A i :PidlAio:?: x) is a weakly meromorphic function on X(Pio,AiO)’with values in SX(P*,,A,,)8 &cus(6,
oio)*,
a complete LF-space, whose singularities lie along admissible hyperplanes. Form Res,(,,o,A,o)t(Q(x: PIAil PioIAi,,: ?: x)), which, by definition, is an element of S B ( P ~ ~ x(P~,. . A ~A,,) ~ )8 - SX(P~,. ~ i ~ l Ccus(6, @ - SB(P,~.A,,) 8 Ccud6, oi,Jv* This being the case, let
oio)”
%esB(E)(X: P I A , : PioIAio:To:A!o: x) be the complex-valued function on H O ~ ( S ~ ( P , ~Ccus(6, , A , ~ )OiJ) , x $(Pi,, A d t x G defined by the prescription
%esB(E)(3: PIAi: P i o ( A i oTo: : A!,: x) = (Resg(~,~, ,,,,)t(Q(X: PIAi: PioIAio: ?: x))(AI0), To). On $(Pi,, Ai0)’, the Laurent expansion gives
E(X: PIAi: PioIAio:?: A/,
m
+ CAI: X) = 1 En(?:AJ0:X)Y, -N n=
the En(?: Ab: x) being meromorphic functions of A t in $(Pi,, Aio)t whose singularitieslie along admissible hyperplanes and differentiable functions of x in G. We have
%es,(E)(X: P I A , : PiolAio:To:AT,:
X)
Therefore %es5(E)(X:PIAi:PioIAio:To:Ato: x)
147
EISENSTEIN SYSTEMS
is a meromorphic function of A!o in B(Pi0,Ai0)+whose singularities lie along admissible hyperplanes.
Remark Suppose that is a point in $(Pi,, A d t . Let be a holomorphic function on some neighborhood of A!o in the complexification of ti!o with values in dcus(6, 0,)-then the definitions imply that P J A , : PioIAio:d@~o(A!o):A!o: x) resb(p,o,A,o)t(E(X: PIA,: PioIAio:dolo(?): ?: x))(Ait0).
%ee,(E)(X: -
It remains to define
%es,(V)(X:
P I ( A j , Ai): P j o l A j o :P i , ) A i o :wJoio:A!o).
There is a canonical injection from SX(P,,.Ai,) @ gcus(6, 0io)" @ S x ( p , , , Ajo),YjoioO acus(8, 0,)
into Hom(Hom(Sx(Pio,A,,)
9
gcus(8,
0iJX S x ( P j o ,Ajo),..:,,, 8 g c u s ( 6 , 0jJ),
the range of which is the linear transformations of finite degree. In particular, therefore,
V(X: P I ( A j ,AJ: PjolAjo:PioIAio:wJoio:?) can be viewed as a weakly meromorphic function on X(Pio,Ai0)' with values in S X ( P ~ ~ ,@ A ~'cus(8, ,)
OiJv O S X ( P j o . A j o ) j w ~ oO , O gcus(6,
OjoL
a complete LF-space (even though it is a question of algebraic tensor products) whose singularities lie along admissible hyperplanes. Form ReS6(Pio.A,o)t(V(X: P l ( A j , Ai): P j o I A j O :
P i o I A i o : Wjoio:
which, by definition, is an element of the tensor product of S 6 ( P i 0 .4
0 )
- X(Pi,. 4,)
with S X ( P ~ , . A6 ~ , gcus(6, ) OiJv @ S x ( P j o ,AjO),,,J,,,, 0 &'cus(d, 0jJ* The Hopf algebra morphism 's(pio. Aio)-x(pios ~
i o+ )
6S
S ~ ( p i A0 i ~o ) - x ( P i o , ~ i o )
~ p i Aoi o~) - x ( P i o * ~ i o ,
followed by the morphism ' B ( p i o . ~ , o ) - x ( ~ i~oi , o @ ) SS(Pio,Aio)-X(Pio.Aio) +
' B ( p i o , A i o ) - z ( p i o , Aio)
@ 'b(Pj0,
Ajo),,.J,,,- x ( P j 0 . AjO)rvjoio
148
CHAPTER 5
induced by ID 8 wJoio then leads, in the obvious way, to a map from the tensor product of
to
by the symbol %es~(V)(wJoio:?).
On fi(Pio,Aio)t, the Laurent expansion gives V(3: P I ( A j , Ai): PjoIAjo:PioIAio:wJoio:A!o
+ CAI)
the’V, being weakly meromorphic functions on fi(Pio,Aio)t with values in
and so
149
EISENSTEIN SYSTEMS
is a meromorphic function of Al0 in $(Pi,, Ai0)+whose singularities lie along admissible hyperplanes. There is a surjective morphism of restriction ResX-6: W&k(X;A j o , Aio) This being the case, let
+
W&,(B; Ajo,Aio)*
%es5(V)(X: PI(A,, A i ) : PjolAjo:PioIAio:wJoio:,A!o)
be the Hom(Hom(Sg(~i~, Ai,) valued function on
9
g c u s ( 6 , o i o ) ) , S 5 ( p j 0 , A , ~ ) , . . : ~O ~ , gcus(8,
W&k(sj;A j o , AiJ x defined by the prescription
$(Pi03
ojJ)
AiJt
%ee5(V)(X: PI(A,, A i ) : PjolAjo:PioIAio:wjoio:A!o) %es5(V)(*: A!o).
= Resx
We have
$(*) = w:oio
(%es5(V)(X: PI(A,, A i ) : P j o l A j o :PioIAio:wtjoio:A!o)To, To)
*(V-,-1(*: A!o)((Al)k v
To),(*AI)"-k v To).
Therefore (%es,(V)(X: PI(A,, Ai): PjolAjo:PioIAio:w : ~ AJo)T0, ~ ~ : qo) is a meromorphic function of A!o in @(Pi,, Ai0)+whose singularities lie along admissible hyperplanes. Remark Suppose that A!o is a point in $(Pi,, Let be a holomorphic function on some neighborhood of A!" in the complexification of 6!o with values in ECu,(6,Oio);let Y :o be a holomorphic function on some open set in the complexification of tijocontaining { - wjoio A!o : wjoio E w&k(sj; A j o AiJI 7
with values in 8,,,(h, Ojo)-then the definitions imply that (%es,(V)(X:
PI(A,, Ai): PjolAjo:PioIAio:wjoi0:
d@!o(A!o), dY~o(-~~oio~~o)), when summed over the wjoioin W&,(sj;A,,, Aio),is equal to
PI(Aj, Ai): PjolAjo: PioIAio:wJfoi0:?) d@,!,(?),dYJ0(- ~ j ~ i ~ ~ ) ) ) ( A ! ~ ) ~~S~(P,,,A,,)+((V(X:
when summed over the wJoioin W&,(X; Ajo,Aio).
150
CHAPTER 5
Using the explicit formulas developed above, it is a straightforward exercise to verify that { %eeB(E),%es5(V)}is an Eisenstein system belonging to !+ (provided j the data are nontrivial). There is nothing to be gained by providing the details. On the basis of the assumptions alone, it is possible to relate the singularities of the E-functions with those of the V-functions. In this connection, recall that we have introduced sets
i
D ( X : E : PIAi: Pi,IAi,) D ( X : v : P I A , : PiolAio).
Our assumptions imply that both of these sets are open and dense in X(Pi,, A d t and that the complement of either one is a locally finite set of hyperplanes, which will be referred to as the singular hyperplanes of E or V, as the case may be. Proposition 5.1 Let { E , V} be an Eisenstein system belonging to X-then the singularities of E are contained in those of V. In precise terms, suppose that
(P, S; Ai)
+
(pi07 s i o ;
Aio).
Let H , be a singular hyperplane of E(X: PIA,: PioIAi,:To:A!,: x).
Then there exists a triple (P,,, S j o ; Ajo),with
( P , S ; Aj)
+
(Pj,, s j o ;
Ajo),
and an element wjoi0E W&,(X; A,,, Aio)such that H , is a singular hyperplane of
V(X: P ( ( A j ,Ai): P j o ~ A j PioIAi,: o: wjoio:A!,).
Corollary Let the notations and hypotheses be as above. Then D(X: v : P I A , : P i o ~ A i oc) D ( X : E : P I A , : Pi,IAi").
The proof of this proposition depends in an essential way on the proposition in the Appendix to Chapter 4,whose notation we shall feel free to use. There are but finitely many triples ( P o - l , S o - l ; A 0 - J ( I E I ) , standing in the relation
( P , S; Ai) 3 ( P o - , , So-1; AO-J, whose daggered parabolics are distinct modulo I-,,-conjugacy. [Observe that for distinct indices I' and I",
151
ElSENSTElN SYSTEMS
may very well lie in distinct association classes in M i . This, however, is irrelevant, our point being only that the ( P o - , , So-,;Ao-,), when daggered, describe a set of representatives for the r,,-conjugacy classes in the various association classes which do arise from goper the daggering procedure.] Now take
i
X = Cm(G/(T n P ) A i . N ) Y = fl Cm(K x M i ) (#(l)copies)
i
T:X-+Y
f H { f K x P b - , :
'EO,
X and Y being supplied with their usual topologies. Let Yo* be the weak*dense subspace of Y* consisting of the finite linear combinations of Dirac measures. Choose for U the set D(X: E : PI A i : PiolAio).Put
0 = E(X:PIA,: PioIAio:To). Then, as a map from U to X , 0 is weakly holomorphic, since, by assumption,
E(X: P I A , : PioIAio:To:A!o: x) is a differentiable function of (A!o, x) on D(X: E : P I A , : PioIAio)x G.
ProofofProposition 5.2 Let At be a point on H , which, without loss of generality, may be assumed simple in the sense that it does not lie on any other singular hyperplane of E ( X : PIA,: PioIAio:?: A!o: ?).
To arrive at a contradiction, let us suppose in addition that A: does not belong to any singular hyperplane of the V(X: PI(Ai, A,): P o - , I A o - , : PioIAio:W
A - , ~ ~Ai',). :
Let A be a polydisk around A: whose closure misses all the singular hyperplanes of
E(X: PIA,: P i o ) A i o?:: AJo:?), except for H,, and all the singular hyperplanes of the
V(X: P I ( A i ,Ai): P o - , l A o - , :PiolAio:w;-,~,:A!o), but whose silov boundary &A misses H,.Obviously, then, & A is contained
in U . If we could show that the hypotheses of the proposition referred to above were satisfied by our data, then we would be able to say that
0 = E ( X : PIA,.:PioIAio:To)
152
CHAPTER 5
extends to a weakly holomorphic function on U u A, which is an impossibility. There are, therefore, two things to be established. First, let C be a compact subset of U-then we must verify that T is one-to-one on alg-span{cTn(@(C))}. Thanks to Theorem 3.6, this is easy to do. If in fact
f E alg-span{5n(@(C))}, then f is a slowly increasing differentiable function on K x M i / { 1 ) x rMi (as follows from the growth condition postulated in Assumption l(E)). Assuming now that T(f 1 = 0, Assumption 5(E-V) then implies that the constant term off along any { l } x r,,-cuspidal parabolic subgroup of K x M i is negligible, hence that f = 0. Therefore T is one-to-one. To finish up, let
{ ( k ,miI): I E I} be a collection of points in ( K x M i ) (#(I)copies)-then we must show that E K X p j - , ( XPIA,: : PioIAio:To: k,mi,)
1 I
extends to a holomorphic function on U u A, which, however, is clear in view of our definitions and Assumption 5(E-V). // By a singular hyperplane of W&,(X; Aio,Aio) in X(Pio,Aio)t we mean a set of the form
H, = {A!o :w / ~ ~=~ A/o A (/ w ~
/ E~ W&,(X; ~ ~ Aio,Aio),wibi0 # 1)).
Let
D(X: W: PIA,: P i o [ A i o ) be the complement of the union of the singular hyperplanes of W&,(X; Aio,Aio) in %(Pi,, Aio)+. Proposition 5.2 Let { E , V } be an Eisenstein system belonging to X-then the singularities of V are contained in those of E and W. In precise terms, suppose that ( P , S ; Ail (Pi03s i o ; Aio) ( P , S ; Aj) 3 (Pjo, S j o ; AjJ
{
+
Let H,be a singular hyperplane of V ( X : PI(Aj, Ai): PjolAjo:PioIAio:wloio: A!o)To.
153
EISENSTEIN SYSTEMS
Then H , is either a singular hyperplane of
E(X: P I A , : PioIAio:To:Aio: x) or else is a singular hyperplane of W&,(X; Aio,Aio).
Corollary Let the notations and hypotheses be as above. Then
D(3:E : P ( A , :PioIAio)n D(X:W : PIA,: PiolAio) is contained in
D(X:V: PIA,: PioIAio). The proof of this proposition depends in an essential way on the proposition in the Appendix to Chapter 4, whose notation we shall feel free to use. One has
i
M j = K,, PJo Pjo = MIo. Ajo * iVjo
This being so, take
I
(MIo = M j o ) .
X = C"(K x M j o x Ajo)
Y= K
n Cm(Ajo) Mjo
T:X+ Y
f H {f(kmjo ?) :( k mjo)E K
x Mjol,
X and Y being supplied with their usual topologies. Let T,* be the weak*dense subspace of Y* consisting of the finite linear combinations of the maps which are evaluation on one of the entries of the product. Choose for U the set D(X: V: P I A , : PioIAio).Let @:U-,X
be the map given by the rule
WA!o)(k m j o , a f > = (aJo)w~ono"~o. (V(3E: P I ( A j , A i ) : PjolAjo:PioIAi,:wjoio:A ~ o ) ~ o ) ( k m j o a ~ o ) . It is clear that 0 is weakly holomorphic. ProofofProposition 5.2 Let A! be a point on H , which, without loss of generality, may be assumed simple in the sense that it does not lie on any other singular hyperplane of
V(3: P I ( A j , Ai): PjolAjo:PioIAio:?: AiJ?.
154
CHAPTER 5
To arrive at a contradiction, let us suppose in addition that Af does not belong to any singular hyperplane of E(X:PIA,: PioIAi,: To:A!,: x) and that Ad does not belong to any singular hyperplane of W&,(X;Ai,, Aio). Let A be a polydisk around Af whose closure misses all singular hyperplanes of V(X: PI(A,, A i ) : PjoIAj,:PioIAi,:?: A!,)?, except for H,,and all the singular hyperplanes of E(X: PIA,: PioIAi,:To:A!,: x)
and all the singular hyperplanes of W$,(X ; Aio,Aio),but whose silov boundary dsA misses H,.Obviously, then, dsA is contained in U . If we could show that the hypotheses of the proposition referred to above were satisfied by our data, then we would be able to say that 0extends to a weakly holomorphic function on U u A, which is an impossibility. Since T is certainly one-to-one, we need only show that
extends to a holomorphic function on U u A. Owing to Assumption 5(E-V), one can express
EKxpjo(X: PIA,: PioIAio:To:A!,: k m j 0 a ~ , ) on A
H,as a sum over W$,(X; A,,, A s ) of certain terms, one of which is ( a ~ o ) w ~ o ~.~(V(X A ~ o: P I( A Ai) : PjoI A : Pi, I Aio: wjoio: A/,)T,) (kmj,ujo). Now fix ( k , m,,) E K x Mj,-then, on A - H,, E K x p j 0 ( X P: I A , : PioIAi,:To:A!,: km,,?) is simply an exponential polynomial qua a function of uj, E AJo.Because a -
,,
,,
given A!, E A lies on no singular hyperplane of W&,(X;Aio,Aio),distinct elements of W&,(X; A,,, Aio),when applied to A!o, have distinct images. Since exponentials of distinct linear functions are linearly independent over the ring of polynomials, it is possible to pick off each term of EKXp3,(X:PIA,: PiolAio:To:A!,: km,,?) in a holomorphic way, so long as A!, construct a map
EA -
-
H,.More carefully put, it is easy to
D(wjoi0:A!,): Cm(Ajo) Cm(AIo) (A!, E A), depending holomorphically on A!,, which, when applied to EKx,,j0(X:PIA,: PioIAio:To:A!,: km,,?),
155
ElSENSTElN SYSTEMS
gives t .
t
(?)wJo@’io.
on A
-
( V ( I : P I ( A j , Ai): PjolAjo:PioIAio:wJoio:A ~ o ) ~ o ) ( k m j o ? )
H , . The entity D(wjoio:AIo)(EKXpj$X:P J A i PioJAio: : To:A6: kmj0?))
then furnishes the desired continuation across H,. // It will be necessary to return to the subject of singularities and residues later on. For the time being, however, we shall study Eisenstein systems in general on an essentially “axiomatic” basis in the sense that we shall set forth a series of conditions and then deduce certain consequences when some or all of these conditions are in force, the interdependence of the various conditions also being a part of the investigation. Keeping to the earlier assumptions and notation, write
v,
= io
via.
Suppose there is attached to each Wio a collection xi0
=
{XI
of equivariant systems of admissible affine subspaces of dimension b. One sets
x(v0) = JJxi,. io
Now let %? be another association class. If, as usual, %?k is a G-conjugacy class in %?,then we put xio(%?k)
=
{X :%k c %?(iO; X))
and form x(Wk;
%O)
=
xio(vk>, io
a subset of I(%o) which could, of course, be empty. It will be convenient to introduce an equivalencerelation in x ( % k ; W),. For this purpose, suppose that %?k
Let
c
%?(io;
I)n W ( j o ; ‘I))
with
X per Wio ‘I) per g j o .
156
CHAPTER 5
be the subset of W&,(X;Wjo, %), consisting of those transformations w],,, such that X,jOio = 9.In terms of specific data, if (Pi07
sio;
Aio) E S i o
(Pjo, s j o ; A j J E g j o and if Ajo, AiJ
W&k(YI,
is the subset of W&,(X; A j o ,A,,) consisting of those transformations w],~, such that 3 ( P j o , A j J w J o i o = 9(Pjo*A j J
then there is a canonical bijection
wJk(YI, fi;
wjgjo, %iJ +
W&k(?l, Aj,, AiJ
This said, elements
{9 X
per Wio perwjj,
in x(%k; go)are declared equivalent iff
W&k(FD,X; w j o , ViJ is nonempty. As it stands, it is not clear that this relation is in fact an equivalence relation. Basically, what one needs to know is that
Wtk(3,X; w i o , giJ is nonempty. For the time being, we shall simply assume that this is so through the formulation and proof of the next proposition. Let X be an equivariant system of admissible affine subspaces attached to %,,-then by Dis(X) we understand the X-admissible subspace of X defined as follows. Let (Pie, Si,; A,,) be a member of %,,-then Dis(X)(Pio, AiJ is the distinguished subspace of a,, of maximal dimension which is contained in X(P,,, A,,)". In the event that the complexification of
(Pi, At,) 9
is actually equal to X(Pi,, A d " , then X is termed principal. There corresponds to Dis(X), in a natural way, an association class V(Dis(X)), containing a G-conjugacy class Wk(Dis(X))such that %?k(DiS(X))3 Wi0.
157
EISENSTEIN SYSTEMS
One can write
hi,
=
Dis(X)(Pi,, Ai,)t 0 Dis(SE)(Pio,Aio).
Let (PDis(X)
9
SDis(X); A b i s ( X ) )
denote the parabolic data determined by Dis(X)(P,,, Aio).We then have (PDis(X), SDis(X); A b i s ( X ) )
@
S i o ; Aio),
so that the daggering procedure determines parabolic data (Pbis(X), Sbis(X); Abis(XJ
in M b i s ( X ) . What follows is a list of possible geometric conditions which could be imposed on Xi,.
Geom: I (Local Finiteness) Xi, is locally finite in the sense that for all (Pi,, Sio;Aio)in Wio and every compact subset oioof the complexification of hi,, the set {3E E Xi0 X(Pio,AiJ n mi0 Z
01
is finite. Geom: I1 (Real Compactness) Xi, is real compact in the sense that for all ( P i , , Sio;A,,) in Wio, the set (Re(X(Pi0, Aio)) :X €Xi01 has compact closure.
Geom: 111 (Conical Containment) Xi, is conically contained in the sense that for all (Pie, Sio;Aio)in W i o , - Re(X(Pi0, AiJ) E
(hkis(2))
for every X belonging to Xio. There is also a group-theoretic requirement which might be invoked per gi0.
Gr For every 3E belonging to Xi,, there exists an element : W
E W(K
X; via, UiJ
such that for all (Pie, Sio;Aio)in Wio, w:(Pio I Ai,: Pi0 IAiJ IWPio, AiJ"
is the identity map.
158
so that
CHAPTER 5
WE, if it exists at all, is unique.]
So far we have said nothing about Eisenstein systems. Let us now suppose that for each io there is an Eisenstein system { E , V} belonging to each 3 in Xi,. We consider three conditions. ES: I Let 3 E xio(%?)'). Suppose that
Let wjoioE WZk(X; A,,, A'?); let w]oioE W&,(X; V,,, Vio)correspond to wjoio under the canonical bijection
E-S: I1 Let 3 E Xio(%?k). Suppose that
Let wJoioE W&,(3; A . , A,?; let wJoioE W&,(X; Wjo,wi,) correspond to wJoio J? . under the canonical bijection Then the adjoint
is given by
EISENSTEIN SYSTEMS
159
E-S: III Let X E Xio(Wk).Suppose that pt
= {pJo]is a completely equivariant system of polynomials, ie., to each j , there is assigned a polynomial pJ0e C[liJ,,] such that for all triples
(Pjh, Sjb; Ajb) (Pja, Sj8; A j d
standing in the relation
by means of scalar multiplication. Suppose that
Then
V ( 3 PI (Ai Ail : Pjo I Ajo: PioIAio: wJoio:A/o)(dHomp!o(A!o) ( To>) = d pJo(wJoi0 A!o) (V(X : P I ( A i , A i ): PjoI A j o : Pi, I Aio: wjoio: A!& To). 9
[Observe that any polynomial pt E C[li!o] which is invariant under the !" , gives rise in a natural way to a completely operations of W t k ( A i oAio) equivariant system p' = {pJo}of polynomials.] Proposition 5.3 Suppose that for each io, X i , is subject to
Geom: I Geom: I1 Geom: I11 and E-S: I.
160
CHAPTER 5
Then X(Wo) is a j n i t e set. Moreover, the normal translation in any member of X(Ws,) is real. Finally, given Wk, every equivalence class
(5
= X F k ; WO)
contains at least one principal element.
[This result is due to Langlands [p. 1831.1 We begin with two observations which essentially have nothing to do with Eisenstein systems as they are really consequences of certain general statements about automorphic forms. Suppose that
After grouping terms, we can write E K x p j 0 ( X :PIA,: P i o ( A i oTo: : A/o: km,) =
1aJo(mi)AJ.
Y’h(kmi),
h
where the A,! are distinct elements in the complexification of 6g and the &, are elements in C[a]o]@ 8,,,(6, Ujo), none of which vanish identically. Lz-(l) If for all triples ( P j o ,Sjo;Ajo)standing in the relation ( P , S ; Ail
it is true that for all h,
+ (Pjo,
WAf)E
Sjo;
AjJ
.a,Jo(qo)9
then
qo:~
E(X: P I A , :
7 ? )~E L: ~ ( Kx
M J { I }x
rM,)
for any T O E
H O ~ ( S X ( P , , , AJcus(6, ~~),
@iJ).
L’42) If E(E P I A , :P , ~ I A , T , : ~A:J ~?: ) E L ~ ( K
x M J { x~
rM,)
is nonzero, then for all triples ( P j o ,Sjo; Ajo)standing in the relation
(P, S ; Ail 3 (Pjo, S j o ; Ajoh it is true that A1 is real.
161
EISENSTEIN SYSTEMS
[Keeping in mind Assumption 5(E-V), L2-(1) follows from Langlands [p. 1043 while L2-(2)follows from Langlands [p. 1841.1 The proof of the present proposition, while not difficult,is a bit subtle. Let us break up the argument into a series of simple lemmas.
Lemma Let
a be an equivalence class in JE(Wk; %,).
Assume
xwk = Dis(X)
for eoery 3E E (5. Fix 3E E E, say X E JEi,(Wk). Suppose that
( P , S ; Ail
3 (Pi09 Sio; Ai,,).
Then there exists a neighborhood A'" o f X ( P i , , Aio)in X(Pi,, AiJt such thatfor every A!, in JV at which both the E-jiunction and all V-jiunctions are defined,
E(X: P ~ AP~~ :~ I AT": ~ ,A!,:: ?) E L ~ ( Kx ~ ~ / { xi rMi) } for any
T o E Hom(Sr(pio,A i , )
9
@io))*
Proof Suppose that ( P , S ; Ai) 3 (Pj,, Sjo; AjJ
Then w:O,,eW: 'w:,,,,,
n
(Wjoi0)-
k ( x : w J O *@to)
'(PjOIAj,: P i , I A i o ) ( d ~ ~ , ( 6 j+ J J-16jo)
Exm(wk)
intersected with X(Pi,, Ai0)+is an open subset of X(Pi,, which, in view of Geom: 111, contains X ( P i , , A,,). Let N be the neighborhood of X ( P i o ,A?) in 3E(Pi,, A,,)+ obtained by intersecting these sets over all possible triples (P,,, Sjo;Aj,) standing in the relation ( P , S ; Ail
3 (Pi,, Sjo; ' 4 j J
We explicitly note that N is in fact open, as the intersection can actually be taken over a finite set of indices. If A!, is a point in N , at which both the Efunction and all V-functions are defined, then
E K X p ~ , ( PX (: A i :PiolAio:To:A!,: kmi) =
1 a~o(mi)Al .Th(kmi), h
where, in view of the definitions and E-S: I, R e ( 4 ) E dPJ,,(6J0) for all h. Our assertion thus followsfrom the square integrability criterion cited above. JJ
162
CHAPTER 5
Let us keep for the moment to the assumptions and notation of the preceding lemma. We then claim that, of necessity, the A!o in N, at which both the E-function and all V-functions are defined, are real (and hence that every point in J(r is real). In fact, if this were not the case, then (cf. supra), for some such
E (I : PIA,: PioIAio:To:A!,,: ?) = 0 for any
T o E Ho~(SX(P,,, A,,)
9
&cuds ,
@iJ)*
Consequently, E ( 3 : P I A , : P i l ~ ~ / l?:i l A!(,: , : ?) = 0
for A/o in an open set, thus everywhere. In other words, E ( 3 : PIA,: PioIAio:?: ?: ?) = 0.
By making a suitable replacement of the data if necessary, this fact can be used to contradict the nontriviality of the Eisenstein system belonging to 3. Every point in J(r is therefore real. But then W P i o , A,,)+ =
{X(Pio, AiJ>,
which implies that X ( P i o ,A,,,) is real. Moreover, in the case at hand, 3 ( P i o ,Aio)is actually equal to the complexification of a,, i.e., 3 is principal. To summarize, if the special circumstances of the above lemma are in force, then the normal translation in any member of (E is real, and, in addition, every element of (E is principal. What is needed now is a lemma which will allow us to reduce to this particular case. ~~a Say
Let
3 E %io(%k),
an equioalence class in x ( % k ; W0)--then there exists X E (E, and q k , with
(E be
such that and such that for every X' in the equivalence class (E' of 3 in 3(%;;W0), one has 1%; = Dis(3').
Proof The proof will be by induction on n0(Wk)where, by definition, no(%k)= rank(V0)-rank(Vk)
(rank(Wk)= rank(%)).
Suppose first that n0(Wk)= 0-then, of course, %?k = Wi0 for some io and the assertion is clearly true. Assume, therefore, that n0(Wk)> 0. If %k itself does
163
EISENSTEIN SYSTEMS
not work, then there exists X, E @ such that Dis(Xo) # xy,. Since no(gk(Dis(Xo)))< the induction hypothesis implies that there exists an x(qk(Dis(XO)); WO), per Vi0say, equivalent to 3, in iX(Wk(Dis(X,J); W,), for which the various assertions of the lemma are true. But then, a fortiori, X E @, so we are done. // Corollary Let @ be an equivalence class in x ( % k ; %,)-then the normal translation in any member of Ci is real, and, moreover, @ contains at least one principal element. [It is clear that the normal translation in one member of 6 is real iff this is the case for every member of 6.Thus, there is no loss of generality in considering only an element X of @ having the properties spelled out in the preceding lemma. Taking into account what has been said earlier, we then conclude at once that the normal translation in 3 is real and that X is principal.]
Remark Let @ be an equivalenceclass in iX(Wk; q0).Let 3 be a principal element [email protected] @(Dis(X))be the equivalenceclassofXinI(Wk(Dis(X));Wo)then it is clear that all the members of Ci(Dis(X)) are themselves principal. To complete the proof of our proposition, we need only establish the following lemma. Lemma X(Vo)is a j n i t e set. Proof Fix io-then it will be enough to show that Xi, is finite. For this purpose, fix ( P i o ,Sio;Aio) in Wio. Since the normal translations are real, Geom: I1 implies that the closure oioof the set {X(Pio, Aio): 3 E xio}
is compact. Owing to Geom: I, the set
{X E x i 0 X(Pio,Aio) n wi0 # W } is finite. This set, however, is just Sioitself. // Scholium Let @ be an equivalence class in element of@, say X E xio(%k). Then
%(%?k
; go).Let X be a principal
E(X:PDis(x))Abi,(x):PioIAio:To:X ( P i 0 ,/Iio):?)
164
CHAPTER 5
[As will become apparent later, this fundamental fact (contained, of course, in the preceding discussion) is the point of departure for the construction of residual forms and their associated Eisenstein series.]
Remark We remind the reader that it has been supposed up until now that W&,(% X; %i,, %i,J is nonempty. What Langlands does to ensure this is simply to assume that Gr holds, which, however, is deceptive since the particular nature of , !w plays no role whatsoever in the proof of the proposition. To even state the next result requires a fair amount of preparation. It will be assumed that the assumptions set down in the preceding proposition are in force here. Fix %,'; fix an element ( P , S; A i ) in W k . Let (E be an equivalence class in %(%,'; go).Let X be a principal element of (E, say X E xi,(%,').Fix an element (Pie, Sio; Aio)in Vi, such that
( P , S; Ai) @ (Pi07 Si,; AiJ Suppose that %,' 3 W j 0 . Let Vj,(PIAi)be the subset of Wj,, consisting of those elements (P,,, S j o ; A,,) such that
+
( P , S; Ail
(Pj,,
Sjo; AjJ
Observe that WjO(PIAi)is not empty. Moreover, any two members of Wj0(PI A i ) are necessarily Mi-conjugate. Now it is clear that rM,\ % j o ( P
IAil
is a finite set. We shall want to choose a set of representatives for it. This, however, must be done in such a way as to reflect an additional ingredient, viz., %,'(Dis(X)). The basis for making our selections is the following simple fact.
Lemma Any member of %,'(Dis(X)) which is a dominant successor of some element in Wjo(PIA i ) is necessarily a member ofWk(Dis(X);PI Ai). [We omit the elementary verification.] Let (pDis(X)
SDis(f); A&s(X))
9
be that element in %,'(Dis(X)) such that Let
( p , s;
*
(PDis(Z), SDis(X); Abis(x))
I(Ptlis(x,
9
3
Stlis(x); A#is(x))}
sio;
Aio).
EISENSTEIN SYSTEMS
165
be a set of representatives for r M ,\ wk(Dis(x); P
I Aih
the first of which being just (PDis(T)
itself. Fix k,
EMi
9
SDis(T); Ahis(X))
with the property that k, conjugates (PDis(X), SDis(X)
; Abis(X))
(Pt;is(x)
;A#is(x))*
to
Remark Parabolics r-conjugate.
St;is(x)
corresponding to distinct indices p cannot be
Given Wjo such that %?k@ W j 0 , there are two possibilities: Wk(Dis(x)) # % j O Wk(Dis(X)) 3 W j o . Let {(Pjovor Sjovo; A j o v o ) }
be a set of representatives for
r M i \ W j & P IA i l If the first possibility obtains, then we make no special assumption. If the second possibility obtains, then we assume, as we may, that each (Pjovo, Sjovo; Ajovo)
is a dominated predecessor of some (PL;is(x) SL;is(x);A%s(x))*
This agreement furnishes a partitioning { ( P j o v o * Sjovo; A j o v o ) )
=
{(Pjovo, Sjovo; Ajovo)}p Ir
into nonempty disjoint sets, where, as the notation suggests, each member of {(pjovoy Sjovo; Ajovo)}p
is a dominated predecessor of (Phis(%)St;is(x);
166
CHAPTER 5
Fix an element (Pjo, Sjo; A j J
in {(Pjovo, Sjovo; Ajovo)}
where, when the second possibility obtains, (PDis(I)? SDis(X); Ahis(X))
3 (pjo,
'jo).
Fix kjovoE M i with the property that kjov0 conjugates ( P j o , Sjo; A j o )
to ( P j o v o , S j o v o ; Ajovo).
Remark
r-conjugate.
Parabolics Pjov0corresponding to distinct indices v,, cannot be
Let '2) E a, say '2) E JEjo(Wk)-then we form HOrn(Sg(pjo,
=
9
gcus(6, o j o ) )
@ Hom(Sg(p,,,,, A,,,o),
acus(6, ojovo))
yo
and s4)(Pjo.Ajo)
=
(8&cud6,
Ojo)
0S~(Pjoyo, yo
8 &cus(6-
AjoYo)
QjovJ
It will be convenient to view the elements in either space as column vectors. The next thing to do is to construct transformations
V : Horn -+ @. The philosophy here is entirely analogous to that encountered earlier in the study of the c-function. Let%ll,%E(f,saY '2) 1 JEjb(wk) '2)2 zj$(%k)*
Let
161
EISENSTEIN SYSTEMS
We shall then denote by
V(E: 92:91: PIAi: w;ajb: AJh) the linear transformation from Hom(SQ,(Pj,:.Aj:,), &CUS(~,
to
Ojh))
S Q ~ ~A,;~) A0, & c u s ( 6 , O j $
defined by the matrix CV(91:
PI(Ai, Ail: P j d v 8 I A j a . a :
P j d v b l A j A v d : k j d . 8 . lt3J6jd
.k,l,:~:k j h v h .AJh>I.
Observe that V(E: 0 2 : 91: PIAi: W;;,A: Ajh) is a meromorphic function of A b whose singularities lie along admissible hyperplanes and has a finite dimensional range, which, by definition, is its rank. In order to describe the functional equations which the V-functions must satisfy, it will be best to form still another, larger matrix. For this purpose, we require some additional definitions. Henceforward, we shall suppose in addition that for each i o , Xi, is subject to Gr.
Lemma Let g1,g 2E E, say
i
'9 1 E xjh(wd '92
Then: The products
E xjs(wk)*
wJ?a .b . w 10P . ( wJblo ?.)-', where
w;dio E W&k(TJ 1, x; wjh ViJ wJaioE W&k(O2 x; gj8 WiJ
{
7
9
9
describe the elements of W&k('923
01;gj6, vjjd).
[The point here is that ( w : ~ = ) ~ 1.1 Introduce now
W i k ( Ex: wio) =
jo
WLk(9,x; vjo, viol I)Ex,g(wk)
rgoe
168
CHAPTER 5
and
There is a canonical injection One may attach to each W f E WJk(@:3:W i 0 )
a unique index j,(wt). 3, be subsets of W&,(@:X: gi0).We shall then denote by Let
V(k,: Z , : P J A , :A/o) the linear transformation from to
Observe that
V(3,:3,:P I A , : Ale) is a meromorphic function of A t whose singularities lie along admissible hyperplanes and has a finite dimensional range, which, by definition, is its rank. If k, = S, = 3,then we write V(S: P ( A , :A/o)
in place of
V(k:e: PI A i : A/o). There are two cases of special importance, namely, when
k = W$,(@: 3: qi0), in which case we write V,,((E.: X: PIA,: A:,),
EISENSTEIN SYSTEMS
169
and when =
Wtk(Discr))((F(DiS(3)): 3: %?io),
in which case we write Vwk((F: X : PI AJ.
We hasten to point out that the last piece of notation is justified since the matrix in question really is a constant, as follows from the definitions and our assumptions. Let us convince ourselves that this is in fact so. A typical entry of the matrix in question has the form V(@: Xw;: X,;: P ( A i :w~*w~.(w])-': ( W ] W ~ ~ ( P ~ ~ ( , ~ )PiolAio)AJ0), IA~,(,~): where, of course, Set
w!, wi E W~,(Dis(x))((F(Dis(X)): X: Uio).
There exist indices p, and p2 such that
170
CHAPTER 5
Proposition 5.4
Suppose that for each i o , Xi, i s subject to Geom: I Geom: I1 Geom: 111, Gr,
and
E-S: I E-S: 11. Fix
(%?k,
an equivalence class @ c X((%?k; ‘KO),
and a principal element X of Ct, say X E 3i0((%?k). Then: The ranks of the matrices
{V,,(@: V,,(@:
X: PI A i : AI0) x : PI Ai)
are the same at all points N o
E W p i o , AiJ’
at which
V,,(@:
X: P I A , : AL)
is dejined.
[This result is due to Langlands [p. 1911.1
Remark It will be seen eventually (cf. Chapter 6) that this “equality of the ranks” is the basis for arriving at the functional equations satisfied by the Eisenstein series associated with residual forms. The proof of the present proposition is difficult, the argument being involved and lengthy. Set nx(Wk)= rank(Wk(Dis(X)))-rank((%?k). We shall proceed by induction on nZ(Wk). If nI(Wk)= 0, then %k = gk(Dis(X)), so that there is nothing to prove. Assume, therefore, that n f ( V k )> 0. The case when nI(%?k) = 1 requires special attention and will be treated first. Having disposed of this possibility, it will then be assumed that nX(Wk)> 1, the assertion of the proposition being supposed to hold for all integers less than ndqk). There are some generalities which it will be best to take care of before getting involved in details.
171
EISENSTEIN SYSTEMS
Technical Lemma (T-L) Let 8 , ,8, be subsets of Wtk(@:X: qi0). Assume Elc 8,. Thefollowing statements are then equivalent: (i) (ii) (iii) (iv) (v)
V(8,: 8 , :PIA,: and V(8,: PIA,: At,) have the same rank; haoe the same image; V(E2: 8 , :PIA,: A!o) and V(3,: PIA,: V ( 8 , : 8,: PIA,: A f ) and V(8,: PIA,: A!o) haoe the same rank; V ( 8 , : 8,: PIA,: Ai',) and V(8,: PIA,: At)) have the same kernel; haoe the same rank. V ( 8 , : P J A , :A!J and V(8,: P J A , :
[The verification of the equivalence of these five statements is straightforward, hence will be omitted. It should be noted that the adjoint
V(E2: 8 , :PIA,: A!o)* of V(8,: 8 , :PIA,: Ale) is given by
V ( 8 , : 8,: PIA,: -wi",!o), as can be checked without difficulty (using E-S: II).]
Remark In T-L, it is assumed that A!o ranges through an open dense subset of X(Pl0, Since rank is locally constant, in order to check T-L, it suffices to restrict AL to some nonempty open subset of X(Pio,Aio)t. Let 'I) E @, say g E xjo(%k).Given Tjo= (qoyo) in
Hom(Sw(~,~, A ~ ~&cus(6, ) ,
fljo))?
set
E('I): P I A , : Ti,: AJo:x) = E('1): PIAi: PjOvO~AjovO: 7jov0:kjovo Ajo: x),
1
9
yo
a meromorphic function of AJowhose singularities lie along admissible hyperplanes. Lemma Let 1 ' ) E (5, say 'I) E Ij&%,)-then
,
,
EKxpj6u6('I)l: PIA,: Tjb:Ag: km,)
= c
wZE@nXj@k)
w&
c
EW:~(C()Z,C(I;w J & s j b )
at (mi)kJdd'w~6Jb'h~b J M
x ( E J ~ oV(@: ~'~
where is the canonical projection.
0,:
'I),: P I A , : w J ~ ~AJA)Tj$(kmi), A:
172
CHAPTER 5
[This assertion is a direct consequence of the definitions.] For the sake of simplicity, we shall agree to write
so that
Let 3 be a subset of W$,(@: X: %,)-then g=
U gjo
there is an obvious partitioning 4
jo
into disjoint sets. This said, put Given T = (Tjo(wt))in
Hom(sx, &C",(&
set
flo)),
E(3: PIA,: T: Ajo: x) =
C 1 E(X,+: PIAi: Tj0(wt):W ' W ~ ~ ( P PioIAio)A]o: ~ ~ I A ~ ~ :x), jo zj0
a meromorphic function of A!o whose singularities lie along admissible hyperplanes. Owing to the preceding lemma, we have
EK.P~oYO(S: PIA,: T: ATo:hi) t
t
c
WjoioEWF&:
3E:Yio)jo
atJoVo(mi)k,ovo.WJoio.Ai6
x
( E t P , V ( { W ~ ~ ~8, }::P J A , :AJ0)T)(kmi), 0
where E$joI;o is the canonical projection of
8 &cus(a3
%w~o,,,(~ ~ j oj .
~
)
Ojo)
onto the vo th-factor. Let A!o E X(Pi,, Aio)t-then, for brevity, we shall say that ATo is a general point if the various
.
.
kjovo WIOio A!o
are distinct.
173
EISENSTEIN SYSTEMS
We shall now make a remark preliminary to establishing a criterion which will serve to simplify our problem. Let Uv be a nonempty open subset of X(Pio, Aio)t consisting of points at which Vwk(C?:X: PIA,: AJ0) is defined. Suppose that
E(B: PIA,: T:
?)EL’(K x Mi/{l} x
rM,)
forall A!o E Uv-then,ofnecessity,E = O.InfacttheexponentskjOvOwJoio.A!o appearing in E K ~ P J ~ ~P~J(ABi :T: AL: ?) cannot all be real (since nx(Wk)> 0), so that, varying vo and j o , the assertion follows from L2-(2)supra. Consequently, upon taking for AIo a general point in Up, the V-terms appearing in
EKxp~o,o(B: PIAi: T: A!o: ?) may be split off; thus, varying vo and j o , we then infer that V(?: 8 : PIAi: AL)T = 0 for every subset of W&,(C?:X: Wio). By continuity, this relation continues to hold at any A!o in Up.
Criterion Let U be a nonempty open subset ofX(Pio,Aio)t. Let B be a subset of W&,(C?:X: Wio). Suppose that for every wJOsE W&,(C?:X: %?io)jo
-
Ej0,
AI0 E U,
we have Re(kj0vo
WJoio
. A!J
E~p~~~,(~Jovo)*
Then rank(V,,(C?: 3: PIA,: AIo)) = rank(V(8: P J A , :A!o)) at all points
E
U at which
Vwk(C?: X: PIA,: A!J is defined. [Note that the condition on U is geometric in character.] Proof
Thanks to T-L, with
174
CHAPTER 5
it will be sufficient to prove that
T EKer(V(E: W&,(@:X: Wi0):P I A , : A/o)) * T E Ker(V(W&,(@:X: Wio): P I A i : A!o)). So suppose that
V(E: W&,(@:X: Wio):P I A i : AI0)T = 0. In view of what has been said above, the fact that
V(W&,(@:X: Wio): PI/&: AI0)T = 0 will follow when it is shown that E(W&,(@:3: % i o ) : P I A , : T: A!o: ? ) € L 2 ( K x M i / { l > x
For this purpose, fix j o and
yo.
rMi).
Let wJoioE Ejo-then, of course,
V({wjoio}:W&,((F:X: WiJ: PI.4,: A!o)T = 0. On the other hand, if wJaio E W&,((F: X: %io)jo exponent of
- 8,,, then, by
hypothesis, the
v({ wjOio}: w&,(@: x : wi0):P IA,: A!~)T
E;!;;
in the constant term of E(W&(@: X: Via): P I A i : T: AJ0:?) along K x P ~ o vhas , its real part in &p~OY0(6~ovO). As j , and vo are arbitrary, L2-( 1) supra implies that
E(W&,(@:X: Wi0):PIA,: T: A!o:
?)E
L2(K x Mi/{l> x
rM,),
//
thereby completing the proof.
Let E be a subset of W&,(@:3: Wio). Form
u,
=
X(Pi0, Aio)+
n Ln n n JO
WlOiO
yo
(kjovo.
J
~ j o ~ ~ , ~ ~ ~+ ~~ ~- ~1 ~ ao ~~ o~ ~~ ~ ~j
where,foreachj,, wjOiorunsthrough thecomplementofEjoin W&,((E;:X: Wio)jo. The preceding result is then applicable to U,, provided Usis nonempty. Now this will always be the case for certain 8. For example, let %(%?k)= {w’ E w&,(@:X: %io) Wk(DiS(Xw+))# %k}. must conThen Geom: I11 guarantees that Up,yk)is nonempty (since Us(%,) tain the normal translation X(P,,, &)). Observe that it is enough to verify “equality of the ranks” on some nonempty open subset of X(Pio,AiJt, it being understood, of course, that we
175
ElSENSTElN SYSTEMS
actually consider only those points at which V,,(@: X: P I A , : AI0) is defined. [In this connection, let us bear in mind that rank is lower semicontinuous.] With this preparation, we are ready to begin the proof of our proposition. We shall treat the “rank one” case first, i.e., the case when nx(%k) = 1. Let us indicate the strategy. In this situation, E(%k)can be alternatively described as the subset of W$,(@: X: Via) made up of those wt such that X,+ is principal. Owing to what has been said above, on U=(,,) one has rank(V,,(@: X: PIA,: AI0)) = rank(V(k(Vk): P1.4,: AIo)). We would be done if we knew that there was a nonempty open subset U of Us(%,)on which this relation were true with a(%&replaced by W$,cDis(x))(@(DiS(3)> : since then v(E(%k): P 1 A i : Ai’,) is replaced by V,,(@: X: PIAi), whence rank(V,,(a: X: P I A i : /\Io))
=
rank(V,,(Q: X: PIAJ),
as desired. To be able to draw this conclusion, it would clearly be sufficient to produce a nonempty open subset U of U=(,,) with the property that for every Wjc,io E E(%k)jo-
W$,(Discx))(@(DiS(X)): X: %io)jo,
A!o E U ,
176
CHAPTER 5
is
Because X(Pio AiJ E .('%?k) 9
U is a nonempty open subset of X(Pio, sumptions and notation, suppose that wjoio
=(gk)j0
9
Keeping to the preceding as-
- Wt,(Dis(x))(a(Dis(X)):3E:%iu)j0?
EISENSTEIN SYSTEMS
177
The proofofour proposition in thecase when nr(Wk) = 1 is thereforecomplete.
178
CHAPTER 5
Remark Suppose still that nS(%‘k)= 1-then it is always the case that =(%k)
# Wt,,Di,ci))(@(Dis(XE)): x: %io).
To see that this is so, proceed by contradiction. This time, let U be the intersection of Ul(vk)with {X(Pio, AiJ
+ A!o:
Re(Ait,)E ~P~,.,.,(~t)is(X)))-
Take for $, in the Criterion supra, the empty set. Arguing as before, we find that for every wJoioE W$,(@: X : %‘io)jo,
A!o E U ,
we have Re(kjovo. WJoio . A!J E ~ P ~ ~ ~ , ( ~ J o v o ) * Consequently. V,,(@: X : PIA,: ?)
=
0.
By making a suitable replacement of the data if necessary, this fact can be used to contradict the nontriviality of the Eisenstein system belonging to X . The principle embodied here is important. Indeed, it is the basis for decomposing elements of W&,(@:X: Via) into products of “simple reflections” (cf. infra). We shall now suppose that i?x(%k)> 1 and use induction. It will be convenient to first set up some preliminary machinery. Let W‘ be another association class, Vka G-conjugacy class in V. Suppose that %k 3 W‘k with rank(%‘:) > rank(Wk). Let @‘ be an equivalence class in X(48k; go),contained in @. Let X’ be a principal element of &‘, say X’ E Xib(W:)-then nzp(g:)< n.d%k)*
Given
WTbi0 E WL,(x’,
X ;Wib, Wi0),
introduce
B(@‘: X ’ : wt.. ) = { W t ’ w;. w&io:W t ’ E W&(@’:x’; Wib)} *
1010
and
B(@’(Dis(X’)) : X’ : w&~,) = {Wt‘ . W i W&io: W t ’ E W~~(Djs(x,))(@‘(DiS(X’)): 3’:Wib)}, *
these being subsets of W&,(@:X ; Wi0).
179
ElSENSTElN SYSTEMS
There are three statements basic to our development. Let AL, be a general point in X(Pi,, A,,)+.Then Im(V(E(6’: 3’:w!”,,): E(W(Dis(X’)): 3‘:wii0): PIA,: A!,)) = Im(V(E(@’:3’:w!~,,): PIA,: A!,));
S, SII
Ker(V(H(W: 3‘:w & , ~ )P: J A , :A!,)) = Ker(V(W&,(@:X: Via): =(@’:3’:w&~,):P I A , : A!,));
S,,, Im(V(W$,(@:X: W,,): E(@‘:X’: wJoio):PIA,: A!,)) = Irn(V(W:X: %,,): E(@’(Dis(X’)):X’: w!~,,): PIA,: A!,)). The proofs of S, and S,, are similar, the main difference being that the proof of S, depends ultimately upon the induction hypothesis, while the proof of S,,does not. In order to avoid interrupting the exposition, we shall postpone them for the time being. The final statement, S,,,, is a formal consequence of S, and S,,, so we might just as well dispense with its proof now. The left-hand side clearly must contain the right-hand side, so we need only establish that the left-hand side is contained in the right-hand side. Let T belong to the domain of =(@’:X’: w & , ~ )P: I A , : A!,). V(W&,(@:X: qi0): Then T belongs to the domain of
V(E(Q?: 3’: wki0): P ( A , :A!,), so that, by S,, there exists TDisin the domain of V(E(@’:X’: w / ~ , ~E(@’(Dis(X’)): ): X’: wIbi0):PI A , : A t ) such that V(E(@’:X: w$,,~):P I A , : A!o)T = V(E((%‘:X’: w&,,,):E(C%’(Dk(X‘)): 3‘:witbi,): PI A i : Ajo)TDis. Let TDisbe the column vector with coordinates of TDis in E(G’(Dis(3’)): X’: w&,,) but zero in
E(W:3’:wTbi0) - B(@’(Dis(X’)):X’: wbio). Then T - TDisbelongs to the kernel of V(E(@’:X‘: w!~,,): P I A , : A!o), so that, by S,,,
V(W&,(@:X: %io): =(@’IX’:
w/bi0): PIAi:
AL)(T - TDis)
= 0,
180
CHAPTER 5
from which it follows that v(w&,(@:I:Wio): a(@': x': wIbi0): PIAi: A!o)T = V(W&,(@:3:Wg3: E(@'(Dis(X')): X': w&~): P ( A i :A!o)TDis. Let A!o be a general point in UI('ok)at which VWk(@: X: PIA,: is defined. We shall prove that rank(V,,(@: X: P I A i : A!o)) = rank(V,,(@: X: PIAi)). Modulo SI and SI,,this will finish the proof of our proposition. Let q % ? kPr) : = {w+ E W&,(@:3 : Wi0): JEW+ is principal}. Obviously $(%k: Pr) c E(Wk). It has been seen above that rank(V,,(@: X: P I A i : A!o)) = rank(V(Wk): PlAi: A!o)). This statement can be refined to read rank(V,,(@: X: P J A i :AIo)) = rank(V(E(%',: Pr): PJA,:AI0)). One may argue as follows. In view of T-L, we need only show thaf Im(V,,(@: 3:PIA,: A!o)) = Im(V(W&,(@:X: Wio): E(Vk:Pr): PIA,: A!o)). As we already know that
Im(V,,(@: X: P I A , : AJ0)) = Im(V(W$,(@: X: Wio):E(Wk):PIAi: A!o)), it will be enough to establish that Im(V(W&,(G: X: Wio): { w t } : P I A i : A!o)) is contained in E(Wk:Pr): PIA,: A!o)) Im(V(W&,(@:X: qi0):
for all wt E E(Wk).For this purpose, fix wt E E(Wk) and set wk
= Wk(Dis(X,+)).
181
EISENSTEIN SYSTEMS
Let @’ and X’ be as before. Specify W!bi0 E
W&,(X’, X;Wib, W i 0 )
by first choosing an element W!bjO(W+)
E W,L( +
3 ’ 3
Xwt;
q i b , Wjo(wt))
and then putting
wI010 t . = Wio,o(wt) t,.
*
WjO0(Wt, ’
w+*
Note that W t E S(B:
X’: w!bio).
We now have Im(V(W&,(@:X: Wio): {wt}: PIA,: A/J) c Im(V(W&,((E:X: Wio): S(@’:X’: wJbio): PIA,: AIo)) = Im(V(W&,((E:3: W,,): S(@’(Dis(SE’)):3‘:w&,~):PIA,: A:,)) c Im(V(W&,((E:X: Wio): Ev6k: Pr): PJA,:A!o)), the equality being the thrust of Slll and the final containment being a consequence of the fact that ) s ( W k : Pr). E(@‘(Dis(X’)):X’: w & , ~c
To prove that rank(V,,(@: X: P J A , :A!o))
=
rank(Vy,(@: X: PIAi)),
it suffices, thanks to T-L, to prove that Im(V,,(@: X: P I A , : A!o)) = h(V(W&,(@:3: Wio): W&,(Dis(x))(@(DiS(X)): X: %io): PIAi: A!o)). Because (cf. supra)
Im(V,,(@: X: PJA,:Ab)) = Im(V(W&,(@:X: Wio):
=(%k:
Pr): PIAi: A!o)),
it actually suffices to prove that Im(V(W&,(@:3: Wio): {wt}: P J A , :A!o)) is contained in
182
CHAPTER 5
for all wt E S(%?k: Pr). To this end, we shall argue by induction on the length I(wt) of wt. [Note: wt has length zero iff
wt E Wtk(~is(x,,(@(Dis(X)): X: wi0I.I Fix wt E E(%?k:Pr) of positive length-then
E(qk:Pr),of length I(w7) - 1, and an element
there exists an element w&ioE
w,t E wtk(Xwt, 3’;%jo(wt),
%?Bib),
X’ being the principal element of @ associated with w Wt
/ such ~ that ~ ~
~
= w! * w;o ’ Wlbi0.
Here, w,‘is a “simple reflection” in the sense that w! induces a map in w&k(wk(Dis(Xw* wk(Dis(X’))) which, upon the insertion of specific data, corresponds to a simple reflection in the usual sense (cf. the Appendix to Chapter 2). We then justify the notation by remarking that the implicitly defined simple root determines a G-conk fitting into our general setup; in particular, C5‘ and X are as jugacy class v before. It is clear that Wt E
a(@’:3‘:Wt,).
Furthermore, any member of E(@’(Dis(X’)): X’ : wti0) has length I(wt)
-
1. This said, we have
Im(V(W&k(@: 3: GZi0):{wt}: PIA,: AJo)) c Im(V(W&,(@:3: giJ: B(B: X’: wii0):PIA,: A!o)) = Im(V(W&k(@: X: gi0): E(a‘(Dis(X’)): 3’:wiio): P I A , : AI0)), by S,,,. Since
E(B(Dis(X’)): 3‘:W!bio)
c
E(%k: Pr)
and consists of elements of length I(wt) - 1, the desired assertion follows by applying the induction hypothesis to wt. There remains the task of demonstrating S, and S,,. Let {(Pi.,s:.;A;.)}
be a set of representatives for \vk(P
IAi)*
EISENSTEIN SYSTEMS
I83
Givenjk such that Wk 3 W j b , there are two possibilities:
i"u;
3 %jb Wjb.
Let us suppose that the second possibility obtains. Each (Pjbvb 9 Sjbvb; Aiovb)
is then a dominated predecessor of a r,,-conjugate of some ( P i , , S:. ; A:,),say vb). There exist, therefore, elements yjovb E rMi and niovbE Niovb with index i'(jo, such that (Pi,(jb,v b ) , Si'(j0, vb);
Ai'(j0,v b J
3 ( P J b v b , S J b v b ; AJbvb),
where (PJ[& 9
= yjbvbnhvb(Pjbvb, Sjbvb;A jbvb)n;,!b
This agreement furnishes a partitioning {(PJbvbr
SJbvb;AJbvb)) =
i'
{ ( P J b v b , SJbvb; A J b v b ) l i *
into nonempty disjoint sets, where, as the notation suggests, each member of
184
CHAPTER 5
Recall that there is also an intermediate choice relative to Dis(X’), but it will play no role here. The preceding notation is then applicable in the obvious way, and therefore will be used without further comment. Let
where
Let
where
by specifying their action termwise. For this purpose, let W t ’ E W&((E‘: x’: Vib)
be given-then
It will be enough to define
{L&J!
on the components of the direct sum associated with wt and (i’, w”). Since =
“.*toio
X’,
it is clear that X,t
=
x:,..
185
EISENSTEIN SYSTEMS
Moreover, the symmetric algebras
i
s x vt(P,o"o.
-4jO"O)
ST.,+'(P;bvb. A&bJ
are isomorphic. Of course, here,
j o = jo(w+) = jb(w+') = jb
and vo
=
vb, with i ' ( j o , vb) (dHorn
=
i'. This said, our map will be taken as
QiJ;v;(.A!o)(?))
0
(Yjbvbniovb)-
'
A!J(?)h where . A!o is the appropriate image of A&. Observe that we are permitting ourselves a small abuse of the notation with respect to yibvb and nibvb,since the present yjbYb is in r and not in rM,,the present nhvb then being chosen to reflect this shift in emphasis (bear in mind that rM,= Mi n r - N ) . Let A!o be a general point in X(Pio,Aio)t-then, for each i', A/,, determines a general point A:;,i.in X'(PitIi,,Ai;li.)t and a point in X'(Pii,i,,Aibi,)t:Tacit restrictions will be placed on AIo as necessity dictates. We shall now prove S,.One has (Yjbvbnjbvb)
*
(&
Q:,bV6(*
v(E(e: 2:w&io):PI.4,: A!o): ($)H,,,,,
+ ($)a.
Write
Vw@: 3':P I A , : A t ) for
@ Vw@': X': Pibie1 Afbir:A&.). i'
One has
vw,(a': X': PIAi: A!o):
($')H~,,, + ($')a.
Form the diagram THom(A!)
1 ($')Horn
V V,,(V: 3':PI.4,: A!,)
1 To" ' ($'I@
This diagram is in fact commutative, as can be seen from Assumptions 2(V), 3(V), and 4(V). Consequently, rank(V(Z(@':3': = rank(V,@':
P I A , : A!o))
3':P I A i :A!o)).
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CHAPTER 5
Write
vwL(@':3':P I A J
for
@ VwL(@':x': Pibi'I Albi'). i'
The induction hypothesis implies that rank(V,@':
X': P I A , : A!o))
=
rank(VW,(6': 3':PIA,)).
But, for reasons analogous to those above, the last term is also equal to rank(V(E(6': X': wlbi0):E(@'(Dis(X')):X': wJbi0):PIA,: Ai0)). S, thus follows upon invoking T-L. We shall now prove S,,. One has
E(%(@':X ' :
wlbi0):
PIAi: ?: A/o: ?): ($)Horn
C'-(G/(T n P ) * Ai * N ) .
-P
Write E(W&(@':x': Wib): PIA,: ?: A/o: ?) for
1 E(W&(@':X': Wib): PIA,: ?: A!bir:?). i'
One has
E(W&'@': 3':Wi6): PIA,: ?:
?): ($')Horn
+
Cm(G/(T n P ) . A i . N ) .
Form the diagram
I
E(E(Q': X': wL,J: P [ A , : ?: A;,: ?)
($)Horn
E
THOrn(N")
($')Horn
C'(G/(T n P ) Ai . N )
E(W&(CF': X ' : Y,,,). P I A , : ?: A!,,: ?)
I
C' (G/(T n P ) . A i . N )
This diagram is in fact commutative, as can be seen from Assumptions 2(E) and 3(E). Suppose that V(E(6': X': w & ~ ) :PIA,: Ait,)T = 0. Set
T
=
THorn(A!o)T.
Then, thanks to the commutativity of diagram V, VwL(6': 3':P I A , : A!o)T = 0.
187
EISENSTEIN SYSTEMS
Let TI. be the i’-component of T-then
VwL((f’:3’:PibitI A&, : A/&)Ti,= 0. Consequently,
E(W&‘L(E’: X’: %?ib): PitlAip:Ti,:
?) = 0.
Assumption 4(E) thus implies that
E(W&‘&E’: X’: %?ib): PIA,: Ti,: A/bi,:?) = 0. Therefore, due to the commutativity of diagram E,
E(E(E‘: X‘: w&J: PIAi: T: A!o: ?)
= 0.
But then the usual reasoning leads to the conclusion that
V(W&((f: 3:Wi0): E@‘:X’: w & ~ ~PIA,: ) : A,6))T = 0, as desired.
Remark It is interesting to observe that the only place that E-S: I1 enters into the proof of the proposition is through T-L.
To help put the preceding developments into perspective, it will be a good idea to look at the canonical Eisenstein system. The following simple result, whose verification is completely elementary, provides a convenient framework for getting at certain functional equations (cf. infra); it will also play a technical role later on. Lemma 5.5 Let
be a matrix whose entries are linear transformations offinite rank. Thefollowing statements are then equivalent: (i) A and M have the same rank; (ii) There exist U and V such that B=AU C = VA D = CU = VB.
Corollary Let the notations and hypotheses be as above. Assume in addition that A 2 = A. Then, under either (i) or (ii), D = CB. [Proof: D = CU
=
VAU
=
VAAU = CB.]
188
CHAPTER 5
Example (The Canonical Eisenstein System) We recall that in this situation, X is the equivariant system of admissible affine subspaces attached to 5fio via the prescription X(Pio,Aio)= complexification of 6,,.
We shall now explicate the relevant data as they appear in the present setting. @jA), Viewed as a linear transformation from 8,,,(6, Ojb) to 8c,,(& V(6: 9 2 : g1:PIA,: wJ&jb:A]b) is defined by the matrix
P(O1 : PI(Ai, Ail: PjBvBIAjAvi:P j b v b I A j b v b : k j d v A * w ] A j h .kjb&: k j b v h * A j h ) I , or still, by the matrix
I
C C c u s ( P ~ ~ vA]&vA: A Pfbvb
I AJbvb: kjAvA
*
WjAjh *
kjb:b:
kjbvb
*
AJb)I,
which, of course, is simply C ~ , ~ ( P ] ~ P]hJA]b: I A ~ ~ : ~ ] d j b Ab). :
-
There is a canonical one-to-one correspondence
W$,(@: 3: Via)
jo
W(Ajo, AJo).
Furthermore, since here Vk(Dis(X)) = V,,, W~k(Dis(x))(~(Dis(X)): X: wio) * {lie},
liobeing the identity in W(Ai',,AJo).Fix indicesjh,ji. Let = {w:,
{El 8 2
=
(w:jA)-'I
{wi",,W]&,}.
One has W$,,,i~(x),(~(DiS(3E)):3: %io)
C
El,%2
C
w$,(@:3:
%io).
Therefore rank(V,,(@: X: PI A,)) = rank(V(Z2 : El : PI A,: AJ0)). In the case at hand, V,,(6:
x : PIAJ
is a one-by-one matrix whose single entry is
EISENSTEIN SYSTEMS
189
while V(&:
81:
PIAi: AJ0)
is a two-by-two matrix with left column C , . , , ( P ~ ~ IP!olAi',: A ~ ~ : lio: ATo) {ccuS(Pj~lAJ~: P/olA!o:wj*io:A/o)
and right column
is the identity matrix, it then follows from Lemma 5.5 and its Corollary that
C,,~(PJ~ I AJd: PJhI AJh: wJdi0wtjh: Ajtb) is equal to ccus(PJd1AJ~: P!oI A!o: wJai0:~!~jaAJh) ~ C , ~ , ( P ~PJhl ~ JA A ~ J~w:~! ~:~Aj'd), A:
which is just the daggered version of Theorem 4.4. We stress, however, that these considerations do not serve to give a new proof of the functional equations for c,,, ,since the functional equations and meromorphic continuation are established simultaneously for c,,, ,the latter being assumed here. Proposition 5.6 Suppose that for each io, Xi, is subject to
Geom: I Geom: I1 Geom: 111, Gr, and
E-S: I E-S: I1 E-S: 111.
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CHAPTER 5
Let p' = {pJo} be a completely equivariant system of polynomials. Let X E %(%k; W), say X E fio(%?k)--then E(X: PIA,: PioIAio:dHomp/o(A/o)(TO): A/o: x) = P / ~ ( A / ~ ) E P( X I A: i : P i o ( A i oTo: : AJo:x) and
V(X: PI(Ai, Ail: PjoIAj,: PioIAio: WJOio: ~ / o ) ( d ~ o m P / o ( h / o ) ( ' T ; . o ) ) = P/(A/~)V(X:PI(Ai, A,): PjolAjo:PioIAio:wJoi0: AlfO)TO. [This result is due to Langlands [p. 200].] Let us first consider a special case.
Lemma Suppose that X is, in addition, principal-then the proposition is true with W kreplaced by Wk(Dis(X)). [The proof is not difficult. We shall defer it for the time being simply because some new considerations will be required.] Proof of Proposition 5.6 It is clearly enough to make the verification at the general points of X(Pio,A,,)+.Accordingly, since the two equalities are then equivalent, we need only establish one of them, say the second. For this purpose, in view of E-S: I11 and the fact that
No> = P/o
PJo(WJ0io
it suffices to prove that the operator
d,
PJo
- pJo(wJoioA/o)
annihilates Im(V(X: PI(Ai, A i ) :PjolAjo:PioIAio:wJoio:A!o)). Replacing P, by a r - N,-conjugate if necessary and keeping in mind Assumptions 2(V) and 3(V), matters can be so arranged that the setup surrounding the previous proposition includes P i , and P j , among its data. Naturally, the notation there need not coincide with the notation here, but this is irrelevant. Furthermore, as will become apparent momentarily, there is no loss of generality in replacing the present X by a principal member of its equivalence class E, which, in agreement with our earlier notational principles, will still be denoted by the same symbol. This settled, let
pjo(Aj0) = @ Pjovo(kjov0 . AJo) yo
ElSENSTElN SYSTEMS
191
and form
If 8 be a subset of WJk(@:X: Via), put and then assign to the symbols
the obvious interpretation. Observe that v(9,: 8 2 : PIA,: /\~o)~dHomd2(l\!o) = d,g,pfr,(AI0) 0 V(EI : 8,: P J A , A!o). : The operator Vwk(@: X: PIA,: A!o) contains the initial V among its entries, so our proposition in the case at hand will be established as soon as it is shown that the operator d , P h k ( Q : x :w,")"
annihilates
-
P!o
X: P I A , : A!o)). Im(Vwk(@:
is contained The orthogonal complement of Dis(X)(Pio,AiJt in X(Pio, in X(P,,,,A,,)-. We can therefore write, in a self-explanatory notation,
+ A!o.
AIo = A!o(Dis)
Fix A!o-then,
with AJ0(Dis)varying throughout Dis(X)(P,, Aio)t, the rule pr(AJo(Dis)) = p!o(A!o)
defines a polynomial
pr which is clearly invariant under the operations of
W&k(Dis(x),(AioA,,); thus it determines a completely equivariant system pDis= {pjDd'"} of polynomials. Since the derivatives are taken perpendicular to X(Pio, Aio)-, one has 3
192
CHAPTER 5
on subsets S of Wt,(Dis(a))(@(Dis(X)):X: %iO), in particular on
S = Wt,(Discr,,(@(Dis(3E)): X: Wio) itself. The preceding lemma evidently implies that the operator Dis
d@h ' d ,pi.cz))(B(Dis(X)): I:Wio)(A!o(Dis)) annihilates
- pp(A!o(Dis))
Im(VWk(@: X: PIA,)).
But then the same is true of the operator d,P?J.:,(e:r:Wio,(~ito) - P!o(No).
Since (cf. Proposition 5.4) Im(V,,(@: X : PIA,)) =
Im(V(W$,(Dis(x,,(@(Dis(X)): X: Wio):W$,(@: 3:Wio): P J A , :A!o)),
it follows that Im(d"0m PLt,((E:I:Wi0)"
- d0(Ait,))
is contained in
Ker(V(W$,(Dis(I,,(@(Dis(X)): X: Wio):W&,((E: X: Wio):PI A i : AI0)), i.e. (cf. T-L), is contained in
3: P ( A , :AJo)), Ker(VWk(@: so that the operator d,
&; k(B:x: ,,o,(A!o)
-
dO(Ai9
annihilates Im(V,,(@: X: PIA,: A!o)), as desired. This completes the proof of our proposition. // Remark In a certain sense, one may view the equalities provided by the preceding proposition as the "functional calculus" for the E-functions and the V-functions. It is possible to use them to place a priori restrictions on the range of V; we shall not stop, though, to go into further detail here.
EISENSTEIN SYSTEMS
193
At this juncture, it will be necessary to shift gears so to speak. Up until this point, analysis has played little role in our considerations, but now it will move to the forefront of the discussion.The residue taking process furnishes a method of producing a new Eisenstein system from a given one. Moreover, at least for the Eisenstein systems of interest to us, the geometric conditions Geom: I, 11, and I11 will be preserved upon taking residues; the verification is fairly direct. The same will also be true of Gr and E-S: I, 11, and I11 but, in contrast, the verification is decidedly indirect. To carry it off, a new condition, An (cf. infra), will be formulated. It will be shown that An implies Gr and E-S: I, 11, and 111 (cf. Proposition 5.8). Because An is an analytic condition arising naturally in the “philosophy of residues,” determining when it holds becomes possible in practice. Our first task will be to define a space G 0 ( ( GI P)/r; & @ o )
of functions in LZ(G/(r n P ) A , . N ) , the so-called space of “partial wave packets,” formed from the E-functions in the canonical Eisenstein system X. For this purpose, let (Pi0 3
so that
Sio; Aio) E wio(P I Ail
( P , S ; Ai) 3 (Pi09 s i o ; AiJ
Write It for the common value of the f$ Let is the partial wave packet ciated with
01”E X&(B, Bi,,)-then
asso-
). that Oo;,(P(Ai)is the lift to G of a where A: is a point in ~ , ; o ( 6 , ’ , Observe hence wave packet on K x M i / {1) x rM,; O,;o(PIAi)~LZ(Gn / ( rP ) . A i . N ) , it being understood that LZ(G/(T n P) . A i . N ) is supplied with the inner product
194
CHAPTER 5
As a space of functions on G, Li,((G I P)p-; 40,) is independent of the choice of the split component A i . In the special case when P = G, we recover Li,(G/T; 6, no).It is clear that L : , ( ( G I P ) / ~a,,; 0:) 1L:,((G
Ip)/r;6,, 0 3
unless
6, = 62 = 0;.
{o: The sum
10 LE,((G I P ) / r ; 6, 001, 6
call it
Li,W I p ) l r ; 001, is evidently a G-stable subspace of L 2(G/(rn P ) . Ai N ) . Finally, there is an exhaustive decomposition L2(G/(T n P ) * Ai * N) = %k
C 1 0 Li,((G 1 P ) / r ; 00). >/YO Qo
Remark Fix a real number Rt > IIp[,ll. By the Rt-tube we mean the tube over the ball of radius R' with center zero in 61,).Denoting by SA$Rt) the space of all holomorphic functions in the R '-tube which decay at infinity faster than the inverse of any polynomial, set XAln(6,
R') = XA:,(R')
oio;
There is a strict inclusion oio) 4
u
R'
*Aio(6,
o gcus(6, oiJ oio;R').
Let a!, belong to Oio; Rt)-then, utilizing a limit process, one can show that it is possible to associate with a!, an element Oaro(PIAi)in t2(G/(rn P ) . Ai . N) which is, in fact, the L2-limit of partial wave packets formed from functions in XA;t,(8, Oio). The following inner product formula will play a technically important role during the subsequent discussion.
Lemma 5.7 Let f be a diferentiable function in Let (Pjo
L:,W
I P)/l-; 0,).
Sj,;
A,,) E wjo
9
195
EISENSTEIN SYSTEMS
4,
where the AL ure elements in the complexiJcation of and the Y hare element in C[ajo] 0 b,,, (6, Ojo). Let YjoE XAj0(6, Ojo)-then
(1,@YJ;o(p I Ail) = 1 (rh? h
dy,(- x',)).
Proof We begin the computation with an examination of the left-hand side of the purported equality, which, to within the factor (2n)It, is equal to
or still
It will be enough to show, therefore, that
(2nyt . (Yh, d Yj)(- XI)) is equal to
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CHAPTER 5
For this purpose, fix the index h and drop it from the notation. Write
r = C u j 8 aj i
(uj 8 Qj E CCaJoI6 R A 6 , oj,J)*
Fix the indexj and drop it from the notation. We must then prove that (24'' (u
6 a, dYJo(- At))
is equal to
Since (u 8 a, dYJo(-At)) = (0, d,*YJo( -AT)),
the right-hand side of this equality being, by definition,
we need only establish that
which, however, is a simple exercise in manipulating Fourier transforms. // For the proofs which are to follow, it will be necessary to set up some machinery tailored to the present context, but entirely analogous to that encountered in the theory of Eisenstein series associated with cusp forms (cf. Langlands [pp. 121-1221). Fix a real number R t > ~ ~ pas~above. o ~ ~Because , the IIp/oll are all the same, no matter what the choice of the index i,, we shall denote their common value by llptII. Let f t = {fJo} be a completely equivariant system of bounded holomorphic functions in the Rt-tube, i.e., to each j , there is assigned a bounded holomorphic function j J oin the Rt-tube such that for all triples (Pjb, Sib; Ajb) (Pjd, SjA; Aid)
197
ElSENSTElN SYSTEMS
standing in the relation
one has
fb(w&
j,:
AT$ = fJA(AJA)
for all
AjA). To such an f there is assigned a bounded linear transformation 3(f t, on L$,((G I P)/l-; 6, U,), characterized by the relation WjAjA
w&k(AjA,
3(ft)@,:,(PIAi) = @,jo,lo(PIAi)
(a!,
EzA:,(a,
oio;
io being any index. One has
I13(ft)II 5
IIftolIm.
If
f+' = where '
{fj;},
ff;(qo> = fjo(-Ajo),
then
3(f')* = 3(ft'); hence 3(ft) is actually a bounded normal operator. It is easy to check that 3(f t, intertwines the G-action on L$,((G I P)/r; 0,). A central part in this theory is also played by an essentially self-adjoint operator 0+, usually unbounded, characterized by the relation
Ot@,:,(PIAi) = @,?,?,,:o(PIAi)
(@Io~ z A : , , ( 8 ,Uio; Rt)),
where (
?)@!,)(No) =
No)@t(No)3
i, being any index. In terms of representation theory, the Casimir operator o
is defined on the space of differentiable vectors in L$,((G I P)/l-; 0,)and, as such, is essentially self-adjoint.One has
0'
=0-
Ct(O),
ct(o) being a certain scalar. The spectrum of 0' is contained in the interval 3 - m, (lpt11 2]. This may be seen as follows. Let R ( [ ; Elt) be the resolvent of
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CHAPTER 5
0 at the complex number (. Now fix a complex number c with Re(c) > (R ' ) 2 . Let fjo,cbe multiplication by 1
5 -
ft
wc; 0') = w:),
the arbitrariness of R' implies our spectral assertion. This is a good place to tie up a loose end, viz., that part of Proposition 5.6 which was left open in the form of an unproved lemma.
Lemma Suppose that X is, in addition, principal-then the proposition is true with %k replaced by %k(DiS(X)). Proof In this situation, WPio, Ai0It = {X(Pio, Aio)>, the single point X ( P i o ,Aio)which, of course, is general. The two equalities in question being then equivalent, we need only establish one of them, say the first. Let pt = {pJo}be a completely equivariant system of polynomials-then we must prove that
E(X: PDis(X)IAbis(X):
dHomP!o(X(Pio,Aio))(To): X(pi0, Aio): x) = Pi',(X(Pio,Aio))E(X: PDis(X) I ALis(X): Pi0 I ~ i o T: o : X(Pi0, AiJ: piol A i o :
By hypothesis, the E-functions vanish if ord(Tn) % 0. Select R t large enough to guarantee that X ( P i o , Aio) belongs to the Rt-tube (computed per Dis(X)(Pio,Ain)'). Let f J o be an invariant bounded holomorphic function in the R '-tube with the property that UJo -
PJo)(X(Pio, A i J )
vanishes to such a high degree as to secure the relation
E(3E:'Dis(X)IAbis(X): p i o l A i o : d H o m f f o ( X ( P i o , A i o ) ) ( T o ) : x(pio, Aio): x) = E(3:PDis(X)IAbis(X): PioIAio: dHomP!o(X(Pio,Aio>)(T,,):X(Pio,Aio): X) for all To.Let f = {fJo}be the corresponding completely equivariant system of bounded holomorphic functions in the Rt-tube. We can and will assume in
EISENSTEIN SYSTEMS
is contained in the kernel of
Taken together, these statements yield the lemma.
199
200
CHAPTER 5
Choose a positive integer N such that the degree of
Taking into account the preceding computations, induction on n leads to the equality
20 1
EISENSTEIN SYSTEMS
is orthogonal to every
1 Abis(X)),
oY]o(pDis(X)
hence is zero. But if a vector is in the kernel of some power of a bounded normal operator, then the vector is in the kernel of the operator itself. In particular, IE(3: PDis(X) I which is (1).
Pi, I Aio: To:X ( P i 0 ,A d : ?) = 0,
Proofof(2) Form the difference between the two sides and compute the inner product of the result against any
1 Ahis(%)>.
oY~o(pDis(X)
Using Lemma 5.7, we get the expression
c
w;oioEW:*(Di.,I))(X:
AJO. Ai,)
(da ffo(wJoioX(pio,
(v(3:PDis(X)I(Abis(X), -v(x: pDis(X)I x
(dHom
P i o I A i o : wfoio: x ( p i o ,
Abis(X)): p j o l A j o :
(Abis(X), Abis(X)):
'io))To)
pjol Ajo: PioIAio: WJoio: x(pio, Aio))
f!o(X(pio, AiJ)(To)), dyIo( - WJoioX(Pi0,Aio))),
which, in view of our assumptions and E-S: 111, is zero, thereby proving (2). The lemma is thus established. // Suppose that for each io, JEi0 is subject to Geom: I Geom: I1 Geom: 111. Fix an R with the property that all the normal translations are in the R-tube. [In view of Geom: 11, this is clearly possible. The point, of course, is that one number, viz., R, is being used simultaneouslyfor all gk.] Introduce the set
> 0) consisting, by definition, of all points of the form IB:(Wpio, Aio))
X(Pi0, AiJ
+
(E
-NO,
where A t is in the projection of 3 ( P i o ,Aio)" onto it!o and has norm less than E. Observe that here X ( P i o ,Aio) is not necessarily real (cf. Proposition 5.3). Write bt for the common value of the dim(3(Pi,, Ai0)').
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CHAPTER 5
Let us recall that one may associate with each 010
E xAlo(6,
oio; R)
an element
@,i0(P IAil E L:o((G Ip)/r;6 , 0 0 ) *
If p b is any complex-valued polynomial on JfA:,(d*
oio;
we understand the subspace of which are divisible by pib, i.e., the Set Ae = R 2 - E~
@!o
P!O
*
(E
> 0).
(@o!
then by
R ; P!J Oio;R ) made up of those functions
E Jf'~:,(d,
oio;
R)).
An Suppose that
Then: There exists an orthogonal projection
Q ~L:,((G : I wr;6,0,)
+
L:,((G
w r ; 6,0,)
onto a closed subspace, commuting with the 3(ft), and, for every E > 0, a family of polynomials pi = { P J ~ , (depending ~} on E ) such that
I
P10.e g(pjo,Ajo)t
#
(9 %jo(Vk))
satisfying the following conditions. Let @ o!
E x A : o ( 6 , Oio; R ; ~ 1 0 . e )
If 3E E %io(%k)has the property that
(V(X: PI(Ai,A i ) :PJoIAjo:PiolAio:wfoio:A]o)d@!o(A!o), d ~ J o ( - w J o i o ~ ~ o ) ) is holomorphic in a neighborhood of ZB!(3(Pio, Aio)).In addition: An, The differencebetween
(WC;0')QP
@@jo(P I
@,j0(P I Ail)
EISENSTEIN SYSTEMS
203
and
summed over all X E Xio(Wk)having the property that Re((X(Pi0,
AiJ,
X(Pi0, A i o ) ) ) > 1,
and over all wjoi0E W&,(X; A,, Aio),can be holomorphically continued to the region Re([) > ie. An, The difference between ( Q ~ @ s i ~ ( P l A i ) R(C; ,
and
S'
(2n)b'
nt)@,;o(f'IAi))
(V(X:P I ( A i , A i ) :PjolAjo:PioIAio:wJoio:
I B : ( X ( P , , . Ato))
summed over all X E Xio(Wk)having the property that Re((X(Pi0, AiJ, X(Pi0, A i J ) ) > 1, and over all wJoioE W&,(X; A,, Aio),can be holomorphically continued to the region Re([) > i t . Remark It is tacitly assumed that QPis independent of the indices io and
j , . QP is not necessarily unique.
Some clarifying comments are in order. Write
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CHAPTER 5
Hence, if then
Re((X(Pi0, AiJ, X(Pi0, Aio))) >
)be,
IIWX(Pi0, Aio>)It < Consequently, the normal translations X(Pi,, AiJ figuring in An, and An, must lie in a compact region. It therefore follows from Geom: I that the sums involved are actually finite. As for the analytic conditions, observe that by general properties of the resolvent, the inner products
( R ( l ;0t ) Q @,:,,(P ~ I A i l @,j0(PI Ail) {(QP @.:o(P I Ail, NC; 0')@,j,(P I Ail) are holomorphic in the region Re(l) > A,, except, perhaps, along the segment ]A,, IIpt1I2]. Now ask: When do the integrals become singular? Under the assumption that they actually exist, any integral per an X with Re((X(Pi0, Aio), X(Pi0, Aio)>) 5 A, is automatically holomorphic for Re(5) > A,. Ofcourse, these are the integrals not taken into account by An, and An,. In practice, E will be allowed to approach infinity. Consider Fig. 1. Here the dots represent the various
u x x x x x X. o(xxxxxx
xx xx xx xx xx
1
cwcwMc*
I lptl I
(XXXX.
uxxxxxxx
XX.
XXX.
FIG.1
R2
t
X
ElSENSTEIN SYSTEMS
205
Off of the finitely many thatched segments, the integrals are holomorphic. An, and An, postulate that the difference between the inner products and the integrals can be holomorphically continued across these segments. Remark Since QP commutes with the 3(ft), QP also commutes with R ( [ ; Elt). Obviously, R ([;Ot)* = R ([; Elt).
Therefore the inner products involved are equal. It will appear, a posteriori, that the same is true of the integrals. Proposition 5.8 Suppose thatfor each io, Xi, is subject to
i
Geom: I Geom: I1 Geom: 111
and
An. Then for each io , Xi, is subject to
and
{
E-S: I E-S: I1 E-S: 111
Gr. [This result is due to Langlands b.2033.1 Surprisingly enough, the proof is not especially difficult. We shall preface the discussion with an auxiliary result (cf. Lemma (A) infra) which will serve as the technical vehicle for the demonstration. Let us begin by dispensing with a technicality. Lemma Let X E Xi0(Wk). Suppose that
Let A!o be a general point in X(Pio,A,)'-then
the set
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CHAPTER 5
set
Needless to say, it is still true that Yjz E %A;0.] Since AI0 is, by hypothesis, a general point, the various wjoiohI0are distinct. But then it is not difficult to see that one can choose YJoin such a way that I* t 0# duw:oioy,o (WjoioA!oL wioio
which is a contradiction. //
Lemma (A) Let X E ili0(Vk). Suppose that
EISENSTEIN SYSTEMS
207
Let there be given a function A(wjoio: on w$k(x;Ajo,
AiJ
x X(Pi0, AiJ'
which is a linear trans$ormarion from Hom(S.%(P,o. Ato), gcus(6, 03) to %P,,,
A , ~ ) ~ ; , ,@ , ~gcus(4 o j o )
such that (A(Wjoi0:N J T o y Tj,) is meromorphic on X(Pio,Ai0)+and vanishes identically ifthe order of is suficiently large. Let @!o {vjo
E Jf',-~~~(fi~ %A,,
+
Toor To
o i o ; R ;P ~ O , (6, ojo;
w.
Make the following assumptions: d'J'jo( - w J ~ ~ ~ K ! ~ ) ) (A(wjoi0:AJo)d@]o(A!o)y
is holomorphic in a neighborhood ofIB!(X(Pio,Aio)) and
summed over all wJoioE W&,(X; A,, Aio),can be holomorphically continued to the region Re(() > i t .
Conclusion : If Re((X(Pi0, Aio), X(Pio, Aio)>) > A t ,
then A(?: ?) = 0. Proof We shall proceed by contradiction. Suppose that A(wJoi0: A!o)To ord(To) > N , but that for some wjoioand some Toof order N , A(wJoio: A,',)T0 # 0 on some open subset of X(Pio,Ai0)'. Fix such a To.Select an element = 0 if
Ti0 E CC6!oI @ &cus(fi,
oiJ
208
CHAPTER 5
with the property that ord(K;, - dYio(?)) > N on X(Pi,, AiJt. Given q], E C[d{,,], put
.
~~f~
= pf,, q/o
Qqi0
E HAio(J, oio;
'!)
e("9
. r.
10.
Obviously,
R ; ~ 1 0A* ,
Moreover, it is clear that ord(d@,,l0(?)- p,',,, . q!o . e(?.?) KO) > N on 3(Pio,Ai0)t. Consequently,
is equal to
EISENSTEIN SYSTEMS
tZ
+ (S
-2J-It
- Re((X(Pio, Aio), X(Pio, Aio)))
209
+ (A]o,
Integrate this expression with respect to s from Re(lo) to C-then we get 2J-1 times thedifferenceof
210
CHAPTER 5
Put EO
= (Re((X(Pi0, Aio), X(Pi0, AiJ>)
- Re(CO))"*,
a positive number which is strictly less than E. Then the limit as t 10 of the preceding difference is a.e: equal to - 2 x J - I times the characteristic funcAio)). Consequently, the integral tion of IB~o(X(Pio,
I
t
f(A!o) . e("lo*"d. q/o(A!o)1 dA/oI
~ B t $ ( W t 40)) ~,
vanishes.Duetothearbitrarinessofql0,weinferthatf = Oon IB~o(fi(Pi,,Ad), hence on all of 3E(Pio,AiO)'. Using the density statement supra, it is then a simple matter to contradict the supposition that for some wjoio,A(wjoi0:A/o)To # 0 on some open subset of X(Pi,,,143'. // We shall also need a classical result from operator theory. Spectral Lemma (S-L) Let T be a self-adjoint linear transformation on a Hilbert space V, R(C; T )its resolvent. Let { E(A)}be the resolution ofthe identity for T-then, for all real p and v with v < p,
+ { ( W L9) + (W- O)f,
9)) - m ( v ) f , 9)
+ (E(v - O)f,g)}
is equal to
lim 010
1 ~
2sJ-1
s.(fl,v,",v)
(NC;T)f,9 ) 4,
where &(p, v, u, u ) is the contour
v+J-1ut V
-
G
V-J-IU
U
p-J-rv
p - o u
We shall now tackle the proof of our proposition. Fix X0 E xi0(%k).[The subscript has been appended momentarily in order to allow X itself to be a running variable.] Pick an E > Q with the property that Re((XdPio,Aio), XdPio, AiJ>) > It has been pointed out above that there are but finitely many X E xio(%k)such that Re((X(Pi0, Aio), X(Pi0, Aio)>)>
21 1
ElSENSTElN SYSTEMS
or finitely many 9 E Xjo(Wk) such that Re((Y(Pj0, A,,), Y(Pj0, Ajo)>) > For these X and 9 we can therefore choose a polynomial qJ0E C [ ~ I /whose ~] restriction to Xo(Pio, Aio)tdoes not vanish, but whose restriction to 3(Pio,Ai0)+ (3 # 3,) vanishes to such a high degree that the composition of i b E .
V(X: P ( ( A i ,Ai): PjolAjo:PioIAio:wJoio: with dHom q!o(A/o)
is null and whose restriction to gw;o,o(Pi,,, such a high degree that the composition of
(?)w!o,o
# 3,) vanishes to
d 8 q!,,(w!,ojo~Jo)
with
V ( g : P J ( A i Ai): , PioIAio: PjoIAjo:w!ojo:AJ0) is null. This in place, let us indicate the strategy. We want to verify E-S: I, 11, and 111, as well as Gr, for 3,. Because An involves not just Xo alone, we must devise an artifice to split off To, as it were, from the rest of the pack. The properties of q1!,, in conjunction with a little manipulation, will accomplish exactly this. In An, take,
@lo
where, of course,
{
--
= ~ l ’ , , e. q!o * @o!
YJo= Pjo.E* vjo,
a/o ExAfo(d,
oio;R )
{qjoE xAjo(d, ojo;R). In the second difference(i.e., An?), first interchange the roles of io andj,, then replace [ by c, and finally conjugate the inner products. Subtract the resuft from the first difference (ie., An,) to get, after multiplying through by (2n)”,
V(3: PI(Ai, Ai): PjolAjo:PioIAio:wJoio:A\t)
212
CHAPTER 5
and over all wjoi0E W&,(3;A j o ,A,,), less the conjugate of
summed over all 9 E JEi,(‘iR,) having the property that Re((Y(Pj0, AjJ, Y(Pj0, Ajo))) > and over all w!,~,E W&,((D;A,,, Ajo).By hypothesis, the expression in toto can be holomorphically continued to the region Re(() > &. In view of our assumptions on q1,;in the first part only 3, itselfremains. Taking into account the fact that the adjoint of dB qito*(wit,joA,’J is
dHomq!o( - w!ojoA,’,-,), in view of our assumptions on q!,, in the second part only those ’2) and w t j 0 such that (Dw;?,, = 3, remain. At this point, it will be convenient to change our notation slightly and write 3 for 3,. Recalling that
(V?, *Y)= (V*?II,?), our expression reduces to the integral over IB;(X(P,,, Ai0))of (dBp,’~e(w,’oioA!o) 0 V ( 3 : P I ( A i , A i ) : PjoIA;,: PioIAi,:wjoi0: A!,)
summed over all wjoi0E W&,(x;A,, Aio) such that 3 w ~Eoiit j,, ( W k ) . The verification of E-S: I and E-S: I1 for 3 is now immediate. In the notations of Lemma (A), let us define a function A(wJoi0:A!,) according to the following
ElSENSTElN SYSTEMS
213
prescription. Let wJoioE W&,(X; A,, Aio).IfX,jo,o # xjo(%k), let A(wJoio:AI0)be
V(X: PI(Ai, Ai): PjolAjo:PioIAio:wJoio:A!o), composed on the left with d , p J ~ E ( ~ ~ o i oand A ~ oon ) the right with dHo,q~o(A~o). If XWj0,,E xjo(%?k), let A(wJoio:A!o) be the difference of
V(X: PI(Ai, A i ) : PjolAjo:PioIAio:wJoio:A!o) and V(Xnfoio:PI(&
A J : PioIAio:PjolAjo:(WJoio)-l: -wJoioXit,)*,
composed on the left with d,pJ&(w~oioA!o) and on the right with dHomq!o(A!o). Lemma (A) then implies that Since
r
A(?: ?) = 0.
d , pJZ&(~~Joio~!o) q/o(A!o) are isomorphisms for a.e. AJ0, the validity of E-S: I and E-S: I1 is thereby established. There remains, then, the task of addressing E-S: 111. For this purpose, note first that for any ft, dHom
(R(C;ot)Qp3(ft)0,~o(PlAi), %j0(PIAi)) = (R(C; 0 t ) Q @,fo(P ~ I 3(ft*)@,jo(PI Ai)). Working only with the first difference (i.e., An,), take as data
and then subtract. The terms containing the resolvent thus disappear and so, proceeding as above, after multiplying through by (27~)’~ we are left with
summed over all wJOioE W&,(X; A,, Aio),where A(wJoio:A!o) is, by definition, the difference d ~ f J o ( W J o i o AV(X: ~ o ) ~PI(Ai, Ai): PjoIAjo:PioIAio:wJoi0: - V(X: PI(Ai, A ( ) :P j o l A j o :PioIAio:wJoi0:AL)odHomf!o(A!o) composed on the right with dHomq!o(A!o).It follows from Lemma (A) that A(?: ?) = 0.
2 I4
CHAPTER 5
A simple approximation argument, which has been used explicitly before, then yields E-S: 111. We have yet to consider Gr.
Lemma Let X E 3Eio(vk)-then w t k ( X ,
X; q i o , WiJ
z 0.
Admit the lemma for the moment-then the fact that Gr holds can be seen by the following argument. Note first that Proposition 5.3 is now in force. The equivalence class in 3E(V,; go)to which X belongs thus contains a principal member, say 9 E jejo(%k). One has
wt,($, X; q j o , q i o ) # 0 iw: 9;%joy qjo) k(Dis(g))@,
#
0
9
say
{
3; q j o , WiJ E W t k ( D i s ( g ) ) ( 9 , 9 ; q j o WjJ
WJOio E $0
wtk('D,
3
Because 9 is principal, wyohas the properties postulated in Gr per 9;but then W E = (wjoio)-'WO JO w? Jot0 . has the properties postulated in Gr per X. Proof of Lemma Suppose the lemma to be false. Assuming that io = j , , take in the first difference
-
qlo aq0. For the usual reasons, all the terms not involving X drop out; in addition, the terms involving X itself also drop out, an empty sum being zero. We conclude, therefore, that = p!o,E*
(R(i;
0' ) Q p @,jo(P I Ail, @,jo(P I Ail)
or still
(R(i; n t ) Q p @ ~ ~ o ( P I A i )Q, P @ @ ~ ~Ail) (PI is holomorphic in the region Re(5) > &.Because R 2is not in the spectrum of El1,it follows from S-L that
((1 - U & ) ) Q p @ o t o ( P
is equal to lim lim
1 ~
I A i l Q P @ , ~ ~ ( PA,)) I
(R(C;n t ) Q p @,lo(P I Ail, Q P @,lo(P I Ail) 4
EISENSTEIN SYSTEMS
215
i
R d O = X, FIG.3
where (tc,ais the contour shown in Fig. 3. As the integrand is holomorphic, the integral tends to zero with 6. Consequently, E(Ie)Qp@olo(PIAi)= Q ~ @ g l ~ ( P l A i ) * This, of course, implies that
(R(5; 0 ' ) Q p @,;to(P I Ai), @vJ~(P I Ail) is holomorphic in the region Re([) > Ie for any YJo. Therefore
V(X: PI(Ai, Ai): P j o l A j o :PioIAio:wjoio: A!o)
summed over all wjoi0 E Wf,(X; Ajo,Aio), is holomorphic in the region Re([) > I , . It then follows from Lemma (A) that
V(X: P I ( A i , Ai): PjolAjo:PiolAio:wJoi0: A!o) ~ d ~ , , q ~ o ( A ~ o ) and hence
V(X: P ( ( A i ,Ai): PjolAjo:PioIAio:wJoio: A$ is zero for all wJoioE W$,(X; A,,, Aio). This forces the conclusion that E(X: P I A , : PioIAio:?: ?: ?) = 0.
216
CHAPTER 5
By making a suitable replacement of the data if necessary, this fact can be used to contradict the nontriviality of the Eisenstein system belonging to 3.In other words, the set Wik(3,X; Uio,Wi0)is nonempty, as was to be shown. //
I
To summarize: E-S: I
Geom: I Geom: I1 AND An Geom: I11
E-S: I11
Remark It is more or less obvious that the canonical Eisenstein system satisfies the conditions on the left-hand side of the implication. We shall now consider the problem of constructing certain closed subspaces of L:,((G IP)/l-; &@,) by means of the machinery introduced above. In spirit, the results obtained are entirely analogous to their counterparts in Chapter 4. They play an important role in the proof of a major theorem of Langlands (cf. Theorem 5.12 infra). We place ourselves in the setting of the formulation of An, which we assume to be in force (along, of course, with Geom: I, 11, and 111). Bearing in mind Proposition 5.3, we can then say that 3(Uk;U o )is a finite set. Furthermore, the normal translation in any member of %(%?k; go)is real. Finally, every equivalence class in X ( % ? k ; U,) contains a principal member. This being so, let X w k ; Uo) = a, be a decomposition of X ( % k ; U,) into equivalenceclasses (35,. Select a principal Set element 3,in (f,,say Xi E 3io(i)(Uk). 0,' =
# (wtk(xi
31;
Uio(i), Uio(i)))-
Write
Supposing now that io and E vary, let L:,((G IWl-; 4 00;3)
be the closed subspace of L:,((G I P)/l-; 6,O0) spanned by the ( I - E ( & ) ) Q P @ ~ ~I Ail ~ ( P (@!o E J f ' ~ f ~ (o6i o, ; R ;P ~ O .el), (Pi,),Sio;Aio)being, as always, a generic member of Uio(PIAi).Let
I W r ;&@,)
Li,((G I P ) K ; 40,;2) be the corresponding orthogonal projection.
Qt: L:,((G
+
EISENSTEIN SYSTEMS
Let Given I , let us agree to write
Then the function is holomorphic on
217
218
CHAPTER 5
Let (pE(dX@!,)equal
[This result is due to Langlands [p. 21 11.1 These two results are, of course, closely related. For the moment, let us remark only that the second is proved first. We begin with a spectral computation.
Lemma Suppose that E 9 0. Let @ ,! E ~ f A j , ( 8 90 i o ; R ;P!,,
iPj, € s A f 0 ( 8 , 0,; R). +
Then
A
ElSENSTEl N SYSTEMS
2 19
is equal to
being, as before, the contour shown in Fig. 4. By our assumptions, the difference between
(R(i; and
t)Qp@@!o(P
IAi)?@YJ,,(P I A,))
220
CHAPTER 5
summed over all 3 E xio(%?,() and over all wjoioE W.&,(X;A,, Aio), can be holomorphically continued to the region Re(C) > I,. Observe that it is a question here of all X E Xi0(Wk),as is permissible. Adding and subtracting the obvious quantity, we can therefore say that ((1 - E(U)QP@,;to(P I A i l @,j0(P I Ail)
is equal to 1 lim lim . T+ C I A , 610 (27C)
~. 1 27C-
Jac, ~ L d ( X ( P i oAio)i( . x d[
O "
i- (?, ?>
V(X: PI(A,, A i ) : PjolAjo:PioIAio:wjoio:
](A/o), d Y ~ o ( - w ~ o i o A ~ o )
summed over all 3E E Sio(%?k) and over all wJoioE W$,(X; Ajo,Aio). Using E-S: 111, bring (c - (?, ?))-' to the outside and compute 1
thereby obtaining
EISENSTEIN SYSTEMS
22 1
Observe that there is a natural map d, (sic) which assigns to each element
2122
CHAPTER 5
Point (1) is, of course, a simple consequence of E-S: 11. Points (2) and (3), however, are not so immediate; it is necessary, for a complete verification, to dispose of some rather fulsome analytic details. The key is the spectral computation supra, in conjunction with an approximation argument. Owing to a variant of the Stone-Weierstrass theorem, to be discussed later, it is possible to find an open dense set
D,c I@o(%(pio(,),Aio(i))), clonsistingof general points at which
(v,(A!o(,))?,?"I is defined, with the following property. Let E > 0 be given. Let
F,: zB!(x,(pio(,), Aio(t)))+ Hom(S,, &cus(6, OiJ) be a continuous function, with support contained in D,, whose components
E
wt,(@, : 3,: w i o ( t ) ) i o )
satisfy a symmetry condition, namely, =~w?ow~o(,)~~~(,~~!o~I~~ (Sym), Fwlo(W:(,)w!o(,)A!o(,)) for all w!o(,)
E
wt,(x, 3,;w i o ( r ) * wio(iJ 9
Such symmetry is evidently possessed by the components of the d,@g. This said, it can then be shown that for every N 2 0, F, is the uniform limit of certain d,@/o, modulo Homs of order > N , the d,. ( I ' # I ) having order :z N . By shrinking D,if necessary, one can arrange that the approximation is actually effected by certain d,@lo, being divisible by pl0,&. We shall now establish (2); (3) may be discussed similarly, hence need not he dealt with in detail. Suppose that E 9 0-then E , is defined. Obviously: E , < E. Since I - E ( I E )and QPare both projections which commute with one another, the product ( I - E(AE))Qpis also a projection. Consequently, thanks to the spectral computation supra,
'S
OlC-.
(v,(A!o(,)NI@!o, dI@!o)I dA,to(,) I
d , ( % ( P i o ( ~A ~i os( d )
for all R ; plo, Choose N 9 0 so large that (V?', ?") = 0 when the orders of ?, ?,, are greater than N . If, therefore, d,@!o + F,, then still @ o!
EsAl0(8r
(V,(Aito(,))F,F,) I d N 0 ( l )I . 9
ElSENSTElN SYSTEMS
223
-
2;!4
CHAPTER 5
be the corresponding orthogonal projection. The Qx are expansive since, as will be seen below, c1 > e2 Im(QS)
so that Qx = lim
Qk
t+m
= WQ3,
(strong operator topology).
On the other hand, fix i, and allow E to vary-then by -Go((GIP)/r;6,@0;W i o
we understand the closed subspace of G o ( ( GI P ) / T ; 6900 ;3)
spanned by the (1 - E ( ~ ) ) Q P @ ~ ~I,Ail ,(P
(@!o
Ex
~ j ~ (o 6i o ,; R ; P!O,
A)*
Let G , ( ( G IP)/r; a,@,; 3;E l i 0
be the closed subspace of J%O((G I P)/r;6,@0;W i o
obtained by freezing both io and E. It is clear that
lJLio((GI P)/r; a,@,;
JE; E l i 0
E*O
is dense in G o ( ( GI P)/r; 6,oo;
W i O .
In order to provide an explicit realization of the space G 0 ( ( GI P)/r; 6,@0;JEIi,,
as well as its truncation Lio((GIP)/r; 6300;JE;
Eli0 9
il. will be necessary to introduce another class of Hilbert spaces.
Let
y i o ( ( G I P)/r;6,@0;
Wio
be the Hilbert space made up of those collections F = (F,) of measurable functions F,: lBL(JEI(Pio(i), Ai0(1)))+ Hom(S1, 6 c u d 6 ,
oio))?
EISENSTEIN SYSTEMS
225
whose components are subject to (Sym),, with
the inner product then being the prescription
Let E B 0-then we can repeat the precedingdefinition to form a Hilbert space Yio((GIP)/r;6 7 8 0 ; 3;E)io,
where now the integration is carried out only over the IB,t,(Xi(Pio(i) Aio(iJ). 9
It is clear that is dense in
E
lJ Y i o ( ( G IPYr; 6 9 0 0 ;3;E l i 0 *O
~ i o ( ( G I p ) / r ;6 3 0 0 ; s ) i o *
Given @!o E
x A l o ( 6 , o i o ; R),
let dx@!o
=
(d,@!o).
Then, on the basis of the approximation argument indicated earlier, the divisible by p!o,e, are dense in restrictions di@!o, with 2iO((GIp)/r; 6900;3;E ) i o .
Consider the diagram
226
CHAPTER 5
The spectral computation supra implies that IIQZ@,fo(PIAi)II
=
IIdi@!oII.
Consequently, the dotted arrow may be filled in to establish an isometric isomorphism Lio((G I P)/r; 6 , 0 0;JE;
E l i 0 --t
y $ o ( ( G I P)/r; 6 9 0 0 ;
E)io
9
the space on the right then providing an explicit realization of the space on the left. Thanks to the expansiveness of our projections, these considerations then lead, in the obvious way, to an isometric isomorphism L$o((G I P)/r; 6 , 0 0 ;W i o
+
y$o((G I P)/r; 6,flo; W i O -
Lemma Let @!o E
x A f o ( 6 , Oio; R),
"hen the restriction dfr@i',belongs to p i o ( ( G I P)/r; 6900;JE; [Note: This is not a priori evident since p,',,, need not divide @J0.]
Proof Let f stand for the projection of O,io(P I A i ) onto L$,,((G IP)/r; 6 7 0 0 ;
Eli0
and call F the corresponding element of
Yio((GIP)/r;6 9 0 0 ; JE;
s o that We shall prove that Let
llfll
&)io,
= IlFll.
F = a:@!,, (a.e.). P/o,&(NoE JQo(8r o*o; R ; do,,).
Then we have 0 = (QZ @pro. .ero(P I Ail, f - f ) = (Qi @pio, .m,t,(P IA i l @,r0(P I A,)) - (QB Opto,.&.,t,(PI = ( Q i @ p ~ o , . ~ ~ o ( ~ ~O~ i, ), ,t o ( ~ ~~ (di~i',,c6/0, i)) F), t:hat is
'S
O = COIi . I
~Vl(~ito(l))dlP!o, &&it, dl @;o - Fl) I"1)
l d , ( X ~ ( P i ~ho(nd) ~t~.
I
f)
I
EISENSTEIN SYSTEMS
227
for every pit,.e &o!
EzAlo(d, oio;
R ;pit,, e>.
Given Ti0
E H W S i3
&cuds, oio>),
it then follows from the approximation argument used before that (Vt(A!o(i))Tio,
di@!o
- FJ = 0 (a*e.Aito(i)).
Since functions which agree a.e. are identified, we are done. // Observe that the preceding lemma implies that the QE are expansive. Let @ o!
EXAfo(d, oio;
R)*
Then we have IIQx@~~o(PIAi)I12
2
IIQE@~~o(Pl~i)l12
Because the integrands are nonnegative, we can let E approach infinity to get
2;!8
CHAPTER 5
is integrable (hence holomorphic) on
IBt,(K(Pio(i) Aio(i)))* 7
The inner product (Qx @,,to(P I A i l %l0(P I Ail)
can be evaluated by the limit
I AiL @vl0(f'I4)), lim(Qi @alo(P
E+W
the latter being, however, exactly 1
(Wf.
1
? 3.J-,B:w*o(,)*@lod, 'yjo)I ( v l ( N o ( l )9 ~ l
Ai0(,)))
"1,
I
*
Proposition 5.9(V) is therefore established. It remains to consider Proposition 5.9(E). Let
@loE
Oio; R).
That
E,(d,@!o:A!o,,,: x) is holomorphic on IBL(K(pio(i), Aio(i)))
ciin be seen by employing an argument similar to that used in the proof of Proposition 5.1, thus need not be considered further. Our only responsibility, then, is the inversion formula, viz., Q x @ a ~ o ( PA(i ) = 1.i.m. qE(dX@/o). E-W
We begin with a preliminary observation. Let
F E Y i 0 ( ( GI P ) / r ;6,oo ; W i o . Assume two things: For each index
I,
(i) F, is continuous, x) is continuous. (ii) E,(F,:
Naturally, the symbol in (ii) is to be assigned the obvious interpretation. Let f be the element of G 0 ( ( GIP ) / r ;6,oo;
corresponding to F, so that
llfll
=
IlFll.
3)iO
229
ElSENSTElN SYSTEMS
We then claim that
for a.e. x. Observe that the left-hand side is square integrable, while the righthand side is slowly increasing. Proceeding in the usual way, form the difference A and, using Lemma 5.7 (or rather an obvious variant thereof), integrate it against any partial wave packet, the result being, as a short computation reveals, zero. In order to conclude that A = 0, one need only verify that a * A = 0 for every LY E IF(G),which can be done by imitating the argument of Lemma 4.1 1. [Note: Tacitly, Assumption Rep($ has been invoked here.] Hence the claim. The inversion formula is now immediate: Just take F = dx@loand let E approach infinity. But the discussion actually gives somewhat more, namely, G,((GI
w r ;4
=
00;
c 0 Go(@I w r ;
6900;
@I),
I
where
G , ( ( G I n/r;4 0 0 ; @ I ) is the closed subspace of
G,((G I spanned by the lim &+m
S
w r ;4 0 0 ; J%d, @lo No(,) -4IdNo(,,I
9
f B J ( & ( P i o ( t ) *Aio(a1))
where, of course, io is allowed to vary. This is best seen by taking this time
F:
{F,
= d,@,
F,. = 0
(1’
# r).
Observe that the decomposition does not involve the projection QPin a direct manner. The proof of Proposition 5.9(E) is now complete. Remark There are a couple of subtleties implicit in the foregoing which should be commented on. First of all, it is perfectly possible that some of the terms figuring in
@lo
(Vl(N,(I))dl
7
dl
y;,)
may have singularities on IBL(Wpio(i) Aio(i)))9
230
CHAPTER 5
For a numerical example, cf. Langlands [p. 313). The point, therefore, is that alter a suitable combination, the possible singularities are removed. Second of all, there is no reason to expect that
w,:Go(,):x)
will be well behaved on
IBt,(Xi(Pio(i) Aioci)))? 9
even, e.g., if F, is differentiable with compact support. Before continuing, we had best fill in the details of the approximation argument left open above. Stone-Weierstrass Theorem Let X be a compact Hausdor- space. Let B be a Jinite-dimensional commutative *-algebra containing C which is generated by C arid the self-adjoint nilpotent elements. Let 8 be a subset of C ( X ,a).Assume 8 is a *-subalgebra of C ( X , a)containing the constants with the property that for all points p, q E X , p # q,
{f(PI : f ( 4 ) = 0,f E 81. Then: 8 is unformly dense in C ( X , a). =
[The proof is not difficult.In order to avoid interrupting the exposition, we shall defer it to the Appendix of this chapter.]
W0),say 3 E Xio(%k). Consider, in the usual Let X be a member of z(%k; notation, H O ~ ( S . XA~, ~P) ',&cus(d, ~, OiJ). Denote by A ~ ~ ) Ho~/N(SX(P~,,
-
OiJ)
the quotient of HO~(S.X(P,~, A(,) &cus(d, 9
OiJ)
b y the elements of order > N. Needless to say,
HO~/N(SX(PI,, Ccudd, OiJ) 9
is finite dimensional. Lemma Let
E
> 0 be given-then the set of restrictions to IBt(X(Pi0 AiJ)
23 1
ElSENSTElN SYSTEMS
IB,t(x(Pio AiJ) 9
of the
d@!,
(@it, E J f A l , ( R ) )
is uniformly dense in C(IBZ(X(Pi0 AirJ), Hom/N(Sx(p,,,A,,) C))* 9
9
To begin with, note that Hom/N(SX(P,o,A , o )
9
c>
is a finite-dimensional commutative *-algebra containing C. Here, of course, the *-operation is the obvious one, viz.,
T * ( u ) = T(u*).
Moreover, HOm/N(S3E(P,,,Aio)7C) is generated by C and the self-adjoint nilpotent elements. This being so, put
a = Hom/N(SX(P,o,A,o)' c)*
Let
x = IBJ(X(Pi,, Aio))u ( 0 0 ) .
Then X is a compact Hausdorff space. Let 8 be the direct sum of the constants and those continuous functions f:X-+%
such that: ( 0 f ( m )= 0; (ii) f I IB:(X(P,,, Aio))= do!, (some @loE HA,*,(R)). It is a straightforward exercise to verify that 5 is a *-subalgebra of C ( X , a) with the properties stipulated in the Stone-Weierstrass theorem. Therefore 8 is uniformly dense in C ( X , a),and this, as is easily seen, implies our assertion. //
232
CHAPTER 5
We can now supply the details for the approximation argument. For each 3 E Xio(%',,), choose a polynomial p i which does not vanish on X(Pio,A,,)+ but which does vanish on all r l ( P i o ,Ai0)+,X' E Xio(%J (rl # X), to order N + 1. Fix an index 1. Given
3E
n JEio(%'J,
let Y(p1)be the set of all No(,)
E ZB!
(xt(Pio(t),
Aio(,)))
with the property that
+
P1(w/oio(~)wiO,(t)~!o~~0 ~)
for every w,foio(,) in W&,(X, X,; Aio, Aio(,J.Call D,the subset of
nx pi)
consisting of the general points at which
(V,(N0(,)),K T) is defined. It is then clear that D,is an open dense subset of ZBL(X,(Pio(,), Aioc,,)). Let Fi lB!(Xt(Pio(,),Aio(i)))
-,H W S ,
3
gcus(8,
OiJ)
be a continuous function, with support contained in D,,whose components are subject to (Sym),. The components of F, indexed by the W/oio(r)E W&k(X,
X,; vio
3
%'io(rJ
may be planted on IB,t(X(Pio AiJ) 9
according to the rule
FXNJ
=
Fw:o,oc,,((Wibio(i)
wiO,(tJ-
'No).
Thanks to (Sym),, the definition of FIE is independent of the choice of witoio(,). Since SPt(FIE) = {No : Pi
E z A f o ( 8 , oio; R)
such that F: is the uniform limit of the d ( p 1 . > N . This being the case, set =
1x P1
*
(W0)x.
modulo Horn's of order
ElSENSTElN SYSTEMS
233
It then follows from our definitions that F, is the uniform limit of the d,@,!,, modulo Hom’s of order > N, the d,, (I’ # z) having order > N. The approximation argument is therefore established. Let X be a member of %(%?k; W0), say X E xio(%?k).By definition, IBL(X(Pi0, AiJ)
= E
IJ IBf(X(Pi0, A i o ) ) * >O
As these sets are ultimately the ones of basic importance, we shall take, at this point, a closer look at the singularities of the E-functions and V-functions along them. In so doing, we keep to the preceding assumptions and notations. Proposition 5.10 Let X E xio(%?k)- Let A!,, be ageneral point in IBL(X(P,,,, A,”)) lying on no X’(Pio,Aio)t (X’ # 3, X ’ equioalent to X)-then V(X: PI(& A i ) : PjoIAjo:PioIAi,:wJoio:?) is holomorphic at A!o.
[Since this result plays but a secondary role in what follows, we shall merely indicate the line of proof (which is technically a bit messy but not really very difficult). Let ( X I be the equivalence class of X,XIa principal element in (3,. According to Proposition 5.9(V), the function ( v l ( ~ ! o ~ l@it,, ) ) ~ ldl VJo) is holomorphic on IBt,(Xi(Pi,,(i),A i o ( i J )
for all
i
E xAlo(d, oio; R )
YJoE x a j o ( d , ojo;R ) ,
The main thing now to realize is that if the assertion were false, then by choosing and Vj,, judiciously, one can pick off the V-term in the sum defining (vl(~!o(l))d, dl YJ0l which, in so doing, leads to a contradiction.] Proposition 5.11 Let X E X i 0 ( q k ) . Let H , be a singular hyperplane of
E(X: PIA,: PioIAio:T o :A!o: x) which meets IBL(X(P,,, A,,))-then (Hs
- X(Pi0, AiJ)
+ Ai
3
Dis(X)(Pio, AiJ.
[This result is due to Langlands [p. 2173.1
234
CHAPTER 5
An equivalent statement is this: H , - X(Pio,Aio) contains the projection of Dis(X)(Pio,Aio) onto li!o.Necessarily, then,
Consider, in particular, the special case when X is principal-then the dimension of the projection of Dis(X)(Pio,AiJ onto &I,, is the same as that of X(Pi,, Ai0)+.On the other hand, H , is a hyperplane in X(Pio,Aio)t; thus it has dimension one less than that of X(Pio,Ai0)+.Conclusion: If X is principal, then
E(X: P I A i : PioIAio:To:ATo: x) is liolomorphic on IBL(X(Pio,Aio)).This, in fact, is the first step in the proof of the proposition. Lemma Let X E Xio(qk).Suppose that X is principal-then E ( X : PIAi: PioIAi,: To:A!o: x)
is holornorphic on IBt,(X(Pio,Aio)).
Proof It will be convenient to change the notation a little and write XI for X, io(i) for i o . In this way, we are in a position to make free use of the machinery developed above for Proposition 5.9(E and V). Proceeding by contradiction, let H , be a singular hyperplane of
Aio,,))). Let A1 be a real unit normal to H , in which meets IBL(Xl(Pjo(l), X,(Pio(,),Aio,,,)t-then, on H , , the Laurent expansion gives E(Xi: PIAi: p i o ( i ) I A i o ( i ) : m
To(l):ATo(,)+ [Al:
X>
235
EISENSTEIN SYSTEMS
with the following property: For all wib(i)E Wt,(xi,
xi; Aio(i)r
AiOct))
such that the restriction of w:(,)wito(,)to H , is not the identity, w:(t)w!o,t)A!l# A & -
Otherwise said, AL is in general position with respect to H , . In addition, we can and will assume that AX lies on no other singular hyperplane of E(xi: PIAi: Pio(i)IAio(i): To(i): A,!,,,):
XI*
Finally, it may be supposed that
E - N ( T ~ (Ah: ~ ) :X )
# 0.
This said, let B be a small ball contained in
Hs n lBL(xi(pi,,(t),Aio(i))) and centered at AA. B will be allowed to shrink in due course. Let 4 be a nonnegative continuous function on
Hs n l B L ( ~ t ( p i o Aioti))) (i~~ with support contained in B such that the measure of the set ofx E G for which - N(
/$(A!o(i))E
I
: A!~(i):
&I)
I#
dA!~(i)
+
is positive. Let be a nonnegative continuous function on the line with support contained in ]1/2, 1[ such that Jii2 3 = 1-then, putting $mcV)
we determine a sequence {$,} that
=
m* N
~Y),
of nonnegative continuous functions $, such
{;m&,:l;y%
1/mC
The points
No(l) +fiYA1, where
N0(i) E Hs n lBL(xi(Pio(i),AiociJ)
236
CHAPTER 5
defines a sequence { I,} of continuous compactly supported functions Jn:
IEt,(xi(pio(i)yAio(i)))
+
R.
Extend To(,) to an element T i o ( i )E
Horn(% gcus(6, 9
@io(i)))
by requiring that all the components of Tio(l) be zero save the one corresponding to w & ~ , ,which we take as To(l) itself. With this preparation, let {F,} be the sequence of functions defined by specifying that the components of F, are zero except for the one associated with I, in which case we put
FXA&i)) =
1
,fm((wL(t)w!o(i))-
'A!o(i))
*
( ~ i ' o ( r*) x i o ( a J -
W?O(l,
Claim: If m B 0, and E is sufficiently small, then F m E z i o ( ( G I p)/r;6,oo ;J E ) i o ( i ) .
Indeed, as is easily seen, under these circumstances, the integral
'.I
(Vl(N0(l))F~9 F 3 IdNo(,)I
~ & W P t o ( t ) Ato(,))) .
is equal to the integral of fm(A~oc,,)2 times (V(3i: PI (Ai, Ai): pio(i)I Aio(i): Pio(r)I A i o ( i ) : w&i): A!o(i))TT;.o(i)>To(,))
taken over
E x 11/2m, l / m [ . J - l ~ l .
Owing to Assumption 4(V), that part of the integrand coming from the Vfunction is actually constant; in fact, it is just
I p i o ( i ) I Aio(i): p i o ( i ) I Aio(i): w & ~ ) X : , ( P i o ( i ) , Aio(i))) applied to To(1)and paired with To(l).Denote this number by Cv and let pDis(X,) (A&s(E,)>
C=
sB 42.It is then clear that the norm of F, is finite, being given by 1
rl/m
EISENSTEIN SYSTEMS
237
or still
Hence the claim. We are now in a position to derive a contradiction. First note that lim IIF,II
= 0.
m+m
On the other hand, let f, be the element of G o ( ( G I p)/r;6 3 0 0 ;JE)io(,)
corresponding to F,,,, so that Ilfmll
= IIFrnll.
If m 9 0 and B is sufficiently small, then the set B x 11/2m, l/mc. f
i ~ l
is contained in the domain of holomorphy of
E ( 3 , : PIAi: p i o ( t ) I A i o ( t ) : To(,):A/o(i): XI. We can therefore say that for a.e. x, fm(x), multiplied by (27r)’+,is equal to
or still
where
L
c, =
Let m approach infinity in this expression-then all terms but the first drop out. Consequently, 1 lim fm(x) = ~. m-tm (274’’
. J”
I9
~ ( N ~ ( , ) ) E - N ( T ~ ( ~A/~(,): ):
I~A/~(,)I.
238
CHAPTER 5
By construction, the function on the right is nonzero on a set of positive measure, thus contradicting the earlier conclusion that lim llfmll = 0. m-rw
This completes the proof of the lemma. // The remainder of the proof of our proposition depends in an essential way on Proposition 5.4. We can now, of course, suppose that X is not principal. It will then be convenient to change our notation: Write 'I) for X , j , for io, and let @ be the equivalence class of 'I),X a principal member of @, say X E Xio(Wk). Placing ourselves in the setting of Proposition 5.4, we can and will assume that ( P j o ,Sjo;Ajo)is part of the data, as there. Extend To,in the obvious way, to an element I c u d S , ojo))* Tjo E Hom(SQ(pjo,ajo),
Lemma ZfHs is a singular hyperplane of
E ( 9 : P I A , : PjoIAjo:To:AJo:x) which meets ZBL(V(Pjn,A J ) , then H , is a singular hyperplane of'
v(w$k(Dis(x))(@( Dis(X)): X : %io):
{ wJoi0}: P I Ai : w,P( wjoi0)-'Ajo)Tjo.
Proof It amounts to showing that if V(W$,(Dism)(@(DiS(X)):3: %io): {Wjoio}:PI Ai: ?)Tjo is defined at N o
then
E IB',(X(Pio,
Ai,-,)),
E ( 9 : PIA,: Pjo)Ajo:qo:?: x)
is defined at w J o i ~ w ~ AglB',(9(pjo, !o AjO)).
This said, choose a fixed right inverse t: Im(V,,((F:
X: PIA,)) -+ Dom(V,,(@: X: PIA,))
to VWk(@: X: PIA,)). Because Proposition 5.4 implies that
V(W~,(,iscx,,(@(Dis(X)): X: Wi0):{ w ~ ~PI~ A~, :} ?), : when defined, takes values in Im(V,,(@: 3:PIA,)),
239
EISENSTEIN SYSTEMS
it makes sense to form the following meromorphic function zjo of E X(Pio,A,,)+:
A!,
r(V(Wtk(DiScr))(a(Dis(X)): 3: %?lo): { wj0io}: P IAi: A!o)Tjo).
Claim: There is an equality of meromorphic functions:
E(g:P I A , : PjoIAjo:To:w ~ o i o w ~ Ax)~ o : = E(Wt,(Dis(x,)(a(DiS(X)): 3: gi0): PI Ai: zj0(A!,):A!o: x).
Admit the claim for the moment. Owing to the preceding lemma, the righthand side of this relation is defined at A!, E IBt,(JE(Pi,,A,))
whenever this is the case for tj,;as this is so only if V(Wtk(Dis(r))(a(DiS(X)): X: %?i,):
{ W j o i o } : P ( A i : ?)Tio
is defined at A!,
IBt,(X(Pio, AiJ),
the lemma will follow. To prove our claim, we can take A!, to be a general point at which everything is defined. Set T;~(A!~)= rjo(AT0)0 ( -Tjo),
an element of
@ Hom(Sxwt(P,,l,+).A j o I w t ) ) ' 8cus(6,
'joOvt))),
3 = Wt,(Discx,,(a(DiS(X)):x: %io)
U (W;,i,}.
E
where Observe that
E(%:PIA,: T>~(A[~): A!,: x) is thedifference between the right- and left-hand sidesofthepurported equality. Bearing in mind the expression for the constant term of E(%:. . .) along K x PJovo, in order to show that E($: . . .) = 0, it suffices, thanks to Theorem 3.6, to verify that V(Wt,(@: X: Wi0): %: PIA,: A/,)$,(A!,)
=0
or still (cf. Proposition 5.4), to verify that V(Wtk(Discx,,((F(DiS(X)): X: via): %: P I A i : A!,)t>,(l\~,)
=
0.
240
CHAPTER 5
But the latter is equal to the sum of V,,((€: 3: PIAi)tjo(A,6)
and -V(Wtk(Dis(x))(Q(DiS(X)): X: %to): {Wjoio}: P J A i :h/o)Tjo.
Since, by construction,
V,,(@: X: PIAi)tjo(A/o) is equal to V(Wt,(Dis(x),(Q(DiS(x)): 3 :gio): {WJoio}: PIAi: A!o)Tjo,
we are done. //
Lemma I f H , is a singular hyperplane of V(W&k,Dis(x,,(Q(Dis(X)): X: Wi0): {wJoio}: PI A i : wi",(wfio)-'AJo)Tj0 which meets IB',(g(Pjo, Ajo)),then H,is a singular hyperplane of
V ( g : P I ( A i , Ai): PjolAjo:PjolAjo:wyo:Ajo)To. Proof By inspecting the terms which make up v(Wt,(Dis(x),(~(Dis(x)): X : %io): { wJoi0}: 'I 1 Ai: A!o)Tjo, one can say that there exists an index I and an element w!o(i)jo such that H, is a singular hyperplane of V(r): PI(Ai, Ail: PiO(i)IAio(i):P j o I A j o : w!o(i)jo:AJJqO. Needless to say,
xi = 'Dw!,,, , ,,,~E @(Dis(X)), hence is principal, thereby justifying our notation. Let us agree to write qoinside
'.
M ~ J ~ for ~ ~ (w!o,l~jo)-' (~) Place
Hom(Si, &cus(6, OjJ) by stipulating that all its components be zero except the one corresponding to M J ) ~ ~ ~ which ,~), we take to be qoitself. Let E Hom(Sx,(pto(,), A , ~ ( , )9 )gcus(6,
Place To(1) inside Hom(Si, g c u s ( 6 , Oio(i)))
Oio(i)))-
EISENSTEIN SYSTEMS
24 1
by stipulating that all its components be zero except the one corresponding to w:,(~,, which we take to be q,,(l) itself. Now specialize inequality (3) of page 221 to this particular situation and derive that I(V(g: p I ( ~ iAil: , pio(t)IAio(i):P j o IAjo: Wito(,)jo:Wjtoio(i,W,P(i)Ait,ci))Tjo, T o ( t J 1 2 5 I(V(g: PI(Ai, Ai): P j O I A j o : P j o I A j o : wyo: wjnio(i)w,P(i)Ar!o(i))Tjo, TjJI x I(V(xi: P I ( A i , Ai): pin(i)IAio(i):pio(i)IAio(i):w,P(i):‘!o(iJTo(i), To(iJI*
It should, perhaps, be recalled that Aitoci) E lBL(x,(Pio(t), Aio(tJ)*
The second factor on the right-hand side of this inequality was encountered a short while ago; it is a constant (cf. Assumption 4(V)). Our assertion is therefore manifest. // Proof of Proposition 5.11 Keep to the notational changes agreed to earlier. Let H , be a singular hyperplane of
E ( 9 : P I A , : P j o l A j o :To:Ajo:x) which meets l B L ( g ( P j o ,Ajo))-then it follows from the two lemmas above that H sis a singular hyperplane of
V ( g : P I ( A i , Ai): PjoIAjo:PjoIAjo:wy0: Aj0)IT;.,. In view of Assumption 4(V), this function depends only on the projection of Ajo onto the complexification of Dis(g)(Pjo, A,,)+. Consequently, H,, and therefore H, - Y ( P j o ,Ajo), is a union of cosets of the projection of Dis(Vj)(Pj0, Ajo)onto i5jc,.So, if we can prove that Y ( P j o ,Ajo)E H,,then we will be done. Referring back to Proposition 5.10, we see that Hs
lBL(g(Pj0 A j o ) ) 9
must be contained in the union of the various “w-hyperplanes” H w and the various
g(pjo,AjoIt n g’(Pjo9 Ajo)t. The Baire category theorem implies that
Hs r‘\ lBL(g(Pjo3 AjJ) is actually contained in one of the members of this union. Since the real dimension of
Hs n lB!o(g(pjoT AjJ) is evidently bt - 1, there are two possibilities: Either
Hs n zBL(9(f‘jo, A j o ) )
=
Hw
n lBL(g(Pjo3 AjJ)
242
CHAPTER 5
for some H,, or else Hs n IBL(’2)(Pjo,Ajo))
=
YI’(PjO3
AjJt n ~ ~ L ( ~ (AjJ) P j 0 ~
for some ‘2)’(Pjo,Ajo)t. In either case, it is easily shown that Y(Pj0,A j J E H s ,
as desired. // The theorem toward which the efforts of the present chapter have been directed can now be stated. Theorem 5.12 Given V o ,write =
IJ W i o . io
For any integer b, 0 I b I rank(%:,),
one may then attach to each Via, in a unique way, a collection Xi, of equivariant systems of admissible afJine subspaces of dimension b and an Eisenstein system bdonging to each element oJ’Xio,subject to
I
Geom: I Geom: I1 Geom: I11
and
An, with the following property: For all ( P , S ; A i ) in W k , the space
G o ( ( GI
w-; a,@:,)
can be written as the orthogonal direct sum of the G 0 ( ( GI P ) / r ;d,@o ;
the sum extending over those collections whose dimension lies in the range
rank(%) Ib I rank(%:,). [This result is due to Langlands [p. 2221.1
Let us agree to some notation. Recall that %(go) =
JJ io
Xio.
243
ElSENSTElN SYSTEMS
If it is a question of a collection of dimension b and if this is to be emphasized, then we write JE(V0; b ) = JEio(b)io
Similar principles will be applied whenever it is convenient, e.g., assigning to the symbol L:,((G I wr; 6, oo;3,b) the obvious meaning, our theorem asserts that is equal to
L:,((G I
c
rank(%) 5 b 5 rank(Yo)
w-; 6,00)
o L:,W I wr; 6,00;
Consider, in particular, the special case when (G, G ; { I})-then
%?k
JE,b).
= {C},
(P,S ; Ai)=
G o ( G / r ; 6900)
admits the orthogonal decomposition
1
0 5 3 5 rank(Yo)
0 L:,(GIr; 6, 0,;JE, b),
a fact which, as will be seen in due course, leads to the orthogonal decomposition hinted at near the end of Chapter 4. The uniqueness is relatively easy and will be dispensed with momentarily. The existence, on the other hand, is more difficult. Its proof is inductive in character. To allow induction, it will be necessary to formulate still another set of assumptions and conditions. In this connection, the residue taking process will play an important role. Remark To avoid any possibility of confusion, we stress that %?k ana ( P , S ; A i ) are “running variables”. The collections Xi,and their associated Eisenstein systems are independent of them. Suppose that for each b we have collections 3?(V0;b), 3”(Vo; b) and associated Eisenstein systems {E’, V’}, { E ” , V”} with the properties stated in our theorem. The proof of uniqueness then begins with the following observation. Since the spaces
w-;
raws
L:,((G I d,oo; JE’, i Li O (I wr; ( ~ 6, oo; r,rank(%)) both give the discrete spectrum of 0’ in
L:,W I P ) F ; 6,
00l
244
CHAPTER 5
they must be equal. This said, let ExAlo(S,
@/o
oio;
R)*
Then (cf. Proposition 5.9(E)) the projection of O,;o(P1 A i ) onto the space spanned by the eigenfunctions of 0 is given by either
or
c‘ E’(3E’:P ( A i :Pio(Aio:d@/o(X’(Pio, Aio)):X’(Pio, Aio): ?)
c“E”(3E”: P(Ai:P i o ( A i od@/o(X”(Pio, : Aio)):X”(Pio,AiJ:
?).
Here c‘ or Z’’ stands for a sum over those 3’or 3” in
{
Xi0(Wk; rank(%)) Xy0(Wk;rank(%)).
In the case at hand,
i
X’(Pi0, Aio) = {X’(Pio, o (hi X”(Pi0,Aio) = {X”(Pio,Aio)) o (ai
+ + J-rAi),
so the normal translation determines the affine subspace. Using a simple approximation argument, it then follows readily that Xio(%k;
rank(%‘))= xy0(%‘k; rank(%‘))
for all i o . Furthermore (cf. Assumptions 4(E) and 4(V)), the E‘- and E functions, as well as the V - and V-functions, are the same. Now let 0 Ib I rank(V0). By varying %‘k over those G-conjugacy classes of rank = b, we conclude from the preceding discussion that the collections X’(%‘,,; b), Z”(W0; b) have the same principal elements and that the associated Eisenstein systems coincide. In order to draw similar conclusions in general, we need some preparation. Lemma Let X(%‘o) =
xi0
io
he given. Suppose that for each io, X i , is subject to Geom: I Geom: I1 Geom: I11 and
An.
ElSENSTElN SYSTEMS
245
Taking Wk = ( G } , let be an equivalence class in X(Wo), Fix a principal element X, E 6,.Then: The 3rloio,where
X E @o(Dis(Xo))
{
wjoio
having the property that
E W,,,(X;
W j o 7 WiJ
V(3: Gl({l), {I}): P j o l A j o :PiolAio:wjoi0:Aio) Z 0, describe g o . Proof That such elements belong to @” is a consequence of E-S: I. Proceeding by contradiction, let 9 be an element of@,, say 9 E X,, such that whenever
9 = finJoio
for some
3 E @o(Dis(Xo))
{
Wjoio
V(X: GI((l), (1)):
E W(,j(X; Vjo3
WiJ,
PjolAjo:PioIAio:wjoio:Aio) = 0.
In view of E-S: 11, this would mean that V(9: GI({l), (1)):
PioIAio:PjolAjo:wiojo:Ajo)= 0
provided ~ w , o l o E &,(Dis(X,)). Now fix X E @,(Dis(X,)), say 3E E Xi,. Write 9 = XwJoi0(some wjoio).We can then assert that
V(W,,,,i,cx))(@(DiS(fi)): 3: WiJ: {wjoio) : G I { 1) : Aio) = 0 or still, thanks to Proposition 5.4, that V(W(&: 3 : Wio): {wjoi0}:G l { l } : Aio) = 0.
This relation, when unraveled, implies that V(9: G J ( ( l ) , (1)):
...) = 0,
from which it follows, in the usual way, that the Eisenstein system belonging to 9 is trivial, which is an impossibility. // Let X0 be a principal element in JE’(Wo; b) or x”(Wo; b). Let Wo be the equivalence class of X, in X’(W0; b), @: the equivalenceclass of Xo in JE”(Wo; b). Claim: Wo = Wh. To see this, note that at least Wo(Dis(Io)) = Wi(Dis(3Eo)). But then our claim follows from the description of @; and Wi provided by the preceding lemma. Consequently, X’(Wo; b) = JE”(W0; b). Denote their common value by X(W,; b).
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CHAPTER 5
It remains to show that the Eisenstein systems {E’,V’}, {E”,V”} belonging to the elements of JE(W:,; b) coincide. For this purpose, let (E be an equivalence class in JE(Wo; b), X a principal element of @, say X E Zio-then it will be enough to show that ViG)((E:X: G l { l } : Aio) = V;b,(E: X: G I { l } : Aio). Let
i
9 2
=
W,,(,is(31))(@(Dis(X)): 3: WiJ
= W,G)(Ci:X: Wio) - W,,,,iscx,,(@(Dis(3E)): X: Wi,).
We then have
i
V’(9, u 9,:9,: G I { l } : Aio) = V”(9, u 9,: E l : G I { l } : Aio) V‘(E1:EI U 9 2 : G l { l } : A i o ) = V”(B,:=I U 9 ~ : G l { l } : A i 0 ) .
Indeed, the second relation is a formal consequence of the first (cf. E-S: II), while the first is a consequence of what we already know to be true. Using 9, and 9,, write, in the obvious way,
On the basis of what has been said so far, A‘ = A”, B’ = B ,
C’ = C”.
Taking into account Lemma 5.5 (applicable because of Proposition 5.4), write B = A”U” B’ = A’u‘ C” = V”A” C’ = V’A’ D“ = C”u” = V“B“ D’ = C‘u‘= V’B’ Then
I
{
D’= C’u’ = V“A“U’ = V“B“ = D”
Therefore ViG)(@:X: GI(1): Aio) = V[Gl(@: X: GI{l}: AiJ, desired. Having taken care of uniqueness, let us pass to the problem of existence. It will be convenient to start off by introducing three types of naturally defined sets, viz., “cylinders,” “cones,” and “real or imaginary balls.”
iis
247
EISENSTEIN SYSTEMS
Let X be an equivariant system of admissible affine subspaces attached to Vio.Given zl, e2 > 0, let C(X(Pio,Aio):61: ~
be the set of all points of the form X ( P i o , Aio)
2 )
+ A:o + - A t ,
where A,!o, A;o
with
E X(Pio,Aio)-
{ wo;;c
n
ai0
61
< E2.
Such a set will sometimes be referred to as a cylinder (of radii and E ~ ) . Given E > 0 and an open set w on the sphere of radius E in X(Pio,Aio) n ai0, let V(X(Pi0,Aio):E : w ) be the set of all points of the form t X ( P i o ,Ado)
where 0 < t < 1,
+ (1 - t)Aio, Aioeu.
Such a set will sometimes be referred to as a cone (of radius X ( P i o ,Aio)and base w). Given A0
and E > 0, let
e X ( P i 0 , AiJ
+ X(Pio,Aio)'
E
with vertex
n aio,
i I:{ 1
RBe(X(Pi0, Aio): zBe(X(pi0, AiJ:
be the set of all points of the form
&Aio,
where
Aio E
X(Pi0, Aio)- n 4,
with IIAio)I< E. Such a set will sometimes be referred to as a real or imaginary ball (of radius E and center Ao). When A. is just the normal translation, we
248
CHAPTER 5
suppress it from the notation. Daggered versions of these three types of sets will be used without comment. We shall now define what we mean by an amalgamation. This is a composite concept, tailored for induction arguments. Let Xi, be a collection of equivariant systems of admissibleaffine subspaces of dimension b attached to Via. It will be assumed that there is an Eisenstein system belonging to each element X of X i o . Let I:, and I:o be two disjoint index sets, Iio = I:, u IL. We shall suppose that there is associated with each index i E Iio an equivariant system XI of admissible affine subspaces of dimension b - 1 attached to Via. It will be assumed that there is an Eisenstein system belonging to each X, (i E Iio).In addition, for each X E Zi0,we suppose given a nonincreasing function EX:
R + + R+,
and for each index i E Iio, we suppose given a nonincreasing function
R + + R+. Finally, for each 3 E Xi, and E > 0, we suppose given a nonempty open subset W P i o IA i o : E )
of X(Pi0, AiJ
+ Wf'io,
AiJ
- n hi0
such that 81
> ~2
KdPiolAio:&~)c
and for each index i E Iio and subset
E
h(PiOlAi0: ~ 2 1 9
> 0, we suppose given a nonempty open
UPio I Aio : E )
of XdPio, AiJ
+ %(Pi,,
Ai0)-
n hi0
such that El
> E2
* P p i o l A i o :E l )
c lgPiolAio:E Z ) .
In both cases, we assume that v ~ ( x P i o x - l I x A i o x -El): = x . Vx(PioIAio: E) ~ ( x P i o x - f l x A i o x - E' :) = x . &(PioIAio:E )
for all x E G for which x P i o x - is again r-cuspidal.
249
EISENSTEIN SYSTEMS
The specification for each io of data of the preceding type is said to be an amalgamation of dimension b if all of the following assumptions and conditions are met. To begin with, we shall suppose that itio, as well as the collection determined by Iio,is subject to Geom: I Geom: I1 Geom: 111. Fix an R with the property that all the normal translations are in the R-tube. [In view of Geom: 11, this is clearly possible. The point, of course, is that one number, viz., R, is being used simultaneously for all Wk.]We impose the blanket hypothesis that the various cylinders, cones, balls (real and imaginary), or whatever that appear below are all contained in the R-tube. The functions E~ and E, play a technical role vis a vis these sets (cf. infra). [Actually, enters does not. We work with both only for notational consistency.] in but There are some additional geometric requirements which must also be fulfilled. Am-Geom: A For every X E itio, For every i E Iio, Am-Geom: B Let
E
> 0. For every X E itio, U P i o I Aio : E )
is a convex cone of radius E ~ ( E )(with vertex X(Pio,Aio)). For every i’ E Iio,
I
I/;*(PioAio: E )
is a convex cone of radius
(with vertex XI,(Pio,Aio)).For every i” E Ira, I/;..(Pi0I Ai,: E )
is convex and
Re( V,.(PioI Aio:E ) )
is contained in the interior of the convex hull of and - ~ p , o ( ( ~ i * * ( PAio) i o - n 4J1)* Apart from these geometric assumptions, we shall impose a series of conditions on the location of the singularities of the E-functions and the V-functions. -~
P i o ( ~ i J
3
250
CHAPTER 5
Am-Sing: I For every X @ X i , and each A,
E
VE(PiolAio: E),
IBd WPio AiJ : A d meets no singular hyperplane of 3
V(X: GI({l}, (1)): ?I?: PioIAio:?: For every i E Iio and each Ai E v ( P i oI Aio:E), IBe(Xi(Pi0 A i o ) 9
?).
4)
meets no singular hyperplane of V(Xi: GI({l}, 11)):
?I?:PioIAio:?:
?).
Am-Sing: I1 For every X E X i 0 , any singular hyperplane of
V(X: Gl({l},
(1)):
?I?:Piol.4io: ?: ?)
mhich meets C(X(Pi0 Aio): E X ( & ) : E ) 7
also meets For every i’ E I;,, any singular hyperplane of which meets also meets IBm(Xi4Pio AiJ)* 9
Am-Sing: 111 For every i’ E Iio, any singular hyperplane H, of
E(3,v: PIA,: PioIAio:?: ?: ?) which meets IBt,(Xi,(Pio,AiJ)
has the property that (Hs - Xi,(Pio,AiJ)
+ 6i
3
Dis(JEv)(Pi0,Ai,J
Remark Am-Sing: I and Am-Sing: I1 have been formulated in the special case: WRk= {G}. But then, in view of Assumption 4(V), this serves to cover all the cases. Let us also bear in mind Propositions 5.1 and 5.2. Am-Sing: 111is, of
25 1
EISENSTEIN SYSTEMS
course, suggested by Proposition 5.11. It will appear shortly that this condition is automatic for the elements of Xi,; hence no assumption in this direction is necessary (thus accounting for the apparent asymmetry in its formulation). Finally, there is an analytic condition which must be met. Am-An
Suppose that
Then: There exists an orthogonal projection
Q ~L:,W : I wr ; 6,00)
-+
L:,((G
I v r ; d,oo)
onto a closed subspace, commuting with the 3(ft),such that for all
r0
+
(6, oio;R )
vjoE2Aio %AJ,(~,
oj,;
R)
and all E > 0 the following hold. Am-An,
The difference between (R(C;0 t ) Q @O!,(P ~ I Ail, @YJ~(P I Ail)
and
the first expression being summed over all X E iTi0(Wk) having the property that Re((X(Pi0,
A i o ) , X ( P i 0 , AiJ>)
and over all wJoioE W&,(X; A,,, Aio),where AX E
I
vS(f'i0 Ai,:
EL
>
-&'
252
CHAPTER 5
the second expression being summed over all i E Iio(Wk)having the property that Re((Xi(Pi0, AiJ,
Xi(PiO3 A i O ) > )
> -c2
and over all wJoioE W&,(Xi; A j o ,Aio),where Ai E
V,(PioI Aio: E ) ,
can be holomorphically continued to the region Am-An2 The difference between
and
the first expression being summed over all X E Xio(Wk) having the property that t Re((X(Pi0, Aio), X(Pi0, AiJ>) > -c2 and over all wJoioE W&,(X; A,, Aio),where AX E VXPiO I Aio: E),
the second expression being summed over all i E Iio(Wk)having the property that Re((Xi(Pi0, Ai,J, Xi(Pi0, Aio))) > and over all wJoioE W&,(Xi; A j o ,Aio),where
4 E UPio IAio: E), can be holomorphically continued to the region
Wi) > 1,.
-8’
EISENSTEIN SYSTEMS
253
Remark It is tacitly assumed that QPis independent of the indices io and
j,. QP is necessarily unique.
The integrals figuring in Am-An, and Am-An2 are evidently holomorphic in the region Re(() > R2. [Needless to say, the sums involved are finite.] Elementary considerations imply that the possible singularities of the integrals in the strip R2 2 Re(c) > & lie in certain parabolic sectors, with vertex the inner product of the conal point and focus the inner product of the normal translation (see Fig. 5.) In the figure the circles represent the various (A!& L A ! , A!>,
while the dots represent the various (X(Pi0, A i o h X(Pi0, Aio)) (XdPio, Aio), Xi(Pio, Aio)), Y
FIG.5
254
CHAPTER 5
Off of the indicated parabolic sectors, the integrals are holomorphic to the right of the line Re([) = A,. It should also be noted that the integrals involving the X or XI such that
i
Re((X(Pio, AiJ, X(Pi,, AiJ>) I-E' Re(
are automatically holomorphic for Re([) > AE. We shall denote an amalgamation by the symbol ( ( x i o ~ l ~ o ~ lErXo, ,~
i ,
6 ,{ Q P } ) ) .
Example (The Canonical Amalgamation) Put b = rank(gi,) + 1. We then set X i , = @,I;, = 0,ly0 = I@}. Assign to the unique index i E li, the equivariant system Xi of admissible affine subspaces attached to Vi0via the prescription XI(Pio,Aio) = complexification of hi0
and take for the Eisenstein system belonging to Xi the canonical one. This said, let be any nonincreasing function. Moreover, for all E > 0, let &(PiolAio:E ) be the points in Fplo(&io) lying in the R-tube. Finally, let QP be the identity map. It is then clear that the data
Iio, constitutes an amalgamation. ((Xi09
IYo, E X ,
~
i
VX, r 6
9
{Qp}))
In what follows, we shall frequently find it convenient to write
A(X: wToio:A6: c)
in place of
V(X: P I ( A i , A i ) : PjolAjo:PioIAio:wjoio:Ai',)
Similar abbreviations with 3: replaced by some other symbol, e.g. Xi or fi, will be used without comment. Lemma Let ((xi09
EX, ~ i VX, ,
VI, {Qpl))
be an amalgamation of dimension b-then, for each i , , Xi, satisjes An. Proof Naturally, the projection QP appearing in Am-An will be the projection Q P appearing in An. The main point, therefore, is to produce the
255
EISENSTEIN SYSTEMS
requisite family of polynomials. So let E > 0 be given-then we can certainly find a family of polynomials p! = {pJo,,}(depending on E ) such that pJo,&IWPjo,A j J + + 0
but such that
('2) E xjo(wk))
~ X ~=( 0, ~io, V(3i: PI(Ai, A{):f'joIAjo: Pio(Aio:wJoi0:?)0 d ~ o m ~ ~ o , E A,,)+ where Re((Xi(Pio, Aio)r Xi(Pi0, Ai,)>) > - E 2 (i E Iio(wk))* In addition, it may be supposed that V(3: PI(& AJ: Pjopljo:PioIAio:wJoio:?) 0 d"ompio.&, t where is holomorphic on
Re((X(Pi0,
Ai,J
X(Pio, Aio)>) > - c 2
IBd(X(Pi0 AiJ) 3
Now let
A ! ~ ( oio; S , R; {yloE s A f o ( d r ojo; R). @ o!
Then, by construction,
( 3 E Xio(wA)*
E~
A
(V(X: PI(Ai, Ai): PjolAjo:PioIAio:wJoio:Ale) d@!o(A/o), dYjo(- w ~ o i o ~ ~ o ) ) is holomorphic in a neighborhood of IB!(3(Pio, Aio))for all 3E E xiO(%k)with Re((X(Pi0, AiJ, X(Pi0, AiJ>) > -cZ, hence a fortiori for all 3 E Xio(wk)with Re((X(Pi0, Aio), X(Pio, Aio)>) > R2 - ez. Furthermore, in Am-An, and Am-An, ,only the terms involving t h e 3 E Xio(Wk) having the property that Re((X(Pi0, AiJ, X(Pi0, AiO)>) > - c 2 remain. This said, let us verify An, and An,. We need only deal explicitly with An,, the discussion of An, being entirely analogous. Claim: If then the difference
s
Re(
lB~(.%(Pio, Aio): A s )
A(X: wJoio:A!o: 4') IdA!ol
256
CHAPTER 5
can be holomorphically continued to the region Re([) > A,. Admit the claim. For An,, it is a question of showing that the difference between
(NC;')QP
%t0(P I Ail, @vJ~(P IAil)
and
summed over all 3E E Xio(Wk)having the property that Re((X(Pi0, AiJ X(Pi0, AiJ>) > A, and over all wJoioE W&,(X; Ajo, Aio),can be holomorphically continued to the region Re([) > A,. The strategy therefore is obvious: First, add and subtract the sum over X and wJoio of the integrals
I
A(X: wJfoio:A!o:
I B t (%PI,, Aio): Ax)
where
Re((X(Pi,,
Air,),
[) IdAfoI,
X(Pi0, A i o ) > ) >
-E';
second, add and subtract the sum over X and wjoioof the integrals
s
A(X: wJoio:A!o: c) IdAfoI,
IBd(X(Pi0. a d )
where Ae
2 Re((X(Pi0, Aio), X(Pio, A J > ) >
-E'*
Bear in mind that the latter integrals are automatically holomorphic in the region Re(<) > A,. An, then follows upon quoting Am-An, and the claim. It remains to consider the claim. Note that
- X(Pi0, AiJ is real. Let be the set of all points in the projection of 3E(Pio,Aio)- onto ?i!o which are orthogonal to AI
-
X(Pi0, Ai,J
and have norm less than E. If 0 I c I 1, then
A(X: wfoio:A!o:
[)
EISENSTEIN SYSTEMS
257
Y
FIG.6
and, by Fubini’s theorem, its integral over this region can be written in the form -
~
I
I -Ax ( P~i l , ,Aio)lt
x A(X: wjoi0:X(Pio,Aio)
j’ J
+
D!.
I
W: A to)
+ z(A&- X(Pio, Aio)):i)dz IdA!oI,
the inner contour being as shown in Fig. 6. Call this integral I(c: c)-then our claim amounts to the assertion that
I(1: C)
-
I(0:[)
can be holomorphically continued to the region Re(c) > 1,. Suppose that Re([) > R2-then Cauchy’s theorem enables us to write I(1: [) - I(0: i)
as
258
CHAPTER 5 Y
Our claim will be settled when it is shown that the real part of this expression is II, on C(AI0). Write
Then Re((X(Pi0, Aio), X(Pi0, AiJ>) - IINoII’ + Re(zZ)IIA&- X(Pi0, AiJII’ = Re((X(Pi0, A i o L X(Pi0, AiJ>) - E~ + x211A&- X(Pi0, AiJII’ 5 (Re(X(Pi0, AiJ), Re(X(f‘io, Ai,-,))> - c2 + IIAL - X(Pi0, AiJI12 = IIRe(Al)II’ - E’ IRZ - c2 = I , . Hence the claim. The proof of the lemma is thus complete. //
Of course, not all the assumptions defining an amalgamation were used in the preceding argument. The point is that now for each io, JEio is subject to Geom: I Geom: I1 Geom: 111 and
An.
Therefore this part of the data may be fitted into the picture developed earlier. The following lemma is the main result on amalgamations.
Main Lemma Let ((xio,
KO,EX, &I,
VX, V,,
{Qp)))
259
EISENSTEIN SYSTEMS
be an amalgamation of dimension b-then there exists an amalgamation ReS{((Xio,
&o,
I;o,
EX, EI,
VX, 6 , {QP}))}
of dimension b - 1 such that
RNQP) = QP - Qx. [The objects of Res{(. . .)) are denoted by placing the symbol Res in front of the corresponding objects in (. . .).I Grant the lemma for the time being-then one has
XI. Im(Qp) = Im(Res(Qp)) 0 J%,((G I W r ;400; Proof of Theorem 5.12 For integral n 2 0,define by recursion a sequence {Amal(n)}of amalgamations according to the prescription
i
Amal(0) = canonical amalgamation Amal(n) = Res{Amal(n - 1)).
Let us agree to write
. . . , {QP’})).
Amal(n) = ((Xi:’, Observe that
dim(Amal(n)) = rank(%,)
+ 1 - n,
so the amalgamation is trivial when n > rank(%,) 0 I b I rank(W0),
set JE(%o;
b) =
IJ Xio(b), io
where Xi001 =
xi,(rank(Yo)+
For each io, Xi,@) is subject to Geom: 1 Geom: I1 Geom: I11 and An.
1-3).
+ 1. For any integer b,
260
CHAPTER 5
Now let (P, S; A i ) in W kbe given. Claim:
Here, of course, is the closed subspace of
L:,((G I Pyr; 6, oo; W ) G , ( ( G I PYr; 4 00)
associated with Amal(r). This can be proved by induction on n. If, to begin with, n = 0, then QP) is the identity map and the sum is empty. In general, thanks to the lemma supra, we have Im(Q',"))= Im(Q',""+'))0 L$,((GIP)/I'; a,@,; X)O(nJ, thus pushing the induction forward. If b < rank(%,),
i.e., if n > rank(q0) - rank(%,)
+ 1,
then the sums figuring in Am-An are empty, so, as the inner product per R ( ( ; Rt)QP) is entire, Q',") = 0. It therefore follows that G , ( ( G IP)/r; S, 0 0 )
is equal to
c
+1
c
o L:,W I wr; 6, oo;x, b),
0 5 r 5 rank(%,) -fank(Yk)
or still, is equal to
rank(%*) 5 b 5 rank(W0)
0L~,((G I wr;6, oo; XP
the space L$,((G I W r ;4 Oo; WC0) being trivial. The theorem is thereby established. // Remark The space L;,(G/r; 8,oo; w o )
was constructed near the end of Chapter 4. In the present context, it corresponds to L+,(G/r;6, oo;3,
EISENSTEIN SYSTEMS
26 1
i.e., the data per Amal(1). The methods employed are completely different. The initial construction relied on estimates, while the current construction depends on the spectral theory of 0. We must now deal with the lemma. Given the amalgamation , K, {QP))), ((xi09 Iio, I&, EX, ~ i VX,
it is then a question of manufacturing Res(Xio),. . . , Res(Q,) so as to satisfy the various conditions. The key rests with a potentially confounding farrago of integral manipulations. We have, of course, already specified Res(Q,): R ~ ~ Q=P QP ) - Qr. This being so, let
i
R) YJoE .YfAjo(6, (!Ijo; R).
@/o
Then
EzAto(b,
oio;
( R ( l ;0t, R~S(QP)@,J~(P I Ail, @YJ~(PI A,))
) Q @,lo(P ~ IA i l @YJ~(P IAil) - (R(C;0t)Qx @,;t,(P I Aih @YJ~(P I Ail)*
= (R(C; 0t
Suppose that Re(() > R2-then, using Proposition 5.9(V), we can write
( R ( l ;0' ) Q x @@lo(PI A i l @YJ~(PI Ail) as
The integrals taken from E to infinity are evidently holomorphic in the region Re(() > 1,.Consequently, the difference between ( R ( ( ;0t ) Q @,lo(P ~ I Ai), @,f0(P I Ail) and
can be holomorphically continued to the region Re(() > A,. For the sake of brevity, let us agree to write
A(3 : Wjoio(r): w!oio(t) : A!o(j) : i)
262
CHAPTER 5
in place of
applied to
and paired with Let
stand for
summed over all i E Iio(%,Jhaving the property that and over all wfoioE W&,(X,: A,, A,,), where
We can therefore say that the difference between
263
EISENSTEIN SYSTEMS
and
=-
A,. Note that the can be holomorphically continued to the region Re([) extra integrals thrown into the first expression are automatically holomorphic, thus cause no difficulty. Analogous considerations apply when [ is replaced by 4. Consider
{A(Ai: X I . . .) - A(X:. . .)}. By hypothesis, A(A4: X: . . .) is holomorphic throughout the region of integration. On the other hand, A(X: . . .) may well have singularities somewhere in the region of integration. [In this connection, bear in mind that it is the sum of the A(X: . . .) which is holomorphic throughout the region of integration; cf. Proposition 5.9(V).] It will be necessary to move the integrals to another region where the summation signs can be pulled outside the integral sign. This requires some preparation. Because %(go) is a finite set, we can form EX(&)
=
minx.X(Wo) Ex(&),
and then assume without loss of generality that KAPio I Aio : E ) has radius E ~ ( E ) ,so that, henceforward, we work only with C(X(Pio,Aio): E X ( & ) :
E).
Let X E Xiopthen, by definition, Fr will be that subset of Cpio(g,ai0) consisting of those with the property that X(Dis(X)(Pio,Aio) = 0. This said, suppose that $ is a hyperplane in X such that $(Pi,, Aio) meets I B m (X(Pi0 AiJ)* If 9
E(X: PIA,: PioIAio:To:A!o: x)
264
CHAPTER 5
Both RBe(X(Pi0, AiJ: X) RBe(WPi0 A i J : f‘~) 9
admit a finite decomposition into a disjoint union of open convex sets which, heuristically, may be thought of as “chambers.” Of course, distinct chambers per RB,(X(Pio,Aio):3)may lie in the same chamber per RB,(X(Pio,A d : FS). In particular, therefore, we can write WAC:Q
RBs,(e)(WPio,Ai,J: X) = C
the set on the right having cardinality c(X), say, a number depending only on the equivalence class of X in X ( { G } ;W o ) (=X(Wo)). Fix general points A,(&: X) E WC(.e: I). Take now two general points A:o,
E
RBEx(c)(X(Pio,AiJ:
which lie in the same component of RBEX(e)(X(Pio,AiJ: f‘d
.I"ilP (cv
ssal
592
266
CHAPTER 5
the difference can be holomorphically continued to the region Re([) > A,. The same conclusion holds if we multiply through by l/m/ and sum over 1. In contradistinction to the first expression, the summands in the second expression are themselves holomorphic throughout the region of integration; thus, for it, the summation signs can be pulled outside the integral sign. Therefore, making an obvious change of variable, we can say that
is equal to
x
A(X: wJoio:A!o: [) ldA!ol.
For each w,foio(l),a given chamber in RB,,(&)(X(Pio Aio) 3)
contains one and only one of the points w ~ o i o ( l ) w ~ ( X,) l ) ~ c ~ ~(1: 5 c 5 CI).
S o , adding and subtracting the integrals per /Ic(&: X) and taking into account what was said in the preceding paragraph, we conclude that the difference bet ween
and
x
A(X: wjoio:
c) ldA,fol
cttn be holomorphically continued to the region Re([) > I , . Let us recapitulate what has been established up to this point: The difference between (R(4'; 0'1 Res(Q~)@,f~(f'I Ail, @YJ~(P I A,))
EISENSTEIN SYSTEMS
267
and
can be holomorphically continued to the region Re(c) > A&.Analogous considerations apply when is replaced by [. It will now be necessary to work with
<
in order to pick up residues. Write
the set on the right having cardinality c(F,), say, a number depending on 3 (and not just its equivalence class in X({G}; Wo) ( =X(Wo))). A given chamber WC(&:F,) contains at least one and perhaps several chambers of RBs,(e)(X(Pio, AiJ: 31, say C , of them. Fix general points A,(E:F,) Rewrite
in the form 1 I-x C(3)
c c
1 s c s c ( x ) wfo,o
E
WC(&:FX).
268
CHAPTER 5
ex
The notation is slightly deceptive since the first usage of the symbol refers to a sum over @, n Zi0(Vk)whereas the second usage of the symbol Ex refers to a sum over &o(%?k).One should also bear in mind that c(X) = c, for all X E (5,. This explained, employ the principle spelled out earlier to conclude that the difference between
and
[J
I B , ~ ( ~ ( P ,A,,): , , AX)
-s
1
A(3:wjoio:A!,: 5) ldA!ol
I B ~ ( S ( P , , A,,): ,
ME:
F ~ ) )
can be holomorphically continued to the region Re([) > I,. For purposes which will become clear shortly, we shall now introduce a third index set I;, whose elements i l are ordered triples (3,c, fix where 3 E Xi,, 1 Ic I c(F,), and fi is a distinguished hyperplane in 3 with the property that A, and A,(&: F,) are on opposite sides of $(pi0 AiJ n RBc,(,)(X(Pio Ai0))3
9
It can be supposed that 1; is disjoint from I;, and Iy0. There is a certain set of data which must be attached to the elements of 1;. To begin with, associate with (3,c, 8)the Eisenstein system
belonging to fi, {E, V} being the Eisenstein system belonging to 3. [Delete the index if the corresponding Eisenstein system is trivial.] The real unit normal implicit in this construction is chosen so as to point out from S(Pi0, AiJ n RBc,(e)(X(Pio, Aio))
toward A,. Next, assign to (3,c, fi) the “epsilon function” determined by 3 itself, namely E~ (cf. supra). It remains to specify the ambient nonempty open “ I/-subset ” of H(Pi0, Aio)
+ &(Pi,, AiJ“
naio.
269
ElSENSTElN SYSTEMS
Note that H(Pi,, A,,,) = X ( P i , , Aio).Select a polygonal path
10, 11
~ c :
+
RBzr(e)(X(Pio,Aio))
starting at A,(E:F,) and ending at A, which crosses the “ w-hyperplanes” and the “A-hyperplanes” in a semiregular manner. This can be done in such a way that each hyperplane is crossed orthogonally, no hyperplane is crossed twice, and no hyperplane is crossed at all unless Ac(&:F,) and A, are on opposite sides of it. The connected component of the point where yc crosses $(Pi,, AiJ in ($(pi09 Aio)
-
u
$'(Pie, AiJ) n RBzx(e,(X(Pio, Aio)),
8‘+5
8’ being a “w-hyperplane” or a “A-hyperplane,” will be taken as the “ V subset” corresponding to (3,c, 9).Denote this set by the symbol Va(Pi0I Ai,
when no confusion is possible. To secure the shrinking property, it can be assumed, with no loss of generality, that > c2 VXPioI Ai,:
~ 1 = )
I/x(PioIAio: ~ 2 n ) RBc,(e,JX(Pio, 40)).
Suppose given X E Xi,. Fix c, 1 Ic Ic(F,). Let us assume that xyk is 3-admissible, i.e., that X E Xi0(%J. Consider those indices (X, c, $) such that xy, is $-admissible. Write for a sum over the set of fi singled out by this requirement. Since X and c are fixed, the notation is permissible.
c5
Lemma Let As E V5(PioIA,,: &)-then the difference between
A(X: wjoio:A/o Aio): A&: F x ) )
and
A($: wJoio:A/,: () IdA/, can be holomorphically continued to the region Re(() > &. [Note: In the first expression, the sum over wJoiois relative to the elements of W&,(X; A,,, A,,), while in the second the sum over wjoio is relative to the elements of W&,($;A ~ , A,,).] , Proof It should be pointed out that the integrand appearing in the integral relative to 9 is holomorphic throughout the region of integration, so that, for the usual reasons, there is determined a function of (,holomorphic in and A2,s lying on that Re(l) > R 2 . This being the case, choose points part of yc which intersects $(Pi,, A,,) orthogonally. We shall assume that lies in the same half-space as A,(&: F,) and that A2,5 lies in the same
270
CHAPTER 5
half-space as A,. Modulo a function holomorphic in the region Re(() > A,, one can then write
[s ;[s
-I -s
I B . ~ ( x ( P , ,A . , ~ )A: ~ )
as
1 1
A@: wjoio:A!o: i)I dA,toI
I B ~ ~ x ( PA , ~ ).A'(&: : FX))
IBd(X(Pt0.Atoh Az9 8 )
A(3: wJoio:A!o: () ldA~ol.
IBd(W'n0, Atoh A I . 6)
Fixing fi, let A0,$ be the point at which ye crosses b(Pio, Aio).It will be enough to show that wJot0
less
[s
IB~(x(P,,A , , ~ )A*, : 6)
271.
-s
c1
wfoto
1
A ( 3 : wJoio:A!o: i)ldA!ol
IB.~(x(P,,A . ,~):
8)
A(fi: wjoi0:A/o: () IdA!ol
IBt(b(Pio,
Ao, 6)
can be holomorphically continued to the region Re(() > &. Let Al be a real unit normal to !ij(P,,, Ai0)pointing in the direction of A2,5. Determine real constants cl, and c2, by writing
i
+ Cl.bA1 + C2.bAl
< 0) > 0). Fubini's theorem, in conjunction with the residual definitions, then implies that the difference is equal to h . 5 = A0.5 A2.5 = A0.5
x A(X: wjoio:A!,
(C,,$
+ z A 1 : () dz ldAJOl,
the inner contour being as shown in Fig. 8. Here, of course, the summation over wJoiois relative to the elements of W&,(3;Aj,, A+. Under the assumption that Re(() > R2,use Cauchy's theorem to shift the integral over
C(C,,~: c2,&:0: A!o) to the contour shown in Fig. 9. To finish up, we need only show that Re((A/,
+ z A f , AIo f z A 1 ) ) I A,
on the contour. As this kind of calculation has been written out several times before, it can be omitted. //
27 1
ENENSTEIN SYSTEMS
FIG.8
I__
f
+ FIG.9
Remark There is a potential minor difficulty in the preceding computation which we deliberately ignored, viz., in order to apply Fubini's theorem legitimately, it is necessary that the contours stay at least a fixed distance from the singularity. The way out is to put, if need be, a bump of predetermined radius r into the horizontal line segments at the point where they cross the y-axis. This does not affect the argument in any way. Now where are we? We can certainly say that the difference between
( R ( i ; 0') Res(Q~)@~l~(f' I Ai),@vJ~(P I Ail)
272
CHAPTER 5
and 1
ob’.
:co 1
1
cc c
wf,,
X
+ can be holomorphically continued to the region Re(() > &. Analogous considerations apply when [ is replaced by 4. Recall that Iio= 11, u 1s. Change the notation so that this time Iio = I:, u 1r0u 1;;. Suppose we throw away those X such that Re((X(Pio, AiJ X(Pi0, AiO))) I Naturally, nothing is lost in doing this. The preceding lemma then implies that the difference between and
(Nl; 0’)Res(Qp)%io(P I A i l @ Y J ~ (I Ail) P
(2n)bt l -
SIB:(XdPi0,
A(&: wjoi0:A/o: [) ldA!ol,
Ato): AI)
summed over all i E Iio(Wk)having the property that Re((Xi(Pi0, Aio), Xi(Pi,, AiO))) > - c 2 and over all wjoioE W&,(Xi; A j o ,Aio), where E UPio
I Aio: E),
can be holomorphically continued to the region Re([) > &. Analogous considerations apply when is replaced by 4. We are, at last, in a position to take the first step in the construction of the residual amalgamation, viz., the definition of Res(Sio).Tentatively, Re@,) will be the set of all equivariant systems of admissible a f i e subspaces of dimension b - 1 attached to Wi0 for which there exists an index i E I, such that X = Xi. Observe that here it is a question of a finite set of i. We then attach an Eisenstein system {E, V} to each such X according to the prescription
jE(X:. . .) = P ( X : . . .) =
1E(Xi: . . .)
c V ( X , : ..
i:X1=X
i:fr=X
.).
273
EISENSTEIN SYSTEMS
Delete X if this prescription yields a trivial Eisenstein system. [Note: Distinct indices i may very well yield distinct Eisenstein systems even though they give the same X,.] There are, of course, certain conditions which the elements of Res(X,,) must be shown to satisfy. The fact that
i
Geom: I Geom: I1 Geom: I11
are in force is clear. The verification of Am-Geom: A for Res(X,,), i.e., that for every X E Res(Xi,), -Re(X(Pio, Aio)) E dP,o((X(Pio,A i J " n Q'), is simple enough. Indeed, the only possible issue would be if X were equal t o some element coming from 1: but equal to no element coming from I:,, or I:.:,. Suppose, for definiteness, that there exists an index (Xo, co, 90)such that X = Go. Write Jso(Pio,
Aio)
= W P i o , AiJ n (XtAPiO, AiJ
+ ~er(&)),
where 1, E Fxo. Recall that Ho(Pio,A,) = Xo(P,,, Aio). One must show therefore that -Re(Xo(Pi,, AiJ) E dPi,((GO(pio, Aio)" n aio)'). Since it is already the case that -Re(Xo(Pi,, Aio)) E dP,,((WPio, A i J " n hJ'), there exist, by definition, elements I Z E E ~ , , (aio), ~ , with xlXo(Pi,,A,,)" = 0, and nonnegative constants c1 such that
- Re(XO(Pi0 Aio)) = C ~1 1. 9
Because Sjo(Pio,A,,)" is contained in Xo(Pio,A,,)", this fact implies our assertion. Am-Geom: B, Am-Sing: I, and Am-Sing: I1 (Am-Sing: 111 is not applicable) are predicated upon the existence of E~ and Vx(PiolAio:E), which, accordingly, must be defined before anything can be said. What will be done is this: Construct E~ so as to render Am-Sing: I1 true and then construct Vx(PiolAio:E ) so as to render Am-Geom: B and Am-Sing: I true.
274
CHAPTER 5
Let 6,be the set of all hyperplanes missing IBm (3(.. .)) which are singular hyperplanes of some V(3,: . . .), 3' = X . The collection 6,(Pi0IAio)of such in 3E(Pio,Aio)is locally finite, thus their union is closed. Let &x(PiolAi0):
IBm(3(Pi", AiJ)
+
R+
be the corresponding distance function. Put =
t min{min dQI(P,oIA,o)IIB,(3(Pio,Aio)X R - IIRe(X(Pi0, AiJ>II, 11.
Then it is clear that any singular hyperplane of V ( 3 : . . .) which meets C(3(Pi0,Aio):Ex(&):
E)
also meets IBm(X(Pi0, AiO)), and this is the content of Am-Sing :I1 for 3. We must now modify t, to produce and then define V,(Pio I Aio:E). For this purpose, let
E,
bn,(Pioy AiJ be an enumeration of the singular hyperplanes of V ( 3 : . . .) which meet B l ( P i o 9 AiiJ
* * *
9
C(X(Pi0,Aio): 1 : E). As E increases, more hyperplanes may appear. We can and will assume that the enumeration then satisfies an obvious consistency condition. Write
Bi(Pi0, Ai,J = {Aio E X(Pio,Ato): Ji(Aio) = ci}. Put
,in{
I W c i ) - Re(Ji(X(Pi0, Aio))) I . IIiiII
M c i ) # Re(Ji(X(Pi,, Ai,))) (1 Ii In,)
11 .
Needless to say, Am-Sing: I1 is still valid for 3. Take then for Vz(PiolAio:E ) any component of RB,,(,)(WPio AiJ: 3). With this choice, Am-Geom: B for 3 is, of course, immediate. To discuss Am-Sing: I for 3, let 1
AXE VdPio I Aio: E )
275
EISENSTEIN SYSTEMS
and suppose that IBc(X(Pi0 AiJ : AX) meets Bi(Pi0,A,) for some i-then, of necessity, 3
1 Therefore
the first expression being summed over all X E Res(Xio(Vk)) having the property that Re((X(Pi0,
AiJ
X(Pi0, A d > ) > - c 2
and over all wjoioE W&,(X;A j o ,Aio),where E
h(Pi0I Aio:E),
the second expression being summed over all i E Iio(Vk)having the property that Re((Xi(Pi0, and over all w;,]~,] E W',,(X,;
AioX
Xi(Pior AiJ>) >
- E ~
A,,,, Ai,,), where E
&(pi0I Aio: E ) ,
can be holomorphically continued to the region Re((') > 1,.Analogous considerations apply when (' is replaced by 4. [Note: Conventionally, we take A,, = As for all i such that Xi = 3.1
276
CHAPTER 5
Observe that the first expression in the preceding difference is exactly what we want for Am-An in the residual picture. Let us now move on and define Res(Ifo).Tentatively, Res(IIo)will be the set of all ordered pairs
(i. 81, where i E 11, u 1; and 9 is a singular hyperplane of V(&: . . .) meeting ZH,(3Ei(. . .)) with the property that @(Pi,, AiJ n RBca(Xi(Pi0, AiJ) separates A, and AXl. Associate with (i, 8)the Eisenstein system {Resb(E), Resb(V)}belonging to sj, {E, V} being the Eisenstein system belonging to Xi. [Delete the index if the corresponding Eisenstein system is trivial.] The real unit normal implicit in this construction is chosen so as to point out from $(Pi0
9
Aio) n RBca(Xi(Pi0 AiJ)
toward Ai. It is not difficult to concoct E,. and i$(PiolAio:E ) in such a way that the requisite conditions are met. The details are similar to but simpler than those presented above, and hence can be omitted. Performing a familiar manipulation, we can then say that the difference between (R(C;
'1
R~~(QP)@cJ~(P I Ail, @,j0(PI Ail)
and n
n
the first expression being summed over all X E Res(Xio(Wk))having the property that Re(
IAio: E),
the second expression being summed over all i' E Res(Iio(Wk))having the property that Re(
277
ElSENSTElN SYSTEMS
and over all wJoioE W&,(Xit;A j o , A d , where A,. E c.(PioI Aio: E ) ,
the third expression being summed over all i" E If0(Wk) having the property that Re(
Aio),where
A,.. E &,,(Pi0I Aio:E),
can be holomorphically continued to the region Re(c) > 1,. Analogous considerations apply when [ is replaced by 1. Observe that the first and second expressions in the preceding difference are exactly what we want for Am-An in the residual picture. We have yet to come to grips with the most difficult part of the proof, viz., the determination of Res(Ifo).The reader may have wondered to what extent some of our conditions have actually been needed. Of course Am-Sing: I, 11, and 111, as well as Am-An, have all seen service. On the other hand, the role thus far of the geometric conditions Am-Geom: A and B is less clear. In fact, they have scarcely been used at all. But this is the place where their presence will finally be felt. Let us begin by stating a simple geometric criterion. Let fi be an admissible hyperplane in X, say
Lemma
$(Pio, Aio) = {AioE x(Pi0,Aio) : &AiJ = c>.
Let A0 E X(Pio,Aio)
+ X(Pio,A,,)"
n &io*
Then fi(Pi0, AiJ n IB,(X(Pio,Aio):Ao) #
0
?fs (i) (ii)
4 E
~
*
-0X(Pi0, Aio)) = ~ e ( c X(X(Pi0, AiO))), 1 1 0 p2 I ~ m ( c - %X(Pio AiJ)) I*
II1I W P i o AiJ9
9
Corollary Suppose that A,,
are such that
A2
E X(Pi0, Aio)
+ x(Pio,A,,)"
$(Pi,, Aio)n IB,(X(Pio, Aio): Ai) = 0
n 61,
(i = 1,2).
If E .
IIXIWPio, A i J * I O P 2 I I ~ (-C%X(Pio, Aio)))I,
278
CHAPTER 5
then there exists t E 30, 1[ with the property that $(Pi,, AiJ n IB&(X(Pi,,Aio):t A ,
8
i
X(A1
+ (1 - t)Az) # 0'
X(Pio,Ato))- Re(c - X(X(Pi0, Ad)) - X(Pio,A,,)) - Re(c - X(X(Pi0, A,,))) -
have opposite signs.
Fix an index i" E I;,-then sets
attached to i" are two nonempty open convex
i
&(Pi, IAio:E ) Vx,#,(Pio I A,,: E ) .
Bear in mind that Vxl,,(Pio~Aio: E ) is actually a cone, defined earlier in the E ) be the set of all singular hyperplanes $3 of Res(X,,) picture. Let So(?', V(XI.,:.. .) such that $3(Pi,, Ai,) n IB&XdPi", A,,):
for some 12, E Xlt,(Pio,A,,)
Z0
+ XI.,(Pio,A,,)" n tii,.
Since { & ~ o-
X,(Pi,, A,,)): ~o
E xv(~i,,A,,)
fills out R,it is clear that $3 E So(?', E ) iff E.
+ ~ i , , ( ~~i ,i,, ) -n
IIJ-IWPio* Aio)"IIop 2 I I ~ ( C J-(Xi,,(Pio, AjJ))I,
it being assumed, of course, that b(Pi0, AiJ = {Aio E &**(Pi,,A,,) : J-(A,,J = c}.
The union
8o(i") =
u So(i",4
&>O
,
is the set of singular hyperplanes of V(X,..: . . .). This said, fix $3 E So(?',E ) and take points
i
A1
E &,,(Pi,IAi,: E )
A? E Vxl,,(PioIAi,:E).
Owing to Am-Sing: I,
$3(Pio,Aio) n IB&(Xv(Pio,A,,): A,)
=
0
(i = 1, 2).
v
or, there exist A, {A2
E
€
t+(PioIAi0:E ) V~ 1 (Pi0 I A io : E )
and t €10, 1[ with the property that
$(Pi,, Aio) n IBz(XdPi0, Aio):
+ (1 - t)A2) f 0.
Let @(i”, E ) be the set of all fi in &,(i’‘, E ) possessing either one of the two preceding equivalent properties. Suppose that > &,-then
i$,(Pio~Aio: E l ) c I/;..(PioIAi0: E,) vxl~~(Pio 1 A;,: ~ 1 c) vxl,,(f‘io I : E Z ) , so it is clear that
$(if’,
E,)
c $(i”,
E ~ ) .
280
CHAPTER 5
Put
$(i")
=
u $(i",
&
E).
>O
We are now in a position to define Res(Iro).Tentatively, Res(I;,) will be the set of all ordered pairs
V', $1, where i" E Ir0 and $ E $(i"). Associate with (irr,$) the Eisenstein system {Res,(E), ResB(V)}belonging to !+j, {E, V} being the Eisenstein system belonging to &. [Delete the index if the corresponding Eisenstein system is trivial.] The real unit normal implicit in this construction is chosen so as to point out from (120 : $(Pi0 AiJ n IBc(XdPi0 A iJ : Ao) # 0 } 3
7
toward V,4PioIAio: E),
it being assumed, of course, that 8 E $(i", construction of the sets Yi,,, ,,(Pi0
E).
Our first objective will be the
I Aio : 6).
Unfortunately, this is a rather involved and complicated business. Lemma
Let $ be an admissible hyperplane in X, say
@(Pi,, AiJ = {Aio W P i o ,AiJ : 4 A i J = C I . Then there exists a unique point
A d $ ) ~ W p i oAio)" , n 6io with the following property: For all E X(Pi0, AiO)
+ WPiO, AiO)" n 6io
such that A.
+ flAs2($)
$(Pi09 AiJ n IBe(X(Pi0, AiJ: A,)
+ 0,
is that element of
$(Pi09 Aio) n IBe(WPi0, AiJ: Ao) closest to A,.
[Let Bw(Pi0,Aio)stand for the A0 E X(Pi0, AiJ
+ X(Pi0, Aio)" n 6io
28 1
EISENSTEIN SYSTEMS
such that $3(Pio, AiJ n IBco(X(Pi0, A i J :
One then has
i
H(Pi,, AiJ = HR(Pi,, Air,) + H(Pi,, AiJ + $(Pi,, AiJ aio as can be checked without difficulty.]
Let &(i”,
E)
=
+ 0.
fia(PiO3 Air,)
+~ M B ) ,
be the set of all fin,$ belonging to %(if’,E). Put &(if’) =
u &(i”,
E).
&>O
Let 2l(i”, E ) be the set of all !& n !iji !, & and .f& belonging to $jsr(i”,E). The elements % of “(i”, E ) are thus affine subspaces of codimension 2. Split 2l(i”,E ) into two parts by the definition
ia1(i”,
E ) = {aE %(i”, 8 ) : Xl,,(Pi,, Ai,) E %(Pi,, Aio)} 212(i”,E ) = {% E a(?’, E ) : XI..(Pi,,A,,) # %(Pi,, Aio)}.
Put
/
i
a l ( i N )=
u Nl(i”, u a2(i”,
E)
e>O
a2(i”)=
E).
&>O
It should be noted that both Wl(i”,E ) and 212(i”,E ) are finite, that in addition %,(ill)is also finite, and that finally a#’) is at most countable. By our assumptions, each l+(Pi0 1 Aio: E )
has a compact closure. Furthermore, the collection of such is nested with respect to E. Therefore, the intersection &>O
W P i o I Aio
E)
is nonempty, containing, say, A$. Let W2(i”, E)* be the set of all in a2(i”,E ) which meet the line segment joining X,..(Pi,, Aio)to A:,. Let RB(i”,E ) and RB(i”,E)* be real balls centered at XI.,(Pio,Aio)and A:,, respectively, such that the only E &(in, E ) which meet the convex hull of RB(i”,E ) and RB(i”,E)* are in N2(i”,E)*. One can certainly arrange matters so that E,
> E2
*
i
RB(i”,E ~ c) RB(i”, t 2 ) RB(i”,E ~ ) * c RB(i”,t 2 ) * .
28 2
CHAPTER 5
Let ‘illE 212(i”,&)*-then, by definition, Xi4pio AiJ 3
4 ‘WPio, AiJ
Therefore the pair (Xi*,(pio 7
%(pi03 Aio))
determines in a unique way a hyperplane in Xi4piay AiJ
+ %4Pi,,
AiJ“ n hi0
containing both. Since ‘9I meets the line segment joining XIt.(Pio,Aio)to A:,, such a hyperplane necessarily passes through A;,. Varying ‘illthen gives a finite collection of hyperplanes which, in view of what has just been said, slices RB(i”,E)* into chambers. A generic chamber %‘(ir’,E)* is said to persist if
%‘(i”,E)* n i$(PioIAi0:E ) # for all E‘ 2
0
E.
Lemma Persistent chambers exist. Proof Let {gP(i”, E ) * } be the set of chambers in RB(i”, E ) * . If, to the contrary, no chamber persists, then for each p we can choose an.c, 2 E such that
%,,(ifr, c)* n c..(PioIAio:c,,)
=
0.
Put c0 = max
E,,
P
Since I/i.,(Pi,JAi,:E O ) c c*,(PiolAio: E,,)
for all p,
%Ji”, E)* n l+(PiolAio:cO) = 0, from which it follows that RB(i“,E)* n c..(PioIAio:eO) = $3,
the union of the %‘,,(if’, &)* being dense in RB(i”, E ) * . Consequently, A?, 4 &,*(PioI Ai,: EO),
which is a contradiction. // Suppose that cl > &,-then it is easy to see that every persistent chamber of RB(i”,E ~ ) * contains a persistent chamber of RB(i“, As E increases, the number of chambers may also increase, but, of course, only in jumps. Since it
283
EISENSTEIN SYSTEMS
is a question of an at most countable sequence of selections, one can choose a persistent chamber %'(i'', E)* in RB(i", E)* such that c1
> t 2 3 %'(i", E ~ ) * c %'(i",
E~)*.
Put l+(Pio I Aio:E)* = %'(i'',
n l+(PioIAio: E).
E)*
Although vt,(PioI Aio:E)* is a nonempty open convex subset of Xi*,(Pi0, Aio)
+ xi,*(Pio,AiJ"
n
such that E1
> E l = L$(PioIAio:E l ) *
c l$*(PiolAio: Q)*,
it is still not the candidate used for producing
I
q i , , , @)(Pi0 Aio : E).
Further reductions are necessary. Recall that b,**(PioI A i o :
E)
was taken to be any component of RBcXI,.(E)(xi4Pio AiJ : xi,,). 9
We shall now make this choice more definite by requiring that meet the interior of the convex hull of
284
CHAPTER 5
then XdPiO, AiJ + t(Aio - Xi4Pi0, A i J ) E ~ ~ i ~ * ( p i o InA(iXoi )4 p i o , Aio) + &,(Pio, AiJ“
hioh
for t E 10, I]. Our contention is now immediate. Consider ‘Bl(i”,E ) u 912(i”,E ) * . Given % in this union, there are then two possibilities :
i
A? 4 %(Pi0 AiJ A:, E %(Pi,, Aio). 7
If the first eventuality obtains, then the pair (A?*, %(pi03 A i J )
determines in a unique way a hyperplane in X d P i o , Aio)
+ &(f‘io,
Ai0)- n hi0
containing both. If the second eventuality obtains, then this is no longer true: One simply has A:, E %(Pie, Aio),the latter being of codimension 2. Varying E then gives an at most countable collection of affine subspaces. The Baire category theorem implies that there exists a point At,,, which lies in none of these subspaces, but which is in &,,,(pi0I Aio: 1) n ~ ~ ~ ~ I AiJ , ( p i o
Let % ’ E %,(i”)-then, by definition,
At,,!# %(Pi, AiJ 9
Therefore the pair (A&,* %(Pi, AiJ) 9
9
determines in a unique way a hyperplane in Xi”(Pi0, Aio) + &(Pio, A,,)“ n 6io containing both. Varying %, this procedure then yields finitely many hyperplanes which may or may not meet i+(Pio I Ai,: &)*. Consider only those that do. The collection of such slices h,,(Pi01 Aio:E)* into chambers. Fix one and, without changing the notation, label it by the symbol l$(Pi0 I Ai,: E)* as well. We can and will assume that El
> E2
-
~ : . . ( P i , i A iEl)* o : c J$*(PiolAio:&2)*.
For t > 0, let us agree to write A&(t) = Xi*,(Pi,,Ai,)
+ t(A&
-
Xl*,(Pio,Aio)).
285
EISENSTEIN SYSTEMS
Lemma Given Ai.. E i+(Pi0 I Aio:E ) * ,
there exists a T > 0 such that if0 < t < T , then the line segment joining A&(c) to A,. misses every 2I E a(?’,E).
Proof Start the verification by choosing T small enough to secure the relation (0 < t < T),
A$,..(t)E RB(i”,E )
Let 2I E Nl(i”,&)-then, since the pair (A$,**(t), %(Pi0 AiJ) 9
determines the same hyperplane in Xi4pio9
Ai,J
+ xi,,(Pio,Aio)“
n hi,
as the pair
(AI,*. W P i o , AiJL and the latter misses l$,(PiolAia:E)*, it must also be the case that ‘9I itself fails to cut the line segment joining &,(t) to Alp..Recalling next that RB(i”, E ) and RB(i”, E)* were constructed in such a way that the only 2I E Nz(i”,E ) which meet the convex hull of RB(i”,E ) and RB(i”, E)* are in N12(i”, E)*, we see that if 2I E !&(if’, E ) - Nz(i”,E ) * , then again 2I fails to cut the line segment joining A,*,..(t) to Ai... Finally, we must consider what happens when 2I E 2112(i”,E ) * . Let us remind ourselves that 3
i$,(Pi0I Aio:E)* c %(i”, E)*.
In particular: The line segment joining Xi..(Pio,Aio)to A,. misses every element of N2(i”,E)*. Because N12(i”, E)* is a finite union of closed sets, there exists a neighborhood N of Xi@ia., Aio)in RB(i”, E ) with the property that for all Aio E N,the line segment joining Aioto A,. misses every element of ‘U,(i”, E)*. Adjusting T, if necessary, to force the relation A$,.,(t)E Jlr
(0 < t < T )
then finishes the proof. // Remark The preceding argument yields an obvious uniformity, namely, that the assertion is true as it stands when A,. is allowed to vary in a compact subset of V,.,(PioI Aio:E ) * .
Let !ijE &i”, &)-then $,(Pi,, Aio)separates l$(Pi0 IAio: E )
and
&,,,(Pi,IAio: E).
286
CHAPTER 5
Form fig(Pi0,
AiJ -
u (bd,
O(E)
the symbol on the right standing for the union over the complement of !&(Pio, Aio)in [$,,(i”, E ) ] ~(obvious notation). Given Ai,, E F,,(PioI Aio:E ) * ,
choose T as in the lemma supra. There is no loss of generality in supposing that matters have been so arranged that tE
10, T[ * A&,,(t)E V,,,,(PioI Aio:E ) n RB(i”, E).
This said, fix t €10, r[-then segment
the crossing point in !+&(Pi,, Aio) of the line
+ (1 - s)Ai,,
(s E [0, 11)
sAg,..(t)
determines a component of &(Pi09
AiJ -
u (8d.
O(E)
Ostensibly, the component thus singled out depends on the choice of Ai.., T, and 1. But, in fact, this is not the case. Let, say, A,’.., t,,t , , and t 2 ,t 2 be given. The line segment L , , joining A,?.to A:, is compact; hence one can select a 2‘ with the uniformity mentioned above and used below. Denoting by seg(. . .) the line segment joining ..., the union of {seg(AxIw(t), A:,) n SR(Pio, Aio): r/2 I t I t l > {seg(Ax&/2), AiJ n B d P i o , A i o ) : Aio E L I ,21 (seg(A,,..(t), A:,) n 8,(Pio, Aio): r/2 It It 2 } is a polygonal path connecting the respective crossing points in &(Pi03
AiJ
-
u (lid.
Let q i, , , BJPio
I Aio: 8 )
be the intersection of the R-tube with that component of 4SdPi0, A i o )
-
u
($01)
o(e)
distinguished by the preceding data. It is clear that
I
Ci,,,B)(Pio A i o :
EISENSTEIN SYSTEMS
28 7
is a nonempty open convex subset of !&(Pio, Aio)such that
such that There is no need to specify an “epsilon function” since it plays no role in the double prime picture. One must now verify that the axioms are met by Res(I:’,) and associated data. Geom : I and Geom : I1 are immediate, as is Am-Geom : B (which requires no argument at all in the present case). Geom: I11 and Am-Geom: A are interconnected. Let us recall that they assert, respectively, that
Put 6, = 5(Pi,l,Ai0). n hi,,-then 6 , = n ( A = 0), A in some subset of Xp,o(g,Ai0). [Note: Needless to say, 6, is not necessarily distinguished.] Our
288
CHAPTER 5
constructions imply that there exists a point Aio E 5j(Pio,Aio)with the property that - Re(Ai,) belongs to the interior of the convex hull of and
~P,,(6io)
&P,,(a$).
By definition, Q!JP,,(ak)
is contained in 6.; If we project onto 6$, then -Re(Ai,) goes ,to - Re(H(P,,, A,,)), so we can say that - Re(H(Pio,A,,)) belongs to the interior of the convex hull of pro,(~P,o(6io))
and
.aP,o
Prog(Wp,,(lli0))standing for the projection of Wp,,(6io) onto 62. Suppose we could show (i) Prog(Wp,,(6io))is contained in &p,,,(6k), (ii) &p,o(6~) is contained in &pai,J6&(,)). The first point would then imply that - Re(H(Pi,, A,,)) belongs to the interior of the convex hull of &P,,<6$>
and
.aP,o
i.e., to &,,,(6$) itself, thus giving Am-Geom: A. The second point, in conjunction with what has just been established, then leads to Geom: 111.This said, let us proceed to the verification of (i) and (ii). For (i), observe that Pro8(Wp,o(6,,)) (cf. Lemma 6 is contained in WP,,(6$), which, in turn, is contained in &,,(a$) and its Corollary in the Appendix to Chapter 2). For (ii), let A1, A,, . . . be the simple roots in XP,,(g, aio); let A', A2, . . . be the corresponding dual roots (so that (Ai, A j ) = Gi)-then each element of & p , o ( 6 ~can ) be written as a nonnegative linear combination of the Ai., the coefficient of Ai actually being zero iff A' E 6,. Since the span of the A', A' E ,6, is Dis(fi)(P,,, Aio),and since is the orthogonal complement of Dis(fi)(Pio, Aio) in 6io, it therefore follows that &p,,(a;)
c &P~,,.,,,(6&~)),
as desired. Of the singularity conditions, only Am-Sing: I is applicable. Let AV E Yv, g)(PioI Aio: 8 )
(fiE Hi",E))*
The claim is that IB.AJS(Pio, AiJ: Av)
EISENSTEIN SYSTEMS
289
We are now in a position to make the final push, viz., the verification of Am-An in the residual setting. In order to reorient ourselves, let us recall that we know so far that the difference between
( R ( l ;0') Res(Qp)@,:,(P I Ail, @YJ,(PI Ail) and
290
CHAPTER 5
the second expression being summed over all i’ E Res(I~,(‘ek))having the property that Re((Xi*(Pio,AiJ, Xi,(PiO, AiJ>> > - e 2 and over all wJoioE W$,(X,.; A,,, Aio), where Air E
K,(PioI Ai,:
E),
the third expression being summed over all i” E I:,(%?&having the property that Re((Xi“(Pior AiJ Xi,,(Pio,Ai,J>> > - g 2 and over all wjoioE W$,(X,..;A j o ,Aio), where A,,, E &,,(Pi,I Aio: E),
can be holomorphically continued to the region Re(() > 1,. Analogous considerations apply when ( is replaced by f . What we must do, therefore, is manipulate the double prime expression in such a way as to bring it into residual form. Keep in mind that
i
Alp,E &(Pi,1 Aio: E ) Ax,. E V’,,,(Pio I Ai,: E )
are at our disposal. Fix
A,,, E L+(Pio I Ai,: 2 ~ ) *n V,(Pio I Aio).
Let J1’ be a small open neighborhood around A,,, with compact closure P. Select a r > 0 such that if 0 c t c T, then the line segment joining A$,,,(t)to any point in JV misses every % E %(i”, E). There is no loss of generality in supposing that t €10, r[
-
A$,.(t) E Vxl.(Pi,IAi,:E ) n RB(i”, E).
Fixing t €10,T[, set A$,.(r) = Ax,. and let y,., be a polygonal path from Ax,- to A,.. lying in the convex hull of Ax,..and J1’ obtained from the line segment joining Ax,,, to Aip,,altered so as to cross each Bq(Pio,Aio)(9 E Hi’’,E ) ) orthogonally. In passing, note that distinct 5 may very well yield the same $3%. Ultimately, it will be a question of summing over the (i”, $3), xyk being !+admissible, where Re((H(Pi0, AiJ
H(Pio,
Ai,J>) > -c2*
A short calculation implies that for such Sj, one necessarily has
Re((X,..(P,,, Aio), XI..(Pi,, A d ) ) > - c 2 - R 2 . We can, of course, throw in the extra integrals arising from these Xi.. without changing the situation in any essential way.
29 1
ElSENSTElN SYSTEMS
Fix, then, an index i” and assume that the normal translation of the corresponding XI.. has the property that Re((X,,.(P,,, Aio),X,..(Pio, A i o ) ) ) >
-E’
-
R2.
For each 9, coming from an Jj E &i”, E), choose points A1,&%and A’, bs lying on that part of y,.. which intersects !jj,(Pio, Aio)orthogonally. It will be supposed that A,,8pl lies in the same half-space as AX,”and that A2,swlies in the same half-space as Ai-. Call A0,& the point at which y,..crosses @,(Pi,, Aio). On the basis of by now familiar principles, one can then write, modulo a function holomorphic in the region Re([) > I & ,
[s [s
I B b ( ~ i ‘ V r oAt,): 9 Ai*‘)
as 6%
1
-s
I ~ ( X I ” ( P , OA,w l : A2.sy1)
A(&,: wJoio:A!,: [) ldA/,1
IBf(Xi”(J‘t0. 40): AII,,)
-1
I
A(X,..:wJoio:
[) IdA!,l.
IBf(Xi”(pi0. Aio): A I , 6%)
Here the summation is taken over the 43, determined by the J3 E $(i”, E). We explicitly observe that while yi.. may actually meet an !&(Pi,, Aio),where 8 is in the complement ofb(i”, E ) in&(i”), this presents, in reality, no problem at all. Generically, let fil, . . . , &, be the elements of $(if’, E ) such that ($,), = 9,. Naturally, xy, is X,..-adrnissible. However, this may or may not be the case for the 8,.[Note that xyk is $,-admissible iff xrgk is $,,-admissible, 1 I p, v I n.] If xy, is not $,-admissible, then J3, is not a singular hyperplane of V(X,..:. . .) (cf. Assumption 4(V)), so the usual considerations imply that the associated “!&-term” supra is holomorphic in the region Re([) > I , . As for what remains, we have the following lemma.
Lemma
and 2.Ic.
Suppose that xwk is 8,-admissible-then the diflerence between
icJ
r = 1 wfoio
1 ~ f ( 8 , ( ~~ 1i,,) :, ~
o6% , +J?
A(!&,: wJoio: AJ0:[) IdA!,l AW(~,))
can be holomorphically continued to the region Re([) > I , .
[Note: In the first expression, the sum over wJoiois relative to the elements of W&,(X,..; Aj,, Aio), while in the second, the sum over wjoi0is relative to the elements of w&,($,; A,,, A,,).]
292
CHAPTER 5
Let us defer the proof of this result for the moment. The definitions show that
+
~0.5s
f
l
~
~
Ep ~ i) ,b,,)(Pio ,, IAi,:
E)*
Since working with one point in yi”,Br)(Pio I Aio:E ) is the same as working with I Aio:E), it is now an easy matter to fulfill our expectaany point in Yi,,,Br)(Pio tions. No restriction has been placed thus far on Re((Hp(Pi0, AiJ
Hp(Pi0,
Aio)))*
Of course, we are only interested in the $j for , which Re((Hp(Pi0, AiJ, Hp(Pi0, Ai,))) >
- E ~ *
Throw away those not meeting this requirement. Add in the contributions from the $ in the complement of Hi”,E ) in @(i”), where xrg, is $-admissible, with Re((H(Pi0, AiJ H(Pi0, As))) > -c2, justified because Re((?, ?)) of the integrand is < I , . Consolidating our notations, we can then conclude that
less
A($: wjoio: the first expression being summed over all i“ E
C) IdA/,l, having the property that
Re((X,,.(Pio, Aio),Xi,,(Pio,Aio))) > - 2 - R 2 and over all wjoio E Wik(Xi”;A,,, Aio),the second expression being summed over all (i”, $5) E Res(Iyo(Wk))having the property that Re((H(Pi0,
Aioh
H(Pi,,
Aio)))
>
-t2
and over all wJoioE W&,($;A,, Aio),where 41*,,6 E ) 4I”.b)(PioIAi,: EL
can be holomorphically continued to the region Re(C) > A,. Analogous considerations apply when C is replaced by g.
293
EISENSTEIN SYSTEMS
Proof of Lemma Let AL be a real unit normal to $%(Pi,,Aio)pointing in the direction of A2,591.Determine real constants c1,8rand c2,891by writing
I
A,.gs = A O , & A2AI = AO,Br
+ C1,BaA* + C2*&AL
Fubini's theorem then implies that Wfoio
[I
IBL(Xi'.(Pio.Aio):A2.691)
is equal to
-I
IB~(Xi"(PiosAi0): Ai,fm)
x A(Xi.: wJoi0:A!o
(CI,& (c2.591
I
< 0)
'0).
A(Xi2,: wJoio:
5) I dAJ0I
+ zA1: 5) d z I dA!oI,
the inner contour being as shown in Fig. 10. Here, of course, the summation over wJoio is relative to the elements of W&,(X,.,;A j o ,AiJ. Needless to say, BdPiO, AiJ = HR(Pio, Aio) + BdPio, Aio)", where, for any p, SjdPio, AiJ" = $p(Pio, AiJ- n 610, so that the symbol rst!($dPio,
b,aS)
may be assigned the obvious interpretation. Let Am($,)
=
c, .
(C, =
FIG.10
ll~%($u)ll).
294
CHAPTER 5
Then
x A(&: wJoio: AJ0
+ zA1: 5) dz ldA!ol
becomes
x A(&: wJoio: A!o
+ zA1: C) dz ldA!ol,
the inner contour being shown representatively in Fig. 11. A small positively oriented circle Ci,, is drawn around a given C, if
J-1
IC,I < JE2
- - ~ _ _ _
-
IIA!o -
Y
FIG.I I
4,$a1129
EISENSTEIN SYSTEMS
295
but no circle is drawn if The contour is undefined if there exists equality for some p (which is possible, though, only on a set of A/o-measurezero). The contributions arising from the horizontal lines yield functions holomorphic in the region Re(5) > A,, so that they may be set aside. We are thus left with the contributions coming from the circles. Let IB,t(BdPio AiJ: &. 8% : p ) 7
denote the subset of
IBb(BR(Pi0, AiJ: Ao,8%) for which EZ
- llNo - Ai,&J’ > IC,I2.
What remains can then be put in the form
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CHAPTER 5
This is permissible, provided one knows that
+
+
flA&j,,) Re((Ab,,, whenever
Ab,,,
+ -A&(&)
+
( A i o E $ j p ( P i o , AiJ“ n aiJ - ICpI’ But the expression in question is equal to
11A/01122
E’
llRe(Ab,b,)112 - (Im(Ab,,,) or still
+ C, . + A!o, Im(Ab,b,) + C ,
+
IIRe(Ab,bsl)IIZ - (Im(Xr(Pio, Aio)) C, * A 1
A!
+ A!o>
+ AI0, Im(X,,,(Pio,Ato))+ C , .A! + A!o;
or still IIRe(Ab,8,)I12 - IIWXdPio, AiJ)II’ - ICpI’ - IIALII’, which is
in summary, the difference
(R(C; 0’)R~NQP)@,~~(P I Ail, @,j0(P IAil)
and
the first expression being summed over all X E Res(Xi0(Wk)) having the property that Re((X(Pi0, A,,), X(Pi0, Air,)>) >
-8’
291
EISENSTEIN SYSTEMS
and over all wJoioE W&,(X;Am, Aio),where Ax E h(f'i0
IAio : E),
the second expression being summed over all i E Res(Ii0(gk))having the property that Re((Xi(f'i0,
Aio),
XdPiO, AiJ>) >
-E'
and over all w;oioE W&,(X,; A,, Aio),where Ai E Y ( f ' i 0 I Aio: E), can be holomorphically continued to the region Re([) > I,. Analogous considerations apply when [ is replaced by 1. The construction of the residual amalgamation is therefore complete.
APPENDIX We shall establish here the following result. Stone-Weierstrass Theorem Let X be a compact Hausdor-space.Let % be a finite-dimensionalcommutative *-algebra containing C which is generated by C and the selfadjoint nilpotent elements. Let 5 be a subset of C ( X , a).Assume 8 is u wubulgebra of C ( X , %) containing the constants with the property that .for all points p , q E X , p # q, = { f ( P ) : f (4) = 0,f E 5).
Then: 3 is uniformly dense in C ( X , a).
Proof The proof proceeds by induction on dim(%). If dim(%) = 1, then our assertion follows from the usual statement of the Stone-Weierstrass theorem. Suppose, therefore, that dim(%) > 1. Fix a nonzero, noninvertible, self-adjoint element a. E %. Such exist by our hypotheses. Consider %aothen %ao is, via the identification'iNao zz %/Ann(ao),a commutative *-algebra satisfying the same general conditions as % itself except, of course, that now dim(21ao)
-= dim(%).
Similar remarks apply to the quotient %/%ao.Abstractly, there is a direct sum decomposition of vector spaces over C: %
2
%/%ao 0 %ao.
Let P be any projection of % onto %Ia,-then Ker(P)
2
%/%ao.
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CHAPTER 5
Fix a norm on 9l so that llall = Il(1- P)4I
+ IlPaII
(UEW.
Place on 21/21ao the norm determined by restriction to Ker(P). There are canonical maps
i
C(X, C(X,
+
+
C ( X , waao>
w,%ao)
carrying 8 to a *-subalgebra satisfying the same general conditions as 3 itself. This being so, fix E > 0. Let f E C ( X , a) be given. By the induction hypothesis, there exists g E 5 such that Il(1- P)f(X) - (1 - P)g(x)II < E / 2
(XEX).
By the induction hypothesis again, there exists h E 3 such that (x E XI.
IIfYf(x) - g(x)) - a0 hWll < &/2
Hence
IIfW - g(x)
- aoh(x)ll
Since 5 contains the constants, g
(x E X ) .
+ aoh E 5,thus finishing the proof.
//
Remark It is clear that the assumptions on % or 8 are subject to some relaxation, but one cannot push things too far. For instance, simple examples show that the theorem is false if every self-adjoint element in 2l is invertible or if 5 does not contain the constants. The first point may be illustrated by taking for X the closed unit disk in C, for ‘9l the quotient C[t]/(l + t Z )(with (a bt)* = si bt), and for 5 the elements
+
where
+
{”
(4+ J--ruz> =
u1
fz = uz
+f l u z +
+ (4 + f l v z ) t , are holomorphic on X.
The second point may be illustrated by taking for X the closed unit interval in R, for & the quotient C[t]/(t2)(with ( a + bt)* = si + 6t), and for 8 the elements
f + f’t
(fE Crn(COI11)).