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Chapter 9 Harish-Chandra Modules
CHAPTER 9
HARISH-CHANDRA MODULES
Let B be a real semi-simple Lie group with finite centre, X a maximal compact subgroup, go and Eo the Lie algebras of 3 and X , g and € the complexifications of go and to, and z a completely irreducible representation of Y in a Banach space (for example, an irreducible unitary representation of Y in a Hilbert space). Let V be the space of vectors which are X finite under z. Then V is a simple g-module such that V = eeaVQ.The study of z can for the most part be reduced to the study of this g-module. The modules which we shall study in this chapter are generalizations of the g-modules just considered. 9.0. Notation
In this chapter,€ will denote a Lie subalgebra of g and G and K the enveloping algebras of g and €.An essential role will be played by the commufant G' of € in G. If e E f", we shall denote the kernel of e in K by I,. If V is a g-module, we have defined Ve in 1.2.8.
9.1. The case of a Lie subalgebra which is reductive in g 9.1.1. For @,a€€, we set Ge"
= (U E
I
G Zeu c GI"}.
We write Ge instead of Gese.The Gelu are vector subspaces of G . 9.1.2. PROPOSITION (notation as in 9.0). (i) We have 'Ge*'r> GI" and GQ"/GZu= (G/G17e
when we consider GIGI' as a g-module for the left regular representation. (ii) If V is a g-module, then Ge*"(VU) c V,.
296
WI S H -CH ANDRA MODULES
[CH.9 . 8 1
We have PGI" c GI,, hence GI" c Ges". Every representation of K/Ie is semi-simple and is the sum of representations of class e (AL VIII, pp. 47-48). Hence, if V is a g-module, then
v, = { Y E
VI IQV = O}.
Firstly this implies (ii), because IeGe9"(V,) c GI"( V,) hand, (G/GI?@ = {u
+ GI" I u E G
= 0.
On the other
and l e u c GI"},
whence (i). 9.1.3. PROPOSITION (notation as in 9.0). (i) Ge is a subalgebra of G which
contains K as a subalgebra and GIe as a two-sided ideal. (ii) If V is a g-module, then GQ(Ve)c Ve and (GIQ)(Ve)= 0; we can thus consider Ve as a (G@/GF)-module. We have QGQGec GIeGpc GIe,
P K c Ie,
GIeGe c GGIe = G P ,
whence (i). Assertion (ii) follows from 9.1.2. 9.1.4. A g-module V is said to be a Harish-Chandra module relative to t
(or simply a Harish-Chandra module if there is no doubt about f) if V = CQccA Ve, whence V = eeeCVQ. The Verma modules are HarishChandra modules relative to a Cartan subalgebra. In this chapter, the subalgebras f that we have in view are in general very different from Cartan subalgebras.
9.1.5. PROPOSITION (notation as in 9.0). W e assume that f is reductive in g. Let e E 1". Then: (i) G/GIe is a Harish-Chandra 8-module under the left regular representation. (ii) G/GIe = G"*e/GP.
From 1.7.9, W = Xocc.. (G/GZe), is a sub-g-module of G/GIe. Now the class of 1 in G/GIQbelongs to W,and hence W = G/GIe. This proves (i), and (ii) then follows from 9.1.2 (i). 9.1.6. PROPOSITION (notation as in 9.0). W e assume that f is reductive in g.
Let V be a simple g-module, and e E f" such that Ve (i) V is a Harish-Chandra g-module. (ii) The GQ-moduleV, is simple.
+ 0. Then:
A L I E ALGEBRA REDUCTIVE IN
CH. 9, fiI]
297
From 1.7.9, V is a Harish-Chandra g-module. Let v E V, - {O}. Then V = Gv = 2, G'% from 9.1.5 (ii), and GU% c V, from 9.1.2 (ii), hence G@v= V,. 9.1.7. PROPOSITION (notation as in 9.0). We assume that t is reductive in g. Let V be a Harish-Chandra g-module, e E f", P the projection of V onto Ve defined by the decomposition V = cBUCte V,, and W a sub-Ge-module of V,. W e set Wmin -
W"""
GW,
= {vE
VI (Gv) A V, c W } .
Then: (i) W""" = { v E VIP(Gv) c W}. (ii) w"'" V, = wmax ve = W. (iii) Zf X is a sub-g-module of Vsuch that X A V, 2 W (or X A Vpc W ) , then x 3 Win(or x c wmax). For all v € V, we have Gv = ecCtA (Gv) A V,, whence (i). On the other hand, GW = G e W from 9.1.5 (ii), and G"@Wc V,, hence
zcCt.
W"'" A V, = GeW = W . Consequently, WmaX 3 Wmin,and it is then clear that W""" A Ve = W. Assertion (iii) is obvious. 9.1.8. PROPOSITION (notation as in 9.0). We assume that f is reductive in g. Let e E f", and let A and 9 be the sets of the maximal left ideals of G and Ge, respectively, containing GIe. (i) The mapping M H M A Ge is a bijection v of A onto 9. (ii) ZfL €9, then y-'(L) contains every left ideal M of G such that M GQc L. (iii) Let n be the canonical mapping of G onto G/GZe,and P the projection of G/GIe onto Ge/GZedefined by 9.1.5 (ii). Zf L E 9, then
I = {U E G I P(Gn(u)) c n(L)}.
v-'(L) = {U E G (Gu) A Ge c L}
The mapping M I + M/GZeis a bijection of the set of the left ideals of G containing GIe onto the set of sub-g-modules of G/GZe; if M $: G, then 1 $ M and hence M A Ge $: Ge. The mapping L H L/GZe is a bijection of the set of the left ideals of Ge containing GZe onto the set of sub-Ge-modules of Ge/GZe= (G/GZe),. Given this, the assertions of the proposition are easy consequences of 9.1.7 applied to V = G/GIe.
298
[CH. 9, D 1
HARISH-CHANDRA MODULES
9.1.9. PROPOSITION (notation as in 9.0). We assume that f is reductive in g . Let p E €".Let Y be the set of the classes of simple g-modules V such that Ve =+ 0. Let 9'be the set of classes of simple (GQ/GIP)-modules.Then the mapping V H V, dejines a bijection of Y onto 9'. If V E 9, then V, E 9" (9.1.6 (ii)). Let V,WE 9, let f be a G4-isomorphism of V, onto W,, v a non-zero element of V,, and L the annihilator of v in GQ;it is also the annihilator of f ( v ) in GQ,and it is a maximal left ideal of G4. Let M and M' be the annihilators of v and f ( v ) respectively in G. Then M and M' are maximal left ideals of G, and M A Ge = L = M' A Ge, hence M = M' (9.1.8), which proves that V and W are G-isomorphic. Let 2 be a simple Ge-module whose annihilator contains GIP, let z E Z - {0}, and let L be the annihilator of z in Ge. There exists a maximal left ideal M of G such that M A Ge = L (9.1.8). Let V be the simple gmodule GIM. Then G/GZe = @(G/GZe)u,
(G/GZe)Q= Ge/GZe
UEl-
(9.1.2, 9.1.5); hence
so that Ve is G%omorphic to 2. 9.1.10. PROPOSITION (notation as in 9.0). We assume that f is reductive in g. Let g E 'f". Then: (i) (GIe) A K = Ze, so that K/ZQcan be identijied with a subalgebra of Ge/GZe. (ii) G ' c GQ,so that G'/G' A GZe can be identiJied with a subalgebra of GQ/GZe; moreover, G' A GIQ= G' A P G . (iii) Zfg is absolutely simple, the canonical mapping of
(K/Z@) @ (G'/Gf A GIQ) into Ge/GZe is an isomorphism; we have GQ= KG'
+ GI@.
Assertion (i) follows from the fact that G is a free right K-module (2.2.7). Clearly, G' c GQ. Let V be a g-module such that V = @ucI.Vu. Let V* be the dual g-module. Let us identify @uEt-(Vu)*with a sub-€-module of V*. Let u E Gc A ZQG. Then U T E G' A GIQ', and Ze((Ve)*) = 0, hence uT((V,)*) = 0 and= = 0;
299
A LIE ALGEBRA REDUCTIVE IN
CH.9 , s 11
since u(Ve) c V,, we conclude that u(V,) = 0. Applying this to V = G/GZe, we see that uGe c GIe, whence u E GZe. Thus G' A IeG c GIe, and consequently G' A GZe = (Gf A Ie'G)T c (GIe*)T = ZeG.
This proves (ii). The adjoint representation of f in G is semi-simple, and Ge and GIe are sub-f-modules of G ; consequently, every element of Ge/GZeannihilated by the adjoint representation o f f is the canonical image of an element of Ge which commutes with t (1.2.11), i.e., of an element of G'. The commutant of K/Ze in Ge/GZe is thus G ' / d A GZe. Finally, if p is absolutely simple, the algebra K/Ze is central, simple and finite-dimensional, and hence Ge/GZe is the tensor product of K/Ze and its commutant in Ge/GIe (11.2.5). This proves (iii). 9.1.11. LEMMA (notation as in 9.0). Let V be a g-module, Z an absolutely simple f-module, e the corresponding representation, and H the vector space Hom,(Z,V) = Hom,(Z,V,). W e consider H as a left Gf-module by virtue of the action of G' in V (or V,). There exists one and only one isomorphisnt of the vector space Z @ H onto the vector space V, which transforms z @ 11 (z E Z , It E H ) into h(z). If we identifv Z @ H with V, by means of this isomorphism, then, for z E Z, h E H , u E K and v E Gf, u*
(Z @
h) = uz @I h ,
V . ( Z @I
11) = z @ vh.
This follows from AL VIII, pp. 13-15. 9.1.12. THEOREM (notation as in 9.0). W e assume that f is reductive in g. Let a ?-module of class e. Let Y be the set of the classes of simple g-modules V such that V, 0. Let 9- be the set of classes of simple (G'/G' r\ GZ@)-modules. (i) If V is a simple 9-module such that V, =j= 0, let us consider F = Hom'(X,V,) as a left Gt-module by virtue of the action of G' in V,. Then F is a simple G'-module whose annihilator contains G' A GZe. (ii) The mapping V H F dejines a bijection of Y onto 9.
e be an absolutely simple element o f f " , and X
+
Let V be a g-module and F = Hon@ rf,' V). Let us identify F @ X with V, by virtue of 9.1.1 1. If F' is a sub-Gf-module of F, I;' @ X is a sub-G'module and a sub-K-module of V,, and hence a sub-Ge-module of V, (9.1.10 (iii)). If V is simple as a g-module and V, 0, then V, is simple as a Ge-module (9.1.6 (ii)), hence F' @ X = V, and F' = F. This proves (i).
+
300
HARISH-CHANDRA MODULES
[CH. 9 , $ 1
Let V,V'E 9, F = Hom,(X,V), F' = Hom,(X,V'), and let ~1 be a G'isomorphism of F onto F'. Then 8 1 is a K-module and a G'-module isomorphism, and hence a Ge-module isomorphism, of F @ X onto F' 8 X (9.1.10 (iii)). Hence V and V' are isomorphic (9.1.9). Let F be a simple G'-module whose annihilator contains G' A GZe.Then, from 9.1.10 (iii), F 8 X is a simple Ge-module whose annihilator contains GIP. There exists a simple g-module V such that the Ge-module Ve can be identified with F 8 X (9.1.9). Then the G*-module Homc(X,Ve)can be identified with the G'-module F (AL VIII, p. 15). 9.1.13. With the notation of 9.0, let V be a 8-module, e E €", and X a 1module with class e. We consider Hom,(X,V) as a G'-module by virtue of the action of G' in V. Let cr be the corresponding representation of G'. Then u is said to be the spherical function of type e of V. If f is reductive in g, e is absolutely simple and V is a simple Harish-Chandra module such =then the spherical function of type e of V is simple and charthat VP=l0, acterises V up to isomorphism (9.1.f2); we shall see in 9.5.1 that it is finitedimensional. 9.1.14. PROPOSITION (notation as in 9.0). We assume that f is reductive in 9. Let e be an absolutely simple element of t"and X a €-module with class e. Let V,V' be g-modules, with V simple and Ve =I= 0. We assume the existence of sub-G'-modules Y,,Y, of Hom(X,V') such that Y, c Y, and Homt(X,V) is G*-isomorphicto YJY,. Then V is isomorphic to a subquotient of V'. We have V , f 0. By replacing V' by Cra.V:, we can assume that V' is a Harish-Chandra module. We have V, = Hom,(X, V ) @ X ,
Vd = Hom,(X, V')@ X.
Let Z1 = Y, @ X and Z , = Y, @ X. From 9.1.10 (iii), the Ge-module V, is isomorphic to the Ge-module Z , / Z , . Let Z; = GZ,. Applying 9.1.7, let us consider the largest sub-G-module Z ; of Z; such that Z ; A Vi = Z,. Then Z;/Z; is a simple Harish-Chandra g-module and (Z;/Zi)e= Z,/Z,. From 9.1.9, V is isomorphic to Z J Z ; . 9.1.15. We assume that k is algebraically closed. Let tj be a reductive Lie algebra, and u a finite-dimensional semi-simple representation of 0 having the adjoint representation as a subrepresentation. The elements of 8" are said to be congruent modulo u when the following condition is satisfied:
CH.9, 5 21
301
CANONICAL MAPPINGS
if 1, and A, are weights of el and e, respectively, then 1, - 1, belongs to an additive group generated by the weights of u. Since u has the adjoint representation as a subrepresentation, this condition is, from 9.1.8 (ii), independent of the choice of 1, and 1,; from 1.9.1 1, it is independent of the choice of the Cartan subalgebra of lj which had to be made in order to speak of weights of representations. The congruence relation modulo u is an equivalence relation in lj". If lj is semi-simple, there exists in I)" only a finite number of congruence classes modulo u (1 1.1.1l), but this is not the case in general. Each congruence class modulo u is denumerable. If e E lj", all weights of e @ u are sums of weights of e and weights of u, hence all simple subrepresentations of e @ u are congruent to e modulo (1. 9.1.16. Given this, let V be a Harish-Chandra g-module. We assume that k is algebraically closed and that f is reductive in g. Let u be the adjoint be the family of congruence classes representation of f in g. Let modulo u in f". For all i E I, Vi= Beef;V, is a sub-g-module of V. Indeed, let e E f'. Let 8 be the linear mapping of g @ V, into V such that O(x @ w ) = xw for x E g and w E V,. It is a f-module homomorphism from 1.7.9, equation (1). All simple subrepresentations of g @ V, (considered as a f-module) are congruent to e modulo u from 9.1.15, hence g V, c Vi, which proves our assertion. 0 belong to In particular, if V is simple, all the e E 1'' such that V, =I= one and the same congruence class modulo u, termed the class modulo u corresponding to V. References: [64], [891.
9.2. Canonical mappings defined by a symmetrizing subalgebra 9.2.1. In sections 9.2 to 9.5 we shall assume that k is algebraically closed, that g is semi-simple, and that f is a symmetrizing Lie subalgebra of g. We shall then reduce the study of G' (the commutant of f in U(g) = G) to that of algebras which are easier to handle. Let g = f @ p be the symmetric decomposition corresponding to f. With the notation of 1.13.11, we choose lj and B, whence we have a,l,n,m,q,R', B' = B r\ R',R, and R; = R , r\ R'. We set
U(t) = K ,
U(I))= H ,
U(a) = A ,
U(n) = N , U(m)= M . U(1) = L , We identify the algebra H with the algebra L @ A. Every element of lj*
302
HARISH-CHANDRA MODULES
[CH. 9,O 2
(or a*,I*) defines a homomorphism of H (resp. A,L) into k, with which it will be identified. If e E f", the kernel of e in K will always be denoted by Ze. For n E N, we set
U,(g)= G,,
U"(a)= S"(a) = A".
U,(a) = A,,
We denote the commutant of analogously.
M
in G by G'" and K"', GO, etc., are defined
9.2.2. From 2.2.10, the vector space G can be canonically identified with the vector space N 0A 8 K. On the other hand, N = k . 1 @ nN, hence G= AKenG.
This decomposition defines a projection 6 of G onto AK. On the other hand, the vector space AK can be identified with the vector space A 0K; the latter is naturally equipped with an algebra structure. We say that 6 is the canonical mapping of G onto A 0K (defined by the Iwasawa decomposition g = f @ a @ n). 9.2.3. PROPOSITION (notation as in 9.2.1). Let 6 be the canonical mapping of G onto A @ K. Then: (i) For u E G and v E G', we have 63(uv) = 6(v)6(u). (ii) 6(P)c A 8 K"'. (iii) Zn particular, GIG' is an anti-homomorphism of Gt into A Q Km. (iv) For u E G and k E K, we have 6(uk) = 6 ( u ) ( l @ k). (v) Let n E N . If u E G, and k E K, then 6(uk)E A,, 0K . Let u E G and v E G', and let us write u=
c, up; + u",
v
=
i
with ui,vjE A, u;,v; E K , u",v" E nG. Then uv =
c, vj.; + v" J
c,u p ; + u"v c,uivjv;u;+ c,uiv"u; + u"v. i
=
LJ
+
i
Since [a,n] c n (1.13.1l), we have an c na n c nA, and consequently An c nA. Hence u,v"u; E nG. It is clear that u"v E nG. We then have 6(uv) =
~ ( U i V j0 ) i,i
If in addition u E Gm, then, for all m E m,
o = [m,u] = C ui[m,u;]+ [m,u"]. i
CH. 9, 5 21
CANONICAL MAPPINGS
303
Now [m,u"]E [m,n,G] c nG since [m,n] c n (1.13.11). Hence C,u,@ [m,u;] = 0. By taking the u, to be linearly independent over k , we conclude that [m,u;]= 0 for all i, whence 6(u) E A 8K'". Assertion (iv) is obvious. To prove (v), we may, taking (iv) into account, assume that k = 1. If u E G,, then u E NA,K, whence 6(u) E 6 ( A , K ) = A ,
8K .
9.2.4. We say that GIG' is the canonical antihomomorphism of G' into A @ K m (defined by the Iwasawa decomposition g = f 8 a @ n). 9.2.5. Let I be a two-sided ideal of K, n the canonical mapping of K onto K/I, and 6 the canonical mapping of G onto A 8 K. We set 6,= (1 8 n) 0 6 :G + A 8 (K/Z).
To study G' A (Ker GI), we introduce the auxiliary mappings o),oI as follows. Let A be the canonical mapping of S(g) onto G. The bilinear mapping (x,y) I+ A(x)y of S(p) x K into G defines an isomorphism of the vector space S(p) @ K onto the vector space G (2.4.15); let @ be the inverse isomorphism. Let b be the orthogonal subspace of a in p. Then p = a @ b (1.13.3, hence S(p) = A @ bS(p), and this decomposition defines a projection r of S(p) onto A. We set o = (r 8 1) @ : G-+ A 8 K, 0
woI= (1 8 n) 0 o : G - t A
8 (K/Z).
If uE G and u = CA
4) = c, r(u,) 8 u;.
9.2.6. LEMMA(notation as in 9.2.1, 9.2.5). Let u E A(S"(p))K. Then o ( u ) E A" @I K and 6 ( u ) - o ( u ) E An.-l 8 K. Let (al, ..., a,) and (bl, ..., b,) be bases for IX and b respectively. The orthogonal subspace of a in g is b 8 f, but also n @ f (1.13.11). If we write each b, in the form n, k, (hiE n, k, E f), then ( n l , . . ., n,) is a basis for n. We have
+
u
=
CA(ql
d;lb;'l+l
*.*
* * *
ml+-*+ml+,=n
bm'+q) Q km,,...,nli+q,
where the kml,...,ml+,belong to K. Then w(u) =
2 a;tl
ml+'-+ml=n
*
4"' 6 km,,...,ml. 0,....0 E A" 0 K *
304
HARISH-CHANDRA MODULES
On the other hand, l.(ql to
a;"/bY1+1
- - bTl+q) is congruent modulo G,-
C 4" 4"' Q
-
ni1+-+mJ-n
**.
[CH. 9,B '2
kml,...,nq,O,...,o
= w(u).
9.2.7. LEMMA(notation as in 9.2.1, 9.2.5). Let I be a two-sided ideal of K . Then . Gt A Ker
=
G' A GI = G' A Ker w,.
(a) We have Gt A GI c G' A Ker 6,. Indeed GI = (AK
+ ttG)I c A I + nG,
hence &(GI) c A @I I and G1(GI)= 0. (b) We have Gc A Ker w, c Gr A GI. Indeed, let us identify the fmodule G with the t-module S(p) 6 K under the isomorphism of 2.4.15. An element of G' can be identified with a t-invariant element of S(p) @ K. Let us assume that this element is u in Ker a ) , ; its image in S(a) @ ( K / I ) is zero. Let (xi) be a basis for K/I. We write the image of u in S(p) @ ( K / I ) in the form iwi 8 xi, where w iE S(p) for all i. By virtue of the Killing form, we identify S(p) and S(a) with the algebras of polynomial functions over p and a respectively. Then r is precisely the restriction operation. Hence we have Ciw,(a)xi= 0 for all a € a. Let 2'be the smallest algebraic group of automorphisms of g whose Lie algebra is ad,€ (cf. 1.13.13). Then X operates in f , p , K, K / I , S(p) and S(p) @I (K/Z). For all k E X , we have
C
hence
Ci (kwi) @I (kxi) = C wi(k-'a) J
i
wi
O xi,
(kxi) = 0 for all a E a.
CH. 9.8 21
305
CANONICAL MAPPINGS
Now (kxi) is a basis for K/I, hence wi(Xa) = 0 for all i. Since X a is dense in p (1.13.13), we have wi = 0 for all i, hence UE S(p) @I I. (c) G' A Ker 6, c Gr A Ker w I . Indeed, let u E G' A Ker 6,. Write u = u,, u1 ... u,,, with uiE A(Si(p))K. Each subspace I(Si(p))K is stable under the adjoint action of f, hence uiE G' for all i. We give an indirect proof, and assume that the w,(ui) are not all zero; let w,(ui) = 0 for i = n, n - 1, ..., p 1 but w,(up) $; 0. From (a) and (b), we have 6 ( u i ) = 0 for i = n , n - l , . . . , p + 1. For i < p , ~ I ( u ~ ) € A ~ - ~ @ I ( K / Z ) (9.2.3 (v)). From 9.2.6,
+ + +
+
'
whence 6,(u)
61(4E wl
+ Ap
-1
€3 ( K / )
+ 0, which is a contradiction.
9
9.2.8. PROPOSITION (notation as in 9.2.1). The canonical antihomomorphism of G' into A @I P is injective. Indeed, G' A Ker 6 = 0 from 9.2.7. 9.2.9. PROPOSITION (notation as in 9.2.1). Let pl be the Harish-Chandra homomorphism of Ga in to H = A €3 L dejined by B, y the Harish-Chandra homomorphism of M' into L defined by B' and 6 the canonical antihomomorphism of G' into A @I K. We define pl' and y' by plw = (pl(zT)IT,
Then: (0 6(Z(g)) c A €3 z(m). (ii) IfzE Z(g), then
d-4= (1 €3 Y)((6(zT))T),
Y'(4
= (y(zT)IT.
€3 Y ' K W ) . (The operation T in A €3 K is of course defined as the tensor product of the operations T in A and K.) For every OL E R(g,$), let XuE 9" - {O}. Let aI,...,a,,be the positive roots, numbered in such a way that al,., . at are the positive roots which are zero on a. The elements X n r + l ,..., X,, form a basis for n, and X,, X f m r ,E m. Let (H,, ..., H,) be a basis for I, and (H,+l, ..., H,) a basis for a. The elements pl'(4 = (1
u((qi),(mi),(pi)) = X?., .. . x~_",,HY~ . H , ~ ~ X-. : : x:; = XFu,.
.. X:anHyl ... H Y I p ; ... x:;HZ:l ... H,"p$-f ... xz
constitute a basis for G. Let =
2
~~i),~~i),~pi~u((qi),(mi),(Pi))
306
HARISH-CHANDRA MODULES
be an element of Z(g). If implies that hence
PIN,
(Pt+lNt+1
l(qi),(mi),(pi)
+ + Pnal = * *.
f.*'
+
Pnan)
=+
Q1an,
1a=
[CH. 9,s 2
0, then the relation [O,z]
+ *. . +
(4,+lat+I
=0
qnan
+
**.
f
I a.
qnan)
If P : + ~ ... , , p n are not all zero, then u((qi),(mJ,(pi))E Gn. If they are all zero, then q,+l, ..., qn are also zero. Hence ((3(zT))T is the sum, for p t + I = ... = p n = q,+l = ... = qn = 0, of the terms
and so belongs to A @ M. On the other hand, 6 ( z ) E A @ K m from 9.2.3 (ii) X:;) is zero This proves (i). Furthermore, y(XCa, ... X?a,P;"l -.. HyK:; qt p 1 p t > 0, and is equal to H;"' H;"l if if q1 q1 -.- q, p I -.. pt = 0. Hence
-
+ + + + + + + + + + 0
.
.
2
(1 @ ~ ) ( ( G ( z ~ )=) ~ ) ~ ( o ) , ( m i ~ , ( o ) ~.*~* ' H, ~m l S 8 H,"' * . * Hi''' = ~ ( z )
whence
9'(4 = ((1 8W)((w)T))T
= (1
8W'>(&(Z>)
-
9.2.10. Let 0 E t", and let E" be the space of 0. Then K/I" can be canonically identified with End(E"). If (5 is the canonical antihomomorphism of G' into A @ K, we denote the antihomomorphism (1 @ 0 ) (3 of G' into A @ (K/I")= A @ End(E") by (3". Its kernel is 0
G'
n GI" = G' nI"G
(9.1.10 (ii) and 9.2.7). Its image is contained in A
(ii)).
8 ( P / K mA
(9.2.3
9.2.11. Moreover, let e E m", let EQ be the space of e, and let HRp = Homm(Ee,E").From 9.1.12, HQ," is null or is a simple Km-module of dimension mtp(e,a) (we recall that m is reductive in g from 1.13.7).
Let JR+' be the kernel in K m of the corresponding representation of Km. The algebra K'"/Jep can be canonically identified with End(H,,,). 9.2.12. As in 9.2.10, we fix only a and E". From 9.1.11, the vector space
E" can be canonically identified with CBeCma(He,,,@ E p ) (this sum only comprises a finite number of non-zero terms). Under the identification KIP = End(E"), M / M A I" is identified with (1 @ End@@)),and,
nQEm-
CH. 9, 0 21
307
CANONICAL MAPPINGS
from 1.2.1 1, Km/K"' A I" is identified with the commutant of M / M A I", that is, with (End(H,,,) @ 1). Thus
noEmA
K"
n
10
=
nJ Q $ ~ ,
K1"/KmA I" =
Consequently,
&jG
n (K"'/Je*") n End(He,,).
,Em^
@Em^
=
@EmA
is identified with an anti-homomorphism:
6":GC+
nA
pEm^
@ (End(H,,") @ 1) =
n A @ (K1"/Je3").
p€m^
We denote by 6e,o the antihomomorphism of Gc into A @ End Hop, or into A 8 (K'"/Je*"),deduced from 6"by composition with the projection onto the factor with index e. We have
G' A GI" = Ker 6"=
nKer .
p€m^
9.2.13. LEMMA (notation as in 9.2.1, 9.2.12). Let q~ be the Harish-Chandra homomorphism of G', into H = A @ L defined by B. Let 6 E f", e E m", and ,u be the smallest weight of e, identiJiedwith ahomomorphism o f L into k. Then for all zE Z(g), we have G,u(z) = (1 @ ~ ) ( ( q ~ ( z ~ ) ) ~ ) -
Let y and y' be as in 9.2.9, and n,,"the canonical homomorphism of
K"' onto K"'/Jeia;we consider n,,. as a representation of K"' in He,". The elements of He," are m-homomorphisms; hence for y E Z(rn), ne,,(y) is
the scalar defined by ~ ( y )or , by e*(yT), i.e., (- p)(y(yT)) = p(y'(y)) (7.2.8 and 7.4.4). For z E Z(Q),we have 8 ( z ) E A @ Z(m) from 9.2.9(i), and then
6e,a(Z)
= (1 8 n p p ) 6 (z)= (1
= (1 8 p) (v(zT)T)
cl> (1 8 v') 8(z)
from 9.3.9 (ii).
9.2.14. Let us retain the notation of 9.2.1, and let 8 still be the canonical antihomomorphism of G' into A 8 K . If 1 E a*, d E f" and e E m",we set 8 A=
(A @ 1) 8 : G' -+ K"', 0
+
9.2.15. PROPOSITION (notation as in 9.2.1), (i) G = A @ (Gf nG). (ii) Let p be the projection of G onto A defined by the preceding decomposition. Then p(G' is a homomorphism of the algebra G' into the algebra A whose kernel is G' n Gf = G' A tG.
308
[CH. 9.8 3
HARISH-CHANDRA MODULES
We have
G = A K Q nG = A Q ( A H Q nG), Gt c (AK
+ nG)t c AKT + nG,
whence (i). Given this, (ii) follows from 9.1.10 (ii), 9.2.3 (iii) and 9.2.7 for I=H. 9.2.16. With the notation of 9.2.15, PIG' is termed the canonical homomorphism of G' into A . References: [871, [1031.
9.3. The principal series 9.3.1. We retain the notation of 9.2.1. Let e E (m @ a)"; in particular, ela can be identified with a linear form
on a, and elme m". Let T, be the representation of q which extends e and is zero on n. We denote the g-module corresponding to ind(t,,g) by M'(e). The g-module corresponding to coind(T,,g), which is a set of linear mappings of G into the space of e, can be canonically identified with the dual of M'@*)(5.5.4). In this coinduced g-module, the sum of the finite-dimensional simple sub-€-modulesis a sub-g-module (1.7.9), which we denote by A'(@). Let T be the adjoint representation o f t in g. Let F b e the set of congruence classes modulo T in t". For y E F, let X(e,y) be the sum of the simple sub-tmodules of X(e) which belong to y. Then A'(@) is the direct sum of the X(e,y), and each X(e,y) is a sub-g-module of X(e) (9.1.16). The family of g-modules X(e,y), for e E (m Q a)" and y E F, is termed the algebraic principal series of g-modules (relative to t,lj,B). 9.3.2. LEMMA (notation as in 9.2.1). Let p E (m @ a)", let w be the largest Then M'(@)can weight of e (relative to lj and B'), and let 6 = be identified with a quotient module of the Verma module M(w 6 )
CaER+ix.
(constructed relatively to
+
4
and B).
+
Let b = lj C &R+ga be the Bore1 subalgebra of g defined by lj and B; then b c q. Let w' be the linear form on b which extends w and is zero on ~ & R : g m . We set G = ind(w',q),
'G
= ind(a,g) = ind(w',g)
.
Then w', and hence u, is zero on the ideal n of q, and u can be identified &gX,m @ a) (5.1.2). From 7.2.2 (extended to the case with ind(o'llj
+
CH.9, 5 31
309
THE PRINCIPAL SERIES
of reductive Lie algebras), the space V, of u contains a sub-(m @ a)-module Vi such that the quotient representation of m @ a (or of q) in V,/Vi is equivalent to e. Hence the g-module corresponding to t,which is precisely M ( o d), contains a sub-g-module M such that the representation of fl in M ( o d)/M is equivalent to ind(t,,g) (with the notation of 9.3.1).
+
+
9.3.3. THEOREM (notation as in 9.2.1). Let e E (m @ a)". (i) The g-module X(e) is a Harish-Chandra module. (ii) y u E t", the multiplicity of u in X(e) is mtp(elm,u). (iii) Let ,u be the smallest weight of e (relative to Ij and B'), and S=LC a E R , ~ .Let wo be the element of W(g,$) which transforms B into -B. Then X(@)has a central Character, which is equal to XwM+d. Assertion (i) is obvious, and (ii) follows from 5.5.7 and 5.5.8. With the notation of (iii), the largest weight of e* is -,u, hence M'(e*) has the central character X-p+d (9.3.2, 7.1.9). Hence M'(e*)* and a fortiori X(e) have a central character . x ; moreover, for z E Z(g) we have, from 7.4.9,
x(z) = X-p+d(zT)
= xwop+d(z)*
9.3.4. LEMMA(notation as in 9.2.1). Let and E" be the corresponding modules, let
H = Hom,(E",X(@)), and let
E
e E (m @ a)" and
u E t", let Ee
H' = Hom,(Ee,Ee),
be the restriction to X(e) of the canonical projection
(i) For h E H and h' E H', the mapping E h h' of Ee into Ee has the form cp(h,h') 1, where cp(h,h') E k. (ii) The bilinear form cp enables us to identfy each of the vector spaces H and H' with the dual of the other. 0
-
0
If h E H and h' E H', the mapping E h 0 h' is an nt-homomorphism of Ee into Ee; since Ee is a simple m-module, we deduce (i). Let h E H - (0). Then 0
&(E"))
=4
N M m
= E (NAKh(E"))
because
E
is a q-homomorphism
because h is a f-homomorphism
= E(Gh(E")),
and this is non-zero from the uniqueness assertion of 5.5.3 [where we take
310
[CH.9, Ii 4
HARISH-CHANDRA MODULES I
+
Y' = Gh(EQ)].Hence E h 0. Now E" is a semi-simple m-module, and h is an m-homomorphism, so there exists h' E H' such that (E h) h' 0, 0
E
0
0
0
+
whence rp(h,h') $. 0. Since H and H' have the same dimension (9.3.3 (ii)), this proves (ii).
9.3.5. Given this, the spherical functions of X(p) can be calculated in the following way :
PROPOSITION (notation as in 9.2.1). Let g, u, EQ, E", H and H' be as in 9.3.4. We identify each of the vector spaces H and H' with the dual of the
other. Let us consider H as a lejit G'-module and H' as a lejit K"'-module (by virtue of the action of G' in X(p) and of Km in E">. Let 1 be the linear form on a which can be identifed with pla and GA the corresponding antihomomorphism of G' into K'" (9.2.14). (i) For every uE G', the action of u in H and the action of GA(u)in H are transpositions of each other. (ii) The annihilator of the G'-module H is the kernel of GQ,m,o,A (9.2.14).
+
Let u E GI. We write u = up; u" with ui E A, u;E K and u"E nG. Let 7~ be the representation of g defined by X(p). For h E H and h'E H , we have, with the notation of 9.3.4, q(uh,h') * 1 = E Z ( U ) 0 h 0 h' 0
=
C ui(3L)E
0
n(ui) 0 h 0 h'
0
It
i
u,(@
=
0
~ ( u :0)h'
i
because E is a q-homomorphism because It is a €-homomorphism
h 0 u(GA(u))0 12' = t' 0 It 0 (GA(u)h') =E
0
= rp(h,GA(u)h') 1
whence (i). Assertion (ii) follows from (i) by the definition of 9.2.12, where the module H' is denoted by HQ,,,,J.
(cf.
References: 1641, [87].
9.4. The subquotient theorem 9.4.1. We retain the notation of 9.2.1, 9.2.12 and 9.2.14. Let U E €" and u E G'. Then G(u) E A @ K"'. Let us identify K m / K mA I" with J-'JQ,,AEnd(HQ,a) (cf. 9.2.12). For all p E m",G,,(u) can be identified with
with finite basis; we can an endomorphism of the free A-module A @I HQso
CH.9.8 41
311
THE SUBQUOTIENT THEOREM
consider the characteristic polynomial of this endomorphism det (T - GQ,a(u))3 where T denotes an indeterminate. This polynomial belongs to A [ T ] ; it is equal to 1 if mtp(p,a) = 0. Let
Then (1) f(6jQ,a(u);u,u) = 0 for all p E m" such that mtp(p,a) > 0. Let W = W(g,fj),and let q and q' be defined as in 9.2.9. From 7.4.5, there exists a homomorphism w m w" of W onto a group W" of automorphisms of H such that q'IZ(g) is an isomorphism of Z(g) onto HW" (the set of elements of H which are invariant under W"); for every W E W, we also denote by w z . the permutation of f j * corresponding to the automorphism w- of H. Since A[TI c H [ q , we can define ?(T;G,U)=
11w"(f(T;a,u)) E HW"[T].
w€W
Let q be the degree off. There exist zo,...,z4-1 E Z(g) such that (2)
S(T;o,u) = T4
+ v'(z4-l)T4+' + + v'(zl)T + ~ ' ( z o ) .
We set
v
='U
* * -
+
z4-1
u'-'
+ + Z ~ U+ z
~ G*. E
9.4.2. LEMMA (notation as in 9.4.1). We have v E G*A GP. Let
e E m" be such that mtp(p,a) > 0. We set = 6,a(UIq
+ q*(zq-1)6Q,a(uY-' + + v P ( z ~ ) ~ J+~ )~ ' ( z o ) * * *
E H 8 End HQ,a= A 8 L @ End H&.
From (1) and (2) we have
v'
-
=f(hQ,u(u) ;a,u) = 0 .
On the other hand, let p E I* be the smallest weight of p, and let us identify p with a homomorphism of L into k. Then 1 @ p 8 1 : A 8L 8 End He,a+ A 8 End HQBa
312
[CH.9.8
HARISH-CHANDRA MODULES
4
is a homomorphism, and we have
0 = (1 @ ,u @ 1)v' = he,u(uY
+ (1 o
,u)(Ql'(zq-l))cije,u(u)q-l
+ + (1 8 ~ ) ( ~ l ' ( z o ) ) 0 . .
= Ge,o(V)
from 9.2.13. Hence vE Gr A GI" (9.2.10). 9.4.3.
LEMMA(notation as in 9.2.1). Let u E €A, let t be a simple representa-
tion of Gt, and let E and Y be the modules corresponding to u and t . We assume that Ker t 3 d n GI". Then: (i) Y is finite-dimensional. (ii) There exists nE (m @I a)^ such that Y is isomorphic to a subquotient of HomXE,X(n)).
The algebra G' has a filtration such that the associated graded algebra is commutative and of finite type (1.7.10). Hence t(Z(g))c k . 1 (2.6.4). Let ~l and @' be as in 9.2.9. From 7.4.8, there exists ,u E I)* such that
-
t ( z ) = p(q'(z)) 1 for all Z E Z(g).
(1)
With the notation of 9.4.1, let L be the set of the u E G' such that Ge,,,,-(u) = 0
(2)
for all
e E m" and all
W E W.
This is a two-sided ideal of Gf. Let u E L. In f(T;u,u),the coefficients of the monomials in Tq-',Tq-2, ... (with the notation of 9.4.1) are zero if we apply the homomorphism ,u : H - + k to them (because of (2)). Hence ,u(q'(zi)) = 0 for all i. We have
0 =t(v) = t(UY
from 9.4.2
+
,u(Ql'(Zq
-l ) ) W 4 - I
+ ... + p(q'(z0))
from (1 1
= .(U)"
Thus, every element of the two-sided ideal t ( L ) of t(G') is nilpotent. From 3.1.14, t ( L ) is nilpotent. Hence there exists an integer n such that
Since Ker t is primitive, there exist e E mh and w E W such that Ker t 3 Ker cije,u,w-(p),a.
FINITENESS THEOREMS
CH. 9,s 51
313
Let n be the element of (m @ a)" which extends p and w-(,u)la. From 9.3.5 (ii), Ker t contains the annihilator J of the G'-module H = Homc(E,X(n)). From 9.3.3 (ii), H is finite-dimensional, hence Ker t has finite codimension in G', which proves (i). Let
H=H,zH,~***~H,=O be a Jordan-Holder series of the G'-module H. Let Ji be the annihilator of the simple G'-module Hi/Hi+,.Then Ker t 3 J z (A Ji),, hence there exists i such that Ker t 2 Ji.Then Y is isomorphic to the G'-module Hi/Hi+l. 9.4.4. THEOREM (notation as in 9.2.1). Let V be a simple Harish-Chandra 0-module. There exists n E (m @ a)" such that V is isomorphic to a subquotient of X(m). Let us choose rs E t" such that V, $. 0; let E be a €-module of class u. Let Y = Hom,(E,V), which is a simple Gf-module whose annihilator contains d A GI" (9.1.12 (i)). From 9.4.3, there exists n E (m @ a)" such that Y is isomorphic to a subquotient of Hom,(E,X(n)). Then V is isomorphic to a subquotient of X ( n ) (9.1.14). References: 1651, [871, [1031.
9.5. Finiteness theorems
9.5.1. THEOREM (notation as in 9.2.1). Let V be a simple g-module, o E E", and p = supQ,,amtp(@,rs).Then mtp(u, V) 5 p . We can assume that V, $. 0, so that V is a Harish-Chandra module (9.1.6). The theorem then follows from 9.4.4 and 9.3.3 (ii). 9.5.2. LEMMA (notation as in 9.2.1, 9.2.10). Let u E t", so that Then:
G,(Z(g))c GAG') c A €9 (K/I").
(i) A 8 (K/Z? is a module of finite type over Gu(Z(g)). (ii) G'/Gf A GI" is a module of finite type over Z(g)/Z(g)A GI". We introduce the notation of 9.2.9. From 7.4.5 and AC V, p. 33, H is a module of finite type over pl'(Z(g)), which is a Noetherian ring. Now H
A 8 y'(Z(m)) = pl'(Z(Q))
= I
from 9.2.9, hence A 8 y'(Z(m))is a module of finite type over pl'(Z(Q))= (1 8 w ' ) ( ~ ( Z ( e > * >
314
HARISH-CHANDRA MODULES
[CH. 9.5 5
Since y'lZ(nt) is injective (7.4.9, A @I Z(m) is a module of finite type over
am)).
is a module of finite type over A and a fortiori It is clear that A @J (K/Z") over A @I (Z(nt)/Z(nt)A I"). Hence A @J (K/Z?is a module of finite type over &,(Z(g)). A fortiori, GP(G3'is a module of finite type over B,(Z(g)). Since the kernel of 65,01G' is G' A GI" (9.2.10), we have proved the lemma. 9.5.3. THEOREM (notation as in 9.2.1). Let CI E €" and let x be a homomorphism of Z(g) into k . Up to isomorphism, there exist only afinite number of simple g-modules V with the following properties:' (0 V, =/= 0 ; (ii) the central character of V is x.
(a) Let J = (Ker x)G, which is a two-sided ideal of G. From 9.5.2, the algebra G'/G' A (GI" J ) is finite-dimensional. Hence, up to isomorphism, it only has a finite number of simple modules. (b) Let X.be a simple €-module of class c. For every simple g-module V having properties (i) and (ii), let H,- be the G'-module Hom,(X,V). From 9.1.12 (i), H, is simple and its annihilator contains G' A (G' J). On the other hand, if V' is a simple g-module having properties (i) and (ii), and Hv and Hv.are isomorphic as G'-modules, then V and V' are isomorphic as g-modules (9.1.12 (ii)). The theorem then follows from (a).
+
+
9.5.4. We retain the notation of 9.2.1. Let a, be the one-dimensional null representation of €.If V is a g-module, then Vuois the set V' of €-invariant elements of V. We say that V is spherical if dim V' = 1. If V is simple and V'+ 0, then V is spherical (9.5.1). Let e E (nt6 a)". If elm =/= 0, then XQ)' = 0; if elm = 0, then X(p) is spherical. This follows from 9.3.3 (ii). 9.5.5. PROPOSITION (notation as in 9.2.1). Let V be a spherical g-module, and p the canonical homomorphism of G' into A . There exists 1 E a* such that, for all u E G', u,l V' is the homothety with ratio (p(u))(l).
Let v E V' - {O}. There exists a homomorphism C of G' into k such that 2 G' A Gt. From 9.2.15, there exists a homomorphism 5' of p ( d ) into k such that = 5' p. Now A is a p(G')-module of finite type (9.5.1 (i)), hence 5' can be extended to a homomorphism of A into k (AC V, p. 39). Such a homomorphism has the form a I+ a(il), where 1is a fixed element of a*, whence we have proved the proposition. u . v = C(u)v for all uE G' (9.1.3). We have Ker 5
0
SPHERICAL MODULES IN THE DIAGONAL CASE
CH. 9.0 61
315
9.5.6. PROPOSITION (notation as in 9.2.1). Let A E a*. There exists one, and up to isomorphism only one, spherical simple g-module V such that for every uE d , u,JVr is the homothety with ratio (p(u))(A) (where p denotes the canonical homomorphism of Gr into A). This follows from 9.1.12 and 9.2.15. References: [64], [87].
9.6. Spherical modules in the diagonal case 9.6.1. Let us assume that k is algebraically closed. Let lo be a semi-simple Lie algebra, m a Cartan subalgebra of lo, 0 = lo x lo, and let us define 8, f , p, a, m,lj, S,R, C, B, S,, S-, R+, R-, 15, h-, n and n- as in 1.13.14. We set
We denote the isomorphism x e =(x,x) of 0 onto f by i; the elements of lo" are the a i, where a E €". An element of (m a)" is a linear form on lj, i.e. a pair (Al,A2), where Al,A2E tu*. For A l l 2 E tu*, we can thus consider the 0-modules
+
0
+
M(Al,AZ), W A 1 , A Z ) = WAl &,A2 - 4, and X(A1,A2),which is the sum of the finite-dimensional sub-€-modules of M(-&,-A2)*. 9.6.2. PROPOSITION (notation as in 9.6.1). Let Al,A2 E m*, and a E f". Let us define as in 9.3.1. Then: i). 0) mtp(a,X(Al,A2)) = mtp(A, (ii) If A, A24P(S), then X(A,,A,) = 0. (iii) If 1, A2 E P(S), then X(A1,A2) is non-zero, and there exists y E such that X(Al,A2) = X(AI,A2,y). (iv) X(A,,A,) is spherical if and only i f A2 = -Al.
r
+
+ +
0
r
The restriction of (A&) to m is the linear form (x,x) I+ (A, and hence (i) follows from 9.3.3 (ii). Assertion (i) implies that X(A1,AZ)
+
*0
@
1,
+
A2 E
+ A2)(x),
P(S).
Let us assume that A, A2 E P(R). From (i), two finite-dimensional simple sub-€-modules of X(A,,A2) are congruent modulo the adjoint representation of € in g , and hence X(A1,A2)is equal to X(A1,A2,y) for some y E I'. Finally, (iv) follows from 9.5.4. 9.6.3. LEMMA(notation as in 9.6.1). Let A € m*, let m be the canonical E), I the annihilator of m in U(lo), and tpA the linear generator of M(-A
+
316 316
9.8 66 [CH.9.8 [CH.
HARISH-CHANDRA MODULES MODULES HARISH-CHANDRA
form on on O(b) O(b) which which isis the the composite composite of of the the mappings mappings form U(b)jj.. U(b)/Z+ U(b)/Z+ U(x-) U(zJ -+ -+ k. k. U(b)
Here the thefirst first arrow arrow iiss the the canonical canonical mapping,'the mapping,'the second second is is the the isomorphism isomorphism Here defined in 7.1.5, and the third is the mapping which relates every element oof defined in 7.1.5, and the third is the mapping which relates every element f to its its constant constant term.) term.) Then Then UU((LL)) to
KerqA= x-U(b)
+ U(a)x + C. U(b)(h+ A)(h)) hEm
+
We know that Z = U(a)x U(D)= Z @ U(x-) (7.1 S), hence
ZhC,,,
+ A(h)) (7.2.7 (i))
U(b)(h
and that
Ker On the other hand, whence we have the first equality of the lemma. Lastly, if h C m, then
hence and we prove the opposite inclusion in the same way, whence we have the second equality of the lemma. 9.6.4. LEMMA(notation as in 9.6.1, 9.6.3). Let be the linear form on U(g) = u(b) C3 U(b) such that fa(a@ b) = va(baT) for a,b E U(b). Then
h E m, 4 Let :
and hence
X(A, --A). On the other hand,
whence the lemma follows.
Then
CH. 9.8 61
317
SPHERICAL MODULES IN THE DIAGONAL CASE
9.6.5. LEMMA (notation as in 9.6.1,9.6.3,9.6.4). (i) For x,y E b and u E U(b), we set (x,y)u = yu - ux. This de$nes a g-module structure on U(b). (ii) For every u E U(b), let @,(u) be the linearform on U(g) = U(a) @ U(b) such that @,@)(a 63 b) = y,(buaT) for a,b E U(a). Then @, is a g-homomorphism of U(b) onto X(A,-A) such that @,(I) =h. (iii) @,(U(b)) = U(g)& and Ker @, is the annihilator of L(-A E).
+
Assertion (i) is obvious. Let x,y E b and u,a,b E U(a). Then %((x,y)u)(a @ 4 = @,bu - M a @ b) = qJ,(byuaT) - qJ,(buxaT)
+ @ M a x 8b) @ M ( a@ b)(x @ 1 + 1 8YN
= @ M a @ by) =
= ((xYY)@,(u>)(a @
and hence @, is a g-homomorphism. Clearly, @,(1)
.
@,(U(b)) = @,(U(g) 1) = U(g)h c X(-l,--l)
+
=fn,
and hence
from 9.6.4.
Let K be the largest sub-b-module of M(--l E ) which is distinct from M(-A E). Let A be the annihilator of L(--l E ) = M(-A E)/K.If we identify M(--l E ) with U(gJ canonically, then K c g-U(g-) from 7.1.11 (i). Let u E A, and a,b E U(b). Then buaTE A, and hence buaT transforms the canonical generator of M(--l E ) into an element of K and hence of g-U(g-). From 9.6.3we deduce that qJ,(buaT) = 0, whence @,(u)=O and A c Ker @* Finally, Ker @, is a sub-g-module of U(a), i.e., a two-sided ideal of U(a). Hence (Ker@,)- M(--l E) is a sub-a-module of M(--l E). If u E Ker @, and a E U(b), then
+
+
+
+
+
+
+
y,(ua) = @,(u)(aT 8 1)
+
+
= 0;
hence (Ker @), M(-A E ) $. M(--l E ) ; consequently, (Ker @bM(--l E ) c K, whence Ker @, c A.
+
9.6.6. LEMMA (notation as in 9.6.1). Let A E tu*, and let J and J' be the E ) and L(-A E ) , respectively. The following conannihilators of M(-A ditions are equivalent: (i) J = J'; (i) X(-l,--l)' generates the g-module X(A,--l). If these conditions are satisfied, X(A,-A) is isomorphic to the g-module U(b)/J (the g-module structure of U(b) being defined as in 9.6.5 (i)).
+
+
318
HARISH-CHANDRA MODULES
[CH. 9,8 6
From 9.6.5, whose notation we are using, we have a diagram of g-homomorphisms U(b)/J v!-+ U(b)/J' U(g)fA X(A, -A) , (1)
*+
where y1 is surjective, y2 is bijective, and y 3 is injective. Let t". From 9.6.2 (i), mtp(u,X(A,-A)) is the multiplicity of the weight 0 in u. On the other hand, the g-module structure on U(b) defines, by restriction to t, a representation o f t in U(b)which can be identified with the adjoint representation of b in U(b). From 8.4.3 and 8.3.9, mtp(u,U(o)/J) is the multiplicity of the weight 0 in u. Hence (2) mtp(u,U(b)/J) = mtp(u,X(A,-A)). Given this, we have J = J' e y1 injective H y 3 0 y z y 1 injective 0
w y 3 yz yl bijective 0
H y 3 0 y2 0 y1 Hy3
(from (2))
0
surjective
(from (2))
surjective
@ U(elh = m-4. The last assertion then follows at once.
9.6.7. PROPOSITION (notation as in 9.6.1). Let A E to*.
( A - e)(H,)G {-l,-2, then X(A,-A)'
If
...,} for all a E S,,
generates the g-module X(A,-A).
This follows from 7.6.24 and 9.6.6. 9.6.8. LEMMA(notation as in 9.6.1). Let eo be the one-dimensional trivial representation of t. Then X(2~,-2.5)~ is a sub-g-module of X(2&,-2.5).
zeEt-,e+eo
For all A € tu*, let us denote the annihilator of M(A) by JA.If A = 2.5, the conditions of 9.6.6 are satisfied from 7.6.24. Hence the g-modules X(2q-2~) and U(b)/J-= are isomorphic. On the other hand, J, = J-, from 8.4.4, and M(E)has a one-dimensional b-module as a quotient (7.2.6). Hence J-, is contained in a two-sided ideal of codimension 1 of U(o). Consequently, X(2.5,-2~) has a subg-module W of codimension 1. Then W is a sub-€-moduleof X(2&,--2.5), and X(2e7-2e)/ W is a trivial g-module, hence X(2~,-2e) is a complement of W in X(2~,-2.5), and
w = CX(2&,-2&). €erA,e+Qo
CH. 9, J1 61
319
SPHERICAL MODULES IN THE DIAGONAL CASE
9.6.9. LEMMA (notation as in 9.6.1). Let 1E tu*. There exists a g-invariant 2&,A - 2 ~ such ) that: bilinear form y on ~ ( 1 , - 1 x) X(-1 (a) for every e E I”, the restriction of y to X(1,-1& x X(-1 2&,A- 24,. is non-degenerate; (b) i f @ , @ ’ €€”and e’ =I=e*, then X(A,-l)Q and X(-1 243, - 2 ~ ) are ~’ orthogonal to each other with respect to y.
+
+
+
(a) We recall that we defined a multiplication on U(g)* (2.7.4). We have
M’(-A,A)*
- M’(A - 2&,-A + 2&)*c M’(-2&,2&)*.
Indeed, if f E M’(-Ail.)*,
z E n, we have, from 2.7.7,
g E M’(A - 2 ~ -1 ,
‘~0t)Cfg) = ( t u Y ) f lg
+f (‘&)
= (il.,-il.)b>f * g +f(-1
+ 2~)*, x E rn,
y E a and
+ 2&, - 24(y)g ?b
= (2&,-2&)0cfg) and similarly ‘L(x)cfg)= ‘L(z)(fg)= 0 . Again from 2.7.7, we have, for all rE 0, r u g ) = (rfk f(rg) hence X(il.,-il.)X(-A 2&,1 - 2&)c X(2&,-2&).
+
+
(b) Let !P be the sum of the finite-dimensional simple sub-t-modules of U(f)* for the left or right coregular representation (2.7.12). In 2.7.13 we defined the fundamental linear form cpo and the fundamental bilinear form Po on Y and P x Y respectively. Let T be the set of the gE U(€) such that g(rnU(t)) = 0. Let T‘ = T A Y. Then the mapping f w f IU(€), where f takes all values in MI(-&I)*, is a t-isomorphism of M’(-A,A)* onto T endowed with the right coregular representation (5.5.8), and hence defines by restriction a €-isomorphism of X(1,--il.) onto T’. We identify X(A,-1) with T‘ under this isomorphism. Then the multiplication
X(A,-1) x X(-1
+ 2&,il. - 2.5) -+ X(2&,-2&)
can be identified with the multiplication in T’ (2.7.4). The form cpolT’ can be identified with a linear form on X(2.5-2~) which, from 9.6.8, is ginvariant. (c) The bilinear form y :U , g )H cpocfg) on
X(1,-1) x X(-A
+ 2&,1 - 2&)
is hence g-invariant. It can be identified with Po, and hence, from 2.7.15 and 2.7.17, satisfies properties (a) and (b) of the lemma.
320
[CH.9,D 6
HARISH-CHANDRA MODULES
9.6.10. PROPOSITION (notation as in 9.6.1). Let A E m*. If (A-e)(Ha)e{l,2,
...} forall ores,,
then every non-null sub-g-module of X(A, -A) contains X(A,-A)'; in particular, the sub-g-module of X(A,-A) generated by X(A,-A)* is simple. From 9.6.7, X(-A
+ 2 ~il, - 2.5)' X(-A
generates the g-module
+ 2 5 2 -24.
Let W be a sub-g-module of X(A,-A). Then W = W ,? X(A,-A)Q. Let y be as in 9.6.9, and W' the orthogonal subspace of Win X(-A 2 ~A,- 2.5). From 9.6.9, W is the orthogonal subspace of W' in X(A,-A). If X(A,-A)* is not contained in W, then X(-A 2&,1- 2e)' is contained in W,' hence W' = X(-A 2 ~A, - 2 4 , and W = 0.
+
+
+
m*. If ( A - ~)(H,)cjZ - (0) for all a E S,
9.6.11. THEOREM (notation as in 9.6.1). Let A E then X(A,-A)
is simple.
This follows from 9.6.7 and 9.6.10. 9.6.12. THEOREM (k algebraically closed, g semi-simple). Let 0 be a Cartan subalgebra of g, B a basis for R = R(g,Ij), A E $*, whence we have a central character xn, and Jn = U(g) Ker xn. We assume that A(H,) cj Z - {0}for all a C R. Then Jnis the only two-sided ideal J of U(g)such that J A Z(g) = KerXr. In particular, JA is a maximal two-sided ideal of U(g). Indeed, the (g x g)-module U(g)/Jnis simple from 8.4.3, 9.6.6 and 9.6.11. 9.6.13. THEOREM (notation as in 9.6.1). Let x be a homomorphism of Z(b) into k , and let us identifr Z(g) with Z(b) @ Z(b). There exists one, and up to isomorphism only one, spherical simple g-module whose central character x' satisfies x'lZ(g) @ 1 = x. We set G = U(g), and bl = b x 0 c x b = g. Since g = bl Q f, we have G = Gt Q U(a,). For every xE by the inner derivation of G defined by (x,x) leaves Gt and U(b,) stable, and in U(bl) induces the inner derivation defined by x. Hence G* = (G' A GT) @ Z(bl). Let @ be the set of homomorphisms of G' into k which are zero on G* A Gt. Let @' be the set of homomorphisms of Z(bl) into k. Then q~I+ ylZ(b,) is a bijection of @ onto a'. Given this, the theorem follows from 9.1.12. References: [371, (791, [1001.
CH. 9.8 71
SUPPLEMENTARY REMARKS
321
9.7. Supplementary remarks
9.7.1. The representations of the principal series were first considered by Gelfand and Naimark for the classical complex semi-simple groups. For the case of general real semi-simple groups, the theorems are, roughly, of the following type: (1) a representation of the principal series is most often simple; (2) two representations of the principal series are in general equivalent if the initial data are conjugate under the Weyl group (intertwining operators); (3) we give conditions for a representation of the principal series to be unitary. In spite of the number and significance of the papers dedicated to these questions, the situation is not yet very clear. Some special cases (e.g. complex groups, groups with symmetric rank 1, spherical representations, unitary representations) are closer to a complete solution. In particular, we may cite [79], [loo], [131], [133]. In this chapter, we have only given those theorems which can be presented in an entirely algebraic form. The essence is due to Harish-Chandra [64], [65]. We have followed Lepowsky and McCollum [89] in 9.1, and Lepowsky [87] and Rader [lo31 in 9.2, 9.3, 9.4 and 9.5. The canonical mapping of G' into A @ K"' was introduced by Lepowsky and Rader. Some important special cases have been considered previously; for example, the canonical homomorphism of Gf into A was introduced by Harish-Chandra. Theorems 9.6.11 and 9.6.13 are due to Parthasarathy, Ranga Rao and Varadarajan [loo]; the method of proof of 9.6.1 1 given here was indicated to me orally by Duflo. 9.7.2. (a) (b) (c) (d)
We adopt the notation of 9.0, with I reductive in g. Gg,"is the set of the u E G such that u(Vu)c V, for every g-modulz V. Ge+'G",' c Ge9' for all e,o,tE I". Ge is the largest subalgebra of G containing GIg as a two-sided ideal. Let V be a g-module, and S c V,. Then (GS) A V, = G'*g. [89]
9.7.3. (notation as in 9.2.1). Let I' = [€,I], and let f'" be the set of the having the following property: there exists nE g" such that n(f' has a subrepresentation equivalent to 5. Let L be the left ideal of U(€') such that the left regular representation of €' in U(€')/Lbelongs to €'". Then U(g)Lis the intersection of the maximal left ideals of finite codimension of U(g) which contain U(g)L. [65]
[ € €"'
9.7.4. (notation as in 7.0). Assume that k = C and that g is s.imple. Let lj* and let € be a symmetrizing subalgebra of g containing 0. Let c be the centre of f . Then dim c = 0 or 1.
322
HARISH-CHANDRA MODULES
[CH. 9,5 7
(a) If dim c = 0, then L(1) is only a Harish-Chandra module relative to t if L(A) is finite-dimensional, i.e., if 1 - 6 E P++. (b) If dim c = 1, the following conditions are equivalent: (i) L(A) is a Harish-Chandra module relative to t; (ii) ( A - d)(H,)E N for every positive root 01 such that ga c t. [66] 9.7.5 (notation as in 9.2.1). Assume that k = C. Let 8 be the automorphism of g corresponding to f , Wthe Weyl group of (g,O,a) (1.14.14), 6 = $- C aER+ 01, and 6' = 61a. If w E W, let w* be the affine automorphism 1 H w(1 - 6') b' of the dual of a, whence there is an automorphism of A, which is also denoted by w * . Let AW* be the set of the elements of A which are invariant under w* for all w E W. (a) Let u E t", let t be the linear form u H tr u(u) on K, 6 the canonical antihomomorphism of G' into A @I K , and 6 ' ' the linear mapping (1 @ t) 6 of G' into A @I k = A. Then GU(G')= AW*. If B is the canonical mapping of S(g) into U(g), then 6,"IP(S(p)') is a bijection of B(S(p)') onto AW*. (b) In particular, the canonical homomorphism of G* into A has AW* as its image. Then 9.5.6 defines a canonical bijective correspondence between classes of spherical simple g-modules and W,-orbits in a*. ([87], [88])
+
0
9.7.6 (notation as in 9.2.1). Let V be a spherical simple g-module. Then V" has dimension 0 or 1, and m V" = 0. [79] 9.7.7 (notation as in 9.2.1). Assume that k = C. Let L be a left ideal of K . Assume that K / L is finite-dimensional and that the left regular representation of f in K / L is semi-simple. (a) The g-module V = G/GL is a Harish-Chandra module. (b) Let o E t". Then V, is a module of finite type over Z(g). [64] (c) Let @,aE t". Then Ge9a/GPis a module of finite type over Z(g). (This is a special case of (b), as Lepowsky pointed out to me.) 9.7.8 (notation as in 9.2.1). Assume that k = C.Let V be a Harish-Chandra g-module having a central character. Consider the following conditions : (i) for all 5 E f", mtp(t, V ) < +oo; (ii) there exists a constant C such that, for all [ E t", we have mtp(l, V)5 C dim 5 ; (iii) V is a U(g)-module of finite type; (iv) V has a Jordan-Holder series. (a) From 9.5.1 and 9.7.7, we have (iv) + (ii) 3 (i), and (iv) *.(iii) * (i). From 7.8.16, (ii) does not imply (iii) and (i) does not imply (ii). (b) Let c be the centre of f . The weights of the adjoint representation
CH. 9,#71
323
SUPPLEMENTARY REMARKS
of c in g generate a subgroup 0 in c * ; this subgroup is free and of rank dim c. For every integer n > 0, let €,^ be the set of the 6~ t" such that the linear form with which 51c can be identified belongs to (l/n)0. If there exists an n such that V = @w;Vt (a property which is automatically satisfied if f is semi-simple), then conditions (i) to (iv) are equivalent. Cf. problem 34. ([87], [90])
9.7.9 (notation as in 9.2.1). Assume that k = C.Let x be a homomorphism of Z(g) into C, and n a positive integer. Up to isomorphism, there only exist a finite number of simple Harish-Chandra g-modules V such that: (1) the central character of V is x ; (2) with the notation of 9.7.8 (b), V = @eu; V,. Cf. problem 34. ([87], [90]) 9.7.10. We now abandon the notation of 9.0. Let G be a simply connected semi-simple real Lie group, and G = KAN an Iwasawa decomposition of G. Let gR, f,, a,, ,TI be the Lie algebras of G, K , A, N, and g, f, a, 11 the complexifications of gR, f,, a,, .,t Let mR be the commutant of a, in fR, and m the complexification of mR.Let Mo be the connected Lie subgroup of G with Lie algebra m,, let Z be the centre of G, which is contained in K, let Ml = MoZ, and let M, be the centralizer of a in K; then Mo c Ml c M,, and these three groups have mR as their Lie algebra. For i = 0,1,2, let Qi = MiAN, which is a Lie subgroup of G with Lie algebra ~TT, @ a,@ ,t A finite-dimensional irreducible representation e of Qi is trivial over N and hence can be identified with a pair (A,@), where A € a* and oE M:. Let IC be the representation of G induced by e. We shall leave the case where i = 0 aside. (It would then seem reasonable to limit ourselves to the case where e is unitary, taking IC in the Mackey sense. Even in this case, the relations with the algebraic coinduced representation are not satisfactory.) If i = 1, then G/Ql is compact, and we can define IC (which is in general not unitary) in Lz(K/M,). The corresponding representation of U(g) in the space of K-finite vectors gives us a Harish-Chandra module which is one of the modules X(e,y) of 9.3.1. Since we often have M1 M,, the representation IC is often reducible, and consequently the module X(e,y) is also reducible. If i = 2, then IC belongs to what is classically termed the principal series. Let X = X"" be the corresponding Harish-Chandra module. Here are certain results concerning the A'*" (which unfortunately do not apply to
+
324
HARISH-CHANDRA MODULES
[CH. 9.5 7
the modules X(e,y) in the text except in certain cases). We shall restrict ourselves to the case where (Ker 0)A 2 has finite index in 2. (i) There exists a subset A of a*, whose complement is the denumerable union of algebraic manifolds distinct from a*, such that, if A E A, then X"" is simple. (Kostant, Wallach) (ii) Let us assume that u is trivial and one-dimensional. Let us define 6' as in 9.7.5. Let D be the set of the linear forms on a which are real-valued on aR and whose scalar product with the elements of R'+ (with the notation of 9.2.1) is non-negative. If il - 6'E D ia:, then X"" is spherical and (XA3b)'generates X'*" as a U(g)-module. [79] (iii) There exists a non-degenerate bilinear form which is canonically defined on XA*a~X26'-A+'*,is g-invariant, and is such that every t-stable vector subspace of XI*"is equal to its bi-orthogonal subspace. [79] (iv) We adopt the notation of 9.7.5. If M' is the normalizer of a in K, then W can be identified with M'IM,, whence we have an action of W in M:. Then, if W E W, the Jordan-Holder series of X"." and of Xw*A*wn have the same quotients with the same multiplicities. [87] (v) If E E g", then there exist A and 0 such that mtp(S,X".") > 0; we then have mtp(E,Xa,") = 1. [90] (vi) Let G, be the simply connected complex Lie group with Lie algebra g. Then there is a canonical homomorphism r of G into G c . If Ker u 2 Ker r , then XI" is generated as a U(g)-module by a single element. [90]
+
9.7.1 1. For the Harish-Chandra modules' over 31(2,C) (with t = Ch), cf. 7.8.16. For the Harish-Chandra modules over Sl(2,C) x 31(2,C) (with t equal to the diagonal subalgebra), cf. [57]. Cf. also [186].
HARISH-CHANDRA MODULES
Let B be a real semi-simple Lie group with finite centre, X a maximal compact subgroup, go and Eo the Lie algebras of 3 and X , g and € the complexifications of go and to, and z a completely irreducible representation of Y in a Banach space (for example, an irreducible unitary representation of Y in a Hilbert space). Let V be the space of vectors which are X finite under z. Then V is a simple g-module such that V = eeaVQ.The study of z can for the most part be reduced to the study of this g-module. The modules which we shall study in this chapter are generalizations of the g-modules just considered. 9.0. Notation
In this chapter,€ will denote a Lie subalgebra of g and G and K the enveloping algebras of g and €.An essential role will be played by the commufant G' of € in G. If e E f", we shall denote the kernel of e in K by I,. If V is a g-module, we have defined Ve in 1.2.8.
9.1. The case of a Lie subalgebra which is reductive in g 9.1.1. For @,a€€, we set Ge"
= (U E
I
G Zeu c GI"}.
We write Ge instead of Gese.The Gelu are vector subspaces of G . 9.1.2. PROPOSITION (notation as in 9.0). (i) We have 'Ge*'r> GI" and GQ"/GZu= (G/G17e
when we consider GIGI' as a g-module for the left regular representation. (ii) If V is a g-module, then Ge*"(VU) c V,.
296
WI S H -CH ANDRA MODULES
[CH.9 . 8 1
We have PGI" c GI,, hence GI" c Ges". Every representation of K/Ie is semi-simple and is the sum of representations of class e (AL VIII, pp. 47-48). Hence, if V is a g-module, then
v, = { Y E
VI IQV = O}.
Firstly this implies (ii), because IeGe9"(V,) c GI"( V,) hand, (G/GI?@ = {u
+ GI" I u E G
= 0.
On the other
and l e u c GI"},
whence (i). 9.1.3. PROPOSITION (notation as in 9.0). (i) Ge is a subalgebra of G which
contains K as a subalgebra and GIe as a two-sided ideal. (ii) If V is a g-module, then GQ(Ve)c Ve and (GIQ)(Ve)= 0; we can thus consider Ve as a (G@/GF)-module. We have QGQGec GIeGpc GIe,
P K c Ie,
GIeGe c GGIe = G P ,
whence (i). Assertion (ii) follows from 9.1.2. 9.1.4. A g-module V is said to be a Harish-Chandra module relative to t
(or simply a Harish-Chandra module if there is no doubt about f) if V = CQccA Ve, whence V = eeeCVQ. The Verma modules are HarishChandra modules relative to a Cartan subalgebra. In this chapter, the subalgebras f that we have in view are in general very different from Cartan subalgebras.
9.1.5. PROPOSITION (notation as in 9.0). W e assume that f is reductive in g. Let e E 1". Then: (i) G/GIe is a Harish-Chandra 8-module under the left regular representation. (ii) G/GIe = G"*e/GP.
From 1.7.9, W = Xocc.. (G/GZe), is a sub-g-module of G/GIe. Now the class of 1 in G/GIQbelongs to W,and hence W = G/GIe. This proves (i), and (ii) then follows from 9.1.2 (i). 9.1.6. PROPOSITION (notation as in 9.0). W e assume that f is reductive in g.
Let V be a simple g-module, and e E f" such that Ve (i) V is a Harish-Chandra g-module. (ii) The GQ-moduleV, is simple.
+ 0. Then:
A L I E ALGEBRA REDUCTIVE IN
CH. 9, fiI]
297
From 1.7.9, V is a Harish-Chandra g-module. Let v E V, - {O}. Then V = Gv = 2, G'% from 9.1.5 (ii), and GU% c V, from 9.1.2 (ii), hence G@v= V,. 9.1.7. PROPOSITION (notation as in 9.0). We assume that t is reductive in g. Let V be a Harish-Chandra g-module, e E f", P the projection of V onto Ve defined by the decomposition V = cBUCte V,, and W a sub-Ge-module of V,. W e set Wmin -
W"""
GW,
= {vE
VI (Gv) A V, c W } .
Then: (i) W""" = { v E VIP(Gv) c W}. (ii) w"'" V, = wmax ve = W. (iii) Zf X is a sub-g-module of Vsuch that X A V, 2 W (or X A Vpc W ) , then x 3 Win(or x c wmax). For all v € V, we have Gv = ecCtA (Gv) A V,, whence (i). On the other hand, GW = G e W from 9.1.5 (ii), and G"@Wc V,, hence
zcCt.
W"'" A V, = GeW = W . Consequently, WmaX 3 Wmin,and it is then clear that W""" A Ve = W. Assertion (iii) is obvious. 9.1.8. PROPOSITION (notation as in 9.0). We assume that f is reductive in g. Let e E f", and let A and 9 be the sets of the maximal left ideals of G and Ge, respectively, containing GIe. (i) The mapping M H M A Ge is a bijection v of A onto 9. (ii) ZfL €9, then y-'(L) contains every left ideal M of G such that M GQc L. (iii) Let n be the canonical mapping of G onto G/GZe,and P the projection of G/GIe onto Ge/GZedefined by 9.1.5 (ii). Zf L E 9, then
I = {U E G I P(Gn(u)) c n(L)}.
v-'(L) = {U E G (Gu) A Ge c L}
The mapping M I + M/GZeis a bijection of the set of the left ideals of G containing GIe onto the set of sub-g-modules of G/GZe; if M $: G, then 1 $ M and hence M A Ge $: Ge. The mapping L H L/GZe is a bijection of the set of the left ideals of Ge containing GZe onto the set of sub-Ge-modules of Ge/GZe= (G/GZe),. Given this, the assertions of the proposition are easy consequences of 9.1.7 applied to V = G/GIe.
298
[CH. 9, D 1
HARISH-CHANDRA MODULES
9.1.9. PROPOSITION (notation as in 9.0). We assume that f is reductive in g . Let p E €".Let Y be the set of the classes of simple g-modules V such that Ve =+ 0. Let 9'be the set of classes of simple (GQ/GIP)-modules.Then the mapping V H V, dejines a bijection of Y onto 9'. If V E 9, then V, E 9" (9.1.6 (ii)). Let V,WE 9, let f be a G4-isomorphism of V, onto W,, v a non-zero element of V,, and L the annihilator of v in GQ;it is also the annihilator of f ( v ) in GQ,and it is a maximal left ideal of G4. Let M and M' be the annihilators of v and f ( v ) respectively in G. Then M and M' are maximal left ideals of G, and M A Ge = L = M' A Ge, hence M = M' (9.1.8), which proves that V and W are G-isomorphic. Let 2 be a simple Ge-module whose annihilator contains GIP, let z E Z - {0}, and let L be the annihilator of z in Ge. There exists a maximal left ideal M of G such that M A Ge = L (9.1.8). Let V be the simple gmodule GIM. Then G/GZe = @(G/GZe)u,
(G/GZe)Q= Ge/GZe
UEl-
(9.1.2, 9.1.5); hence
so that Ve is G%omorphic to 2. 9.1.10. PROPOSITION (notation as in 9.0). We assume that f is reductive in g. Let g E 'f". Then: (i) (GIe) A K = Ze, so that K/ZQcan be identijied with a subalgebra of Ge/GZe. (ii) G ' c GQ,so that G'/G' A GZe can be identiJied with a subalgebra of GQ/GZe; moreover, G' A GIQ= G' A P G . (iii) Zfg is absolutely simple, the canonical mapping of
(K/Z@) @ (G'/Gf A GIQ) into Ge/GZe is an isomorphism; we have GQ= KG'
+ GI@.
Assertion (i) follows from the fact that G is a free right K-module (2.2.7). Clearly, G' c GQ. Let V be a g-module such that V = @ucI.Vu. Let V* be the dual g-module. Let us identify @uEt-(Vu)*with a sub-€-module of V*. Let u E Gc A ZQG. Then U T E G' A GIQ', and Ze((Ve)*) = 0, hence uT((V,)*) = 0 and
299
A LIE ALGEBRA REDUCTIVE IN
CH.9 , s 11
since u(Ve) c V,, we conclude that u(V,) = 0. Applying this to V = G/GZe, we see that uGe c GIe, whence u E GZe. Thus G' A IeG c GIe, and consequently G' A GZe = (Gf A Ie'G)T c (GIe*)T = ZeG.
This proves (ii). The adjoint representation of f in G is semi-simple, and Ge and GIe are sub-f-modules of G ; consequently, every element of Ge/GZeannihilated by the adjoint representation o f f is the canonical image of an element of Ge which commutes with t (1.2.11), i.e., of an element of G'. The commutant of K/Ze in Ge/GZe is thus G ' / d A GZe. Finally, if p is absolutely simple, the algebra K/Ze is central, simple and finite-dimensional, and hence Ge/GZe is the tensor product of K/Ze and its commutant in Ge/GIe (11.2.5). This proves (iii). 9.1.11. LEMMA (notation as in 9.0). Let V be a g-module, Z an absolutely simple f-module, e the corresponding representation, and H the vector space Hom,(Z,V) = Hom,(Z,V,). W e consider H as a left Gf-module by virtue of the action of G' in V (or V,). There exists one and only one isomorphisnt of the vector space Z @ H onto the vector space V, which transforms z @ 11 (z E Z , It E H ) into h(z). If we identifv Z @ H with V, by means of this isomorphism, then, for z E Z, h E H , u E K and v E Gf, u*
(Z @
h) = uz @I h ,
V . ( Z @I
11) = z @ vh.
This follows from AL VIII, pp. 13-15. 9.1.12. THEOREM (notation as in 9.0). W e assume that f is reductive in g. Let a ?-module of class e. Let Y be the set of the classes of simple g-modules V such that V, 0. Let 9- be the set of classes of simple (G'/G' r\ GZ@)-modules. (i) If V is a simple 9-module such that V, =j= 0, let us consider F = Hom'(X,V,) as a left Gt-module by virtue of the action of G' in V,. Then F is a simple G'-module whose annihilator contains G' A GZe. (ii) The mapping V H F dejines a bijection of Y onto 9.
e be an absolutely simple element o f f " , and X
+
Let V be a g-module and F = Hon@ rf,' V). Let us identify F @ X with V, by virtue of 9.1.1 1. If F' is a sub-Gf-module of F, I;' @ X is a sub-G'module and a sub-K-module of V,, and hence a sub-Ge-module of V, (9.1.10 (iii)). If V is simple as a g-module and V, 0, then V, is simple as a Ge-module (9.1.6 (ii)), hence F' @ X = V, and F' = F. This proves (i).
+
300
HARISH-CHANDRA MODULES
[CH. 9 , $ 1
Let V,V'E 9, F = Hom,(X,V), F' = Hom,(X,V'), and let ~1 be a G'isomorphism of F onto F'. Then 8 1 is a K-module and a G'-module isomorphism, and hence a Ge-module isomorphism, of F @ X onto F' 8 X (9.1.10 (iii)). Hence V and V' are isomorphic (9.1.9). Let F be a simple G'-module whose annihilator contains G' A GZe.Then, from 9.1.10 (iii), F 8 X is a simple Ge-module whose annihilator contains GIP. There exists a simple g-module V such that the Ge-module Ve can be identified with F 8 X (9.1.9). Then the G*-module Homc(X,Ve)can be identified with the G'-module F (AL VIII, p. 15). 9.1.13. With the notation of 9.0, let V be a 8-module, e E €", and X a 1module with class e. We consider Hom,(X,V) as a G'-module by virtue of the action of G' in V. Let cr be the corresponding representation of G'. Then u is said to be the spherical function of type e of V. If f is reductive in g, e is absolutely simple and V is a simple Harish-Chandra module such =then the spherical function of type e of V is simple and charthat VP=l0, acterises V up to isomorphism (9.1.f2); we shall see in 9.5.1 that it is finitedimensional. 9.1.14. PROPOSITION (notation as in 9.0). We assume that f is reductive in 9. Let e be an absolutely simple element of t"and X a €-module with class e. Let V,V' be g-modules, with V simple and Ve =I= 0. We assume the existence of sub-G'-modules Y,,Y, of Hom(X,V') such that Y, c Y, and Homt(X,V) is G*-isomorphicto YJY,. Then V is isomorphic to a subquotient of V'. We have V , f 0. By replacing V' by Cra.V:, we can assume that V' is a Harish-Chandra module. We have V, = Hom,(X, V ) @ X ,
Vd = Hom,(X, V')@ X.
Let Z1 = Y, @ X and Z , = Y, @ X. From 9.1.10 (iii), the Ge-module V, is isomorphic to the Ge-module Z , / Z , . Let Z; = GZ,. Applying 9.1.7, let us consider the largest sub-G-module Z ; of Z; such that Z ; A Vi = Z,. Then Z;/Z; is a simple Harish-Chandra g-module and (Z;/Zi)e= Z,/Z,. From 9.1.9, V is isomorphic to Z J Z ; . 9.1.15. We assume that k is algebraically closed. Let tj be a reductive Lie algebra, and u a finite-dimensional semi-simple representation of 0 having the adjoint representation as a subrepresentation. The elements of 8" are said to be congruent modulo u when the following condition is satisfied:
CH.9, 5 21
301
CANONICAL MAPPINGS
if 1, and A, are weights of el and e, respectively, then 1, - 1, belongs to an additive group generated by the weights of u. Since u has the adjoint representation as a subrepresentation, this condition is, from 9.1.8 (ii), independent of the choice of 1, and 1,; from 1.9.1 1, it is independent of the choice of the Cartan subalgebra of lj which had to be made in order to speak of weights of representations. The congruence relation modulo u is an equivalence relation in lj". If lj is semi-simple, there exists in I)" only a finite number of congruence classes modulo u (1 1.1.1l), but this is not the case in general. Each congruence class modulo u is denumerable. If e E lj", all weights of e @ u are sums of weights of e and weights of u, hence all simple subrepresentations of e @ u are congruent to e modulo (1. 9.1.16. Given this, let V be a Harish-Chandra g-module. We assume that k is algebraically closed and that f is reductive in g. Let u be the adjoint be the family of congruence classes representation of f in g. Let modulo u in f". For all i E I, Vi= Beef;V, is a sub-g-module of V. Indeed, let e E f'. Let 8 be the linear mapping of g @ V, into V such that O(x @ w ) = xw for x E g and w E V,. It is a f-module homomorphism from 1.7.9, equation (1). All simple subrepresentations of g @ V, (considered as a f-module) are congruent to e modulo u from 9.1.15, hence g V, c Vi, which proves our assertion. 0 belong to In particular, if V is simple, all the e E 1'' such that V, =I= one and the same congruence class modulo u, termed the class modulo u corresponding to V. References: [64], [891.
9.2. Canonical mappings defined by a symmetrizing subalgebra 9.2.1. In sections 9.2 to 9.5 we shall assume that k is algebraically closed, that g is semi-simple, and that f is a symmetrizing Lie subalgebra of g. We shall then reduce the study of G' (the commutant of f in U(g) = G) to that of algebras which are easier to handle. Let g = f @ p be the symmetric decomposition corresponding to f. With the notation of 1.13.11, we choose lj and B, whence we have a,l,n,m,q,R', B' = B r\ R',R, and R; = R , r\ R'. We set
U(t) = K ,
U(I))= H ,
U(a) = A ,
U(n) = N , U(m)= M . U(1) = L , We identify the algebra H with the algebra L @ A. Every element of lj*
302
HARISH-CHANDRA MODULES
[CH. 9,O 2
(or a*,I*) defines a homomorphism of H (resp. A,L) into k, with which it will be identified. If e E f", the kernel of e in K will always be denoted by Ze. For n E N, we set
U,(g)= G,,
U"(a)= S"(a) = A".
U,(a) = A,,
We denote the commutant of analogously.
M
in G by G'" and K"', GO, etc., are defined
9.2.2. From 2.2.10, the vector space G can be canonically identified with the vector space N 0A 8 K. On the other hand, N = k . 1 @ nN, hence G= AKenG.
This decomposition defines a projection 6 of G onto AK. On the other hand, the vector space AK can be identified with the vector space A 0K; the latter is naturally equipped with an algebra structure. We say that 6 is the canonical mapping of G onto A 0K (defined by the Iwasawa decomposition g = f @ a @ n). 9.2.3. PROPOSITION (notation as in 9.2.1). Let 6 be the canonical mapping of G onto A @ K. Then: (i) For u E G and v E G', we have 63(uv) = 6(v)6(u). (ii) 6(P)c A 8 K"'. (iii) Zn particular, GIG' is an anti-homomorphism of Gt into A Q Km. (iv) For u E G and k E K, we have 6(uk) = 6 ( u ) ( l @ k). (v) Let n E N . If u E G, and k E K, then 6(uk)E A,, 0K . Let u E G and v E G', and let us write u=
c, up; + u",
v
=
i
with ui,vjE A, u;,v; E K , u",v" E nG. Then uv =
c, vj.; + v" J
c,u p ; + u"v c,uivjv;u;+ c,uiv"u; + u"v. i
=
LJ
+
i
Since [a,n] c n (1.13.1l), we have an c na n c nA, and consequently An c nA. Hence u,v"u; E nG. It is clear that u"v E nG. We then have 6(uv) =
~ ( U i V j0 )
If in addition u E Gm, then, for all m E m,
o = [m,u] = C ui[m,u;]+ [m,u"]. i
CH. 9, 5 21
CANONICAL MAPPINGS
303
Now [m,u"]E [m,n,G] c nG since [m,n] c n (1.13.11). Hence C,u,@ [m,u;] = 0. By taking the u, to be linearly independent over k , we conclude that [m,u;]= 0 for all i, whence 6(u) E A 8K'". Assertion (iv) is obvious. To prove (v), we may, taking (iv) into account, assume that k = 1. If u E G,, then u E NA,K, whence 6(u) E 6 ( A , K ) = A ,
8K .
9.2.4. We say that GIG' is the canonical antihomomorphism of G' into A @ K m (defined by the Iwasawa decomposition g = f 8 a @ n). 9.2.5. Let I be a two-sided ideal of K, n the canonical mapping of K onto K/I, and 6 the canonical mapping of G onto A 8 K. We set 6,= (1 8 n) 0 6 :G + A 8 (K/Z).
To study G' A (Ker GI), we introduce the auxiliary mappings o),oI as follows. Let A be the canonical mapping of S(g) onto G. The bilinear mapping (x,y) I+ A(x)y of S(p) x K into G defines an isomorphism of the vector space S(p) @ K onto the vector space G (2.4.15); let @ be the inverse isomorphism. Let b be the orthogonal subspace of a in p. Then p = a @ b (1.13.3, hence S(p) = A @ bS(p), and this decomposition defines a projection r of S(p) onto A. We set o = (r 8 1) @ : G-+ A 8 K, 0
woI= (1 8 n) 0 o : G - t A
8 (K/Z).
If uE G and u = CA
4) = c, r(u,) 8 u;.
9.2.6. LEMMA(notation as in 9.2.1, 9.2.5). Let u E A(S"(p))K. Then o ( u ) E A" @I K and 6 ( u ) - o ( u ) E An.-l 8 K. Let (al, ..., a,) and (bl, ..., b,) be bases for IX and b respectively. The orthogonal subspace of a in g is b 8 f, but also n @ f (1.13.11). If we write each b, in the form n, k, (hiE n, k, E f), then ( n l , . . ., n,) is a basis for n. We have
+
u
=
CA(ql
d;lb;'l+l
*.*
* * *
ml+-*+ml+,=n
bm'+q) Q km,,...,nli+q,
where the kml,...,ml+,belong to K. Then w(u) =
2 a;tl
ml+'-+ml=n
*
4"' 6 km,,...,ml. 0,....0 E A" 0 K *
304
HARISH-CHANDRA MODULES
On the other hand, l.(ql to
a;"/bY1+1
- - bTl+q) is congruent modulo G,-
C 4" 4"' Q
-
ni1+-+mJ-n
**.
[CH. 9,B '2
kml,...,nq,O,...,o
= w(u).
9.2.7. LEMMA(notation as in 9.2.1, 9.2.5). Let I be a two-sided ideal of K . Then . Gt A Ker
=
G' A GI = G' A Ker w,.
(a) We have Gt A GI c G' A Ker 6,. Indeed GI = (AK
+ ttG)I c A I + nG,
hence &(GI) c A @I I and G1(GI)= 0. (b) We have Gc A Ker w, c Gr A GI. Indeed, let us identify the fmodule G with the t-module S(p) 6 K under the isomorphism of 2.4.15. An element of G' can be identified with a t-invariant element of S(p) @ K. Let us assume that this element is u in Ker a ) , ; its image in S(a) @ ( K / I ) is zero. Let (xi) be a basis for K/I. We write the image of u in S(p) @ ( K / I ) in the form iwi 8 xi, where w iE S(p) for all i. By virtue of the Killing form, we identify S(p) and S(a) with the algebras of polynomial functions over p and a respectively. Then r is precisely the restriction operation. Hence we have Ciw,(a)xi= 0 for all a € a. Let 2'be the smallest algebraic group of automorphisms of g whose Lie algebra is ad,€ (cf. 1.13.13). Then X operates in f , p , K, K / I , S(p) and S(p) @I (K/Z). For all k E X , we have
C
hence
Ci (kwi) @I (kxi) = C wi(k-'a) J
i
wi
O xi,
(kxi) = 0 for all a E a.
CH. 9.8 21
305
CANONICAL MAPPINGS
Now (kxi) is a basis for K/I, hence wi(Xa) = 0 for all i. Since X a is dense in p (1.13.13), we have wi = 0 for all i, hence UE S(p) @I I. (c) G' A Ker 6, c Gr A Ker w I . Indeed, let u E G' A Ker 6,. Write u = u,, u1 ... u,,, with uiE A(Si(p))K. Each subspace I(Si(p))K is stable under the adjoint action of f, hence uiE G' for all i. We give an indirect proof, and assume that the w,(ui) are not all zero; let w,(ui) = 0 for i = n, n - 1, ..., p 1 but w,(up) $; 0. From (a) and (b), we have 6 ( u i ) = 0 for i = n , n - l , . . . , p + 1. For i < p , ~ I ( u ~ ) € A ~ - ~ @ I ( K / Z ) (9.2.3 (v)). From 9.2.6,
+ + +
+
'
whence 6,(u)
61(4E wl
+ Ap
-1
€3 ( K / )
+ 0, which is a contradiction.
9
9.2.8. PROPOSITION (notation as in 9.2.1). The canonical antihomomorphism of G' into A @I P is injective. Indeed, G' A Ker 6 = 0 from 9.2.7. 9.2.9. PROPOSITION (notation as in 9.2.1). Let pl be the Harish-Chandra homomorphism of Ga in to H = A €3 L dejined by B, y the Harish-Chandra homomorphism of M' into L defined by B' and 6 the canonical antihomomorphism of G' into A @I K. We define pl' and y' by plw = (pl(zT)IT,
Then: (0 6(Z(g)) c A €3 z(m). (ii) IfzE Z(g), then
d-4= (1 €3 Y)((6(zT))T),
Y'(4
= (y(zT)IT.
€3 Y ' K W ) . (The operation T in A €3 K is of course defined as the tensor product of the operations T in A and K.) For every OL E R(g,$), let XuE 9" - {O}. Let aI,...,a,,be the positive roots, numbered in such a way that al,., . at are the positive roots which are zero on a. The elements X n r + l ,..., X,, form a basis for n, and X,, X f m r ,E m. Let (H,, ..., H,) be a basis for I, and (H,+l, ..., H,) a basis for a. The elements pl'(4 = (1
u((qi),(mi),(pi)) = X?., .. . x~_",,HY~ . H , ~ ~ X-. : : x:; = XFu,.
.. X:anHyl ... H Y I p ; ... x:;HZ:l ... H,"p$-f ... xz
constitute a basis for G. Let =
2
~~i),~~i),~pi~u((qi),(mi),(Pi))
306
HARISH-CHANDRA MODULES
be an element of Z(g). If implies that hence
PIN,
(Pt+lNt+1
l(qi),(mi),(pi)
+ + Pnal = * *.
f.*'
+
Pnan)
=+
Q1an,
1a=
[CH. 9,s 2
0, then the relation [O,z]
+ *. . +
(4,+lat+I
=0
qnan
+
**.
f
I a.
qnan)
If P : + ~ ... , , p n are not all zero, then u((qi),(mJ,(pi))E Gn. If they are all zero, then q,+l, ..., qn are also zero. Hence ((3(zT))T is the sum, for p t + I = ... = p n = q,+l = ... = qn = 0, of the terms
and so belongs to A @ M. On the other hand, 6 ( z ) E A @ K m from 9.2.3 (ii) X:;) is zero This proves (i). Furthermore, y(XCa, ... X?a,P;"l -.. HyK:; qt p 1 p t > 0, and is equal to H;"' H;"l if if q1 q1 -.- q, p I -.. pt = 0. Hence
-
+ + + + + + + + + + 0
.
.
2
(1 @ ~ ) ( ( G ( z ~ )=) ~ ) ~ ( o ) , ( m i ~ , ( o ) ~.*~* ' H, ~m l S 8 H,"' * . * Hi''' = ~ ( z )
whence
9'(4 = ((1 8W)((w)T))T
= (1
8W'>(&(Z>)
-
9.2.10. Let 0 E t", and let E" be the space of 0. Then K/I" can be canonically identified with End(E"). If (5 is the canonical antihomomorphism of G' into A @ K, we denote the antihomomorphism (1 @ 0 ) (3 of G' into A @ (K/I")= A @ End(E") by (3". Its kernel is 0
G'
n GI" = G' nI"G
(9.1.10 (ii) and 9.2.7). Its image is contained in A
(ii)).
8 ( P / K mA
(9.2.3
9.2.11. Moreover, let e E m", let EQ be the space of e, and let HRp = Homm(Ee,E").From 9.1.12, HQ," is null or is a simple Km-module of dimension mtp(e,a) (we recall that m is reductive in g from 1.13.7).
Let JR+' be the kernel in K m of the corresponding representation of Km. The algebra K'"/Jep can be canonically identified with End(H,,,). 9.2.12. As in 9.2.10, we fix only a and E". From 9.1.11, the vector space
E" can be canonically identified with CBeCma(He,,,@ E p ) (this sum only comprises a finite number of non-zero terms). Under the identification KIP = End(E"), M / M A I" is identified with (1 @ End@@)),and,
nQEm-
CH. 9, 0 21
307
CANONICAL MAPPINGS
from 1.2.1 1, Km/K"' A I" is identified with the commutant of M / M A I", that is, with (End(H,,,) @ 1). Thus
noEmA
K"
n
10
=
nJ Q $ ~ ,
K1"/KmA I" =
Consequently,
&jG
n (K"'/Je*") n End(He,,).
,Em^
@Em^
=
@EmA
is identified with an anti-homomorphism:
6":GC+
nA
pEm^
@ (End(H,,") @ 1) =
n A @ (K1"/Je3").
p€m^
We denote by 6e,o the antihomomorphism of Gc into A @ End Hop, or into A 8 (K'"/Je*"),deduced from 6"by composition with the projection onto the factor with index e. We have
G' A GI" = Ker 6"=
nKer .
p€m^
9.2.13. LEMMA (notation as in 9.2.1, 9.2.12). Let q~ be the Harish-Chandra homomorphism of G', into H = A @ L defined by B. Let 6 E f", e E m", and ,u be the smallest weight of e, identiJiedwith ahomomorphism o f L into k. Then for all zE Z(g), we have G,u(z) = (1 @ ~ ) ( ( q ~ ( z ~ ) ) ~ ) -
Let y and y' be as in 9.2.9, and n,,"the canonical homomorphism of
K"' onto K"'/Jeia;we consider n,,. as a representation of K"' in He,". The elements of He," are m-homomorphisms; hence for y E Z(rn), ne,,(y) is
the scalar defined by ~ ( y )or , by e*(yT), i.e., (- p)(y(yT)) = p(y'(y)) (7.2.8 and 7.4.4). For z E Z(Q),we have 8 ( z ) E A @ Z(m) from 9.2.9(i), and then
6e,a(Z)
= (1 8 n p p ) 6 (z)= (1
= (1 8 p) (v(zT)T)
cl> (1 8 v') 8(z)
from 9.3.9 (ii).
9.2.14. Let us retain the notation of 9.2.1, and let 8 still be the canonical antihomomorphism of G' into A 8 K . If 1 E a*, d E f" and e E m",we set 8 A=
(A @ 1) 8 : G' -+ K"', 0
+
9.2.15. PROPOSITION (notation as in 9.2.1), (i) G = A @ (Gf nG). (ii) Let p be the projection of G onto A defined by the preceding decomposition. Then p(G' is a homomorphism of the algebra G' into the algebra A whose kernel is G' n Gf = G' A tG.
308
[CH. 9.8 3
HARISH-CHANDRA MODULES
We have
G = A K Q nG = A Q ( A H Q nG), Gt c (AK
+ nG)t c AKT + nG,
whence (i). Given this, (ii) follows from 9.1.10 (ii), 9.2.3 (iii) and 9.2.7 for I=H. 9.2.16. With the notation of 9.2.15, PIG' is termed the canonical homomorphism of G' into A . References: [871, [1031.
9.3. The principal series 9.3.1. We retain the notation of 9.2.1. Let e E (m @ a)"; in particular, ela can be identified with a linear form
on a, and elme m". Let T, be the representation of q which extends e and is zero on n. We denote the g-module corresponding to ind(t,,g) by M'(e). The g-module corresponding to coind(T,,g), which is a set of linear mappings of G into the space of e, can be canonically identified with the dual of M'@*)(5.5.4). In this coinduced g-module, the sum of the finite-dimensional simple sub-€-modulesis a sub-g-module (1.7.9), which we denote by A'(@). Let T be the adjoint representation o f t in g. Let F b e the set of congruence classes modulo T in t". For y E F, let X(e,y) be the sum of the simple sub-tmodules of X(e) which belong to y. Then A'(@) is the direct sum of the X(e,y), and each X(e,y) is a sub-g-module of X(e) (9.1.16). The family of g-modules X(e,y), for e E (m Q a)" and y E F, is termed the algebraic principal series of g-modules (relative to t,lj,B). 9.3.2. LEMMA (notation as in 9.2.1). Let p E (m @ a)", let w be the largest Then M'(@)can weight of e (relative to lj and B'), and let 6 = be identified with a quotient module of the Verma module M(w 6 )
CaER+ix.
(constructed relatively to
+
4
and B).
+
Let b = lj C &R+ga be the Bore1 subalgebra of g defined by lj and B; then b c q. Let w' be the linear form on b which extends w and is zero on ~ & R : g m . We set G = ind(w',q),
'G
= ind(a,g) = ind(w',g)
.
Then w', and hence u, is zero on the ideal n of q, and u can be identified &gX,m @ a) (5.1.2). From 7.2.2 (extended to the case with ind(o'llj
+
CH.9, 5 31
309
THE PRINCIPAL SERIES
of reductive Lie algebras), the space V, of u contains a sub-(m @ a)-module Vi such that the quotient representation of m @ a (or of q) in V,/Vi is equivalent to e. Hence the g-module corresponding to t,which is precisely M ( o d), contains a sub-g-module M such that the representation of fl in M ( o d)/M is equivalent to ind(t,,g) (with the notation of 9.3.1).
+
+
9.3.3. THEOREM (notation as in 9.2.1). Let e E (m @ a)". (i) The g-module X(e) is a Harish-Chandra module. (ii) y u E t", the multiplicity of u in X(e) is mtp(elm,u). (iii) Let ,u be the smallest weight of e (relative to Ij and B'), and S=LC a E R , ~ .Let wo be the element of W(g,$) which transforms B into -B. Then X(@)has a central Character, which is equal to XwM+d. Assertion (i) is obvious, and (ii) follows from 5.5.7 and 5.5.8. With the notation of (iii), the largest weight of e* is -,u, hence M'(e*) has the central character X-p+d (9.3.2, 7.1.9). Hence M'(e*)* and a fortiori X(e) have a central character . x ; moreover, for z E Z(g) we have, from 7.4.9,
x(z) = X-p+d(zT)
= xwop+d(z)*
9.3.4. LEMMA(notation as in 9.2.1). Let and E" be the corresponding modules, let
H = Hom,(E",X(@)), and let
E
e E (m @ a)" and
u E t", let Ee
H' = Hom,(Ee,Ee),
be the restriction to X(e) of the canonical projection
(i) For h E H and h' E H', the mapping E h h' of Ee into Ee has the form cp(h,h') 1, where cp(h,h') E k. (ii) The bilinear form cp enables us to identfy each of the vector spaces H and H' with the dual of the other. 0
-
0
If h E H and h' E H', the mapping E h 0 h' is an nt-homomorphism of Ee into Ee; since Ee is a simple m-module, we deduce (i). Let h E H - (0). Then 0
&(E"))
=4
N M m
= E (NAKh(E"))
because
E
is a q-homomorphism
because h is a f-homomorphism
= E(Gh(E")),
and this is non-zero from the uniqueness assertion of 5.5.3 [where we take
310
[CH.9, Ii 4
HARISH-CHANDRA MODULES I
+
Y' = Gh(EQ)].Hence E h 0. Now E" is a semi-simple m-module, and h is an m-homomorphism, so there exists h' E H' such that (E h) h' 0, 0
E
0
0
0
+
whence rp(h,h') $. 0. Since H and H' have the same dimension (9.3.3 (ii)), this proves (ii).
9.3.5. Given this, the spherical functions of X(p) can be calculated in the following way :
PROPOSITION (notation as in 9.2.1). Let g, u, EQ, E", H and H' be as in 9.3.4. We identify each of the vector spaces H and H' with the dual of the
other. Let us consider H as a lejit G'-module and H' as a lejit K"'-module (by virtue of the action of G' in X(p) and of Km in E">. Let 1 be the linear form on a which can be identifed with pla and GA the corresponding antihomomorphism of G' into K'" (9.2.14). (i) For every uE G', the action of u in H and the action of GA(u)in H are transpositions of each other. (ii) The annihilator of the G'-module H is the kernel of GQ,m,o,A (9.2.14).
+
Let u E GI. We write u = up; u" with ui E A, u;E K and u"E nG. Let 7~ be the representation of g defined by X(p). For h E H and h'E H , we have, with the notation of 9.3.4, q(uh,h') * 1 = E Z ( U ) 0 h 0 h' 0
=
C ui(3L)E
0
n(ui) 0 h 0 h'
0
It
i
u,(@
=
0
~ ( u :0)h'
i
because E is a q-homomorphism because It is a €-homomorphism
h 0 u(GA(u))0 12' = t' 0 It 0 (GA(u)h') =E
0
= rp(h,GA(u)h') 1
whence (i). Assertion (ii) follows from (i) by the definition of 9.2.12, where the module H' is denoted by HQ,,,,J.
(cf.
References: 1641, [87].
9.4. The subquotient theorem 9.4.1. We retain the notation of 9.2.1, 9.2.12 and 9.2.14. Let U E €" and u E G'. Then G(u) E A @ K"'. Let us identify K m / K mA I" with J-'JQ,,AEnd(HQ,a) (cf. 9.2.12). For all p E m",G,,(u) can be identified with
with finite basis; we can an endomorphism of the free A-module A @I HQso
CH.9.8 41
311
THE SUBQUOTIENT THEOREM
consider the characteristic polynomial of this endomorphism det (T - GQ,a(u))3 where T denotes an indeterminate. This polynomial belongs to A [ T ] ; it is equal to 1 if mtp(p,a) = 0. Let
Then (1) f(6jQ,a(u);u,u) = 0 for all p E m" such that mtp(p,a) > 0. Let W = W(g,fj),and let q and q' be defined as in 9.2.9. From 7.4.5, there exists a homomorphism w m w" of W onto a group W" of automorphisms of H such that q'IZ(g) is an isomorphism of Z(g) onto HW" (the set of elements of H which are invariant under W"); for every W E W, we also denote by w z . the permutation of f j * corresponding to the automorphism w- of H. Since A[TI c H [ q , we can define ?(T;G,U)=
11w"(f(T;a,u)) E HW"[T].
w€W
Let q be the degree off. There exist zo,...,z4-1 E Z(g) such that (2)
S(T;o,u) = T4
+ v'(z4-l)T4+' + + v'(zl)T + ~ ' ( z o ) .
We set
v
='U
* * -
+
z4-1
u'-'
+ + Z ~ U+ z
~ G*. E
9.4.2. LEMMA (notation as in 9.4.1). We have v E G*A GP. Let
e E m" be such that mtp(p,a) > 0. We set = 6,a(UIq
+ q*(zq-1)6Q,a(uY-' + + v P ( z ~ ) ~ J+~ )~ ' ( z o ) * * *
E H 8 End HQ,a= A 8 L @ End H&.
From (1) and (2) we have
v'
-
=f(hQ,u(u) ;a,u) = 0 .
On the other hand, let p E I* be the smallest weight of p, and let us identify p with a homomorphism of L into k. Then 1 @ p 8 1 : A 8L 8 End He,a+ A 8 End HQBa
312
[CH.9.8
HARISH-CHANDRA MODULES
4
is a homomorphism, and we have
0 = (1 @ ,u @ 1)v' = he,u(uY
+ (1 o
,u)(Ql'(zq-l))cije,u(u)q-l
+ + (1 8 ~ ) ( ~ l ' ( z o ) ) 0 . .
= Ge,o(V)
from 9.2.13. Hence vE Gr A GI" (9.2.10). 9.4.3.
LEMMA(notation as in 9.2.1). Let u E €A, let t be a simple representa-
tion of Gt, and let E and Y be the modules corresponding to u and t . We assume that Ker t 3 d n GI". Then: (i) Y is finite-dimensional. (ii) There exists nE (m @I a)^ such that Y is isomorphic to a subquotient of HomXE,X(n)).
The algebra G' has a filtration such that the associated graded algebra is commutative and of finite type (1.7.10). Hence t(Z(g))c k . 1 (2.6.4). Let ~l and @' be as in 9.2.9. From 7.4.8, there exists ,u E I)* such that
-
t ( z ) = p(q'(z)) 1 for all Z E Z(g).
(1)
With the notation of 9.4.1, let L be the set of the u E G' such that Ge,,,,-(u) = 0
(2)
for all
e E m" and all
W E W.
This is a two-sided ideal of Gf. Let u E L. In f(T;u,u),the coefficients of the monomials in Tq-',Tq-2, ... (with the notation of 9.4.1) are zero if we apply the homomorphism ,u : H - + k to them (because of (2)). Hence ,u(q'(zi)) = 0 for all i. We have
0 =t(v) = t(UY
from 9.4.2
+
,u(Ql'(Zq
-l ) ) W 4 - I
+ ... + p(q'(z0))
from (1 1
= .(U)"
Thus, every element of the two-sided ideal t ( L ) of t(G') is nilpotent. From 3.1.14, t ( L ) is nilpotent. Hence there exists an integer n such that
Since Ker t is primitive, there exist e E mh and w E W such that Ker t 3 Ker cije,u,w-(p),a.
FINITENESS THEOREMS
CH. 9,s 51
313
Let n be the element of (m @ a)" which extends p and w-(,u)la. From 9.3.5 (ii), Ker t contains the annihilator J of the G'-module H = Homc(E,X(n)). From 9.3.3 (ii), H is finite-dimensional, hence Ker t has finite codimension in G', which proves (i). Let
H=H,zH,~***~H,=O be a Jordan-Holder series of the G'-module H. Let Ji be the annihilator of the simple G'-module Hi/Hi+,.Then Ker t 3 J z (A Ji),, hence there exists i such that Ker t 2 Ji.Then Y is isomorphic to the G'-module Hi/Hi+l. 9.4.4. THEOREM (notation as in 9.2.1). Let V be a simple Harish-Chandra 0-module. There exists n E (m @ a)" such that V is isomorphic to a subquotient of X(m). Let us choose rs E t" such that V, $. 0; let E be a €-module of class u. Let Y = Hom,(E,V), which is a simple Gf-module whose annihilator contains d A GI" (9.1.12 (i)). From 9.4.3, there exists n E (m @ a)" such that Y is isomorphic to a subquotient of Hom,(E,X(n)). Then V is isomorphic to a subquotient of X ( n ) (9.1.14). References: 1651, [871, [1031.
9.5. Finiteness theorems
9.5.1. THEOREM (notation as in 9.2.1). Let V be a simple g-module, o E E", and p = supQ,,amtp(@,rs).Then mtp(u, V) 5 p . We can assume that V, $. 0, so that V is a Harish-Chandra module (9.1.6). The theorem then follows from 9.4.4 and 9.3.3 (ii). 9.5.2. LEMMA (notation as in 9.2.1, 9.2.10). Let u E t", so that Then:
G,(Z(g))c GAG') c A €9 (K/I").
(i) A 8 (K/Z? is a module of finite type over Gu(Z(g)). (ii) G'/Gf A GI" is a module of finite type over Z(g)/Z(g)A GI". We introduce the notation of 9.2.9. From 7.4.5 and AC V, p. 33, H is a module of finite type over pl'(Z(g)), which is a Noetherian ring. Now H
A 8 y'(Z(m)) = pl'(Z(Q))
= I
from 9.2.9, hence A 8 y'(Z(m))is a module of finite type over pl'(Z(Q))= (1 8 w ' ) ( ~ ( Z ( e > * >
314
HARISH-CHANDRA MODULES
[CH. 9.5 5
Since y'lZ(nt) is injective (7.4.9, A @I Z(m) is a module of finite type over
am)).
is a module of finite type over A and a fortiori It is clear that A @J (K/Z") over A @I (Z(nt)/Z(nt)A I"). Hence A @J (K/Z?is a module of finite type over &,(Z(g)). A fortiori, GP(G3'is a module of finite type over B,(Z(g)). Since the kernel of 65,01G' is G' A GI" (9.2.10), we have proved the lemma. 9.5.3. THEOREM (notation as in 9.2.1). Let CI E €" and let x be a homomorphism of Z(g) into k . Up to isomorphism, there exist only afinite number of simple g-modules V with the following properties:' (0 V, =/= 0 ; (ii) the central character of V is x.
(a) Let J = (Ker x)G, which is a two-sided ideal of G. From 9.5.2, the algebra G'/G' A (GI" J ) is finite-dimensional. Hence, up to isomorphism, it only has a finite number of simple modules. (b) Let X.be a simple €-module of class c. For every simple g-module V having properties (i) and (ii), let H,- be the G'-module Hom,(X,V). From 9.1.12 (i), H, is simple and its annihilator contains G' A (G' J). On the other hand, if V' is a simple g-module having properties (i) and (ii), and Hv and Hv.are isomorphic as G'-modules, then V and V' are isomorphic as g-modules (9.1.12 (ii)). The theorem then follows from (a).
+
+
9.5.4. We retain the notation of 9.2.1. Let a, be the one-dimensional null representation of €.If V is a g-module, then Vuois the set V' of €-invariant elements of V. We say that V is spherical if dim V' = 1. If V is simple and V'+ 0, then V is spherical (9.5.1). Let e E (nt6 a)". If elm =/= 0, then XQ)' = 0; if elm = 0, then X(p) is spherical. This follows from 9.3.3 (ii). 9.5.5. PROPOSITION (notation as in 9.2.1). Let V be a spherical g-module, and p the canonical homomorphism of G' into A . There exists 1 E a* such that, for all u E G', u,l V' is the homothety with ratio (p(u))(l).
Let v E V' - {O}. There exists a homomorphism C of G' into k such that 2 G' A Gt. From 9.2.15, there exists a homomorphism 5' of p ( d ) into k such that = 5' p. Now A is a p(G')-module of finite type (9.5.1 (i)), hence 5' can be extended to a homomorphism of A into k (AC V, p. 39). Such a homomorphism has the form a I+ a(il), where 1is a fixed element of a*, whence we have proved the proposition. u . v = C(u)v for all uE G' (9.1.3). We have Ker 5
0
SPHERICAL MODULES IN THE DIAGONAL CASE
CH. 9.0 61
315
9.5.6. PROPOSITION (notation as in 9.2.1). Let A E a*. There exists one, and up to isomorphism only one, spherical simple g-module V such that for every uE d , u,JVr is the homothety with ratio (p(u))(A) (where p denotes the canonical homomorphism of Gr into A). This follows from 9.1.12 and 9.2.15. References: [64], [87].
9.6. Spherical modules in the diagonal case 9.6.1. Let us assume that k is algebraically closed. Let lo be a semi-simple Lie algebra, m a Cartan subalgebra of lo, 0 = lo x lo, and let us define 8, f , p, a, m,lj, S,R, C, B, S,, S-, R+, R-, 15, h-, n and n- as in 1.13.14. We set
We denote the isomorphism x e =(x,x) of 0 onto f by i; the elements of lo" are the a i, where a E €". An element of (m a)" is a linear form on lj, i.e. a pair (Al,A2), where Al,A2E tu*. For A l l 2 E tu*, we can thus consider the 0-modules
+
0
+
M(Al,AZ), W A 1 , A Z ) = WAl &,A2 - 4, and X(A1,A2),which is the sum of the finite-dimensional sub-€-modules of M(-&,-A2)*. 9.6.2. PROPOSITION (notation as in 9.6.1). Let Al,A2 E m*, and a E f". Let us define as in 9.3.1. Then: i). 0) mtp(a,X(Al,A2)) = mtp(A, (ii) If A, A24P(S), then X(A,,A,) = 0. (iii) If 1, A2 E P(S), then X(A1,A2) is non-zero, and there exists y E such that X(Al,A2) = X(AI,A2,y). (iv) X(A,,A,) is spherical if and only i f A2 = -Al.
r
+
+ +
0
r
The restriction of (A&) to m is the linear form (x,x) I+ (A, and hence (i) follows from 9.3.3 (ii). Assertion (i) implies that X(A1,AZ)
+
*0
@
1,
+
A2 E
+ A2)(x),
P(S).
Let us assume that A, A2 E P(R). From (i), two finite-dimensional simple sub-€-modules of X(A,,A2) are congruent modulo the adjoint representation of € in g , and hence X(A1,A2)is equal to X(A1,A2,y) for some y E I'. Finally, (iv) follows from 9.5.4. 9.6.3. LEMMA(notation as in 9.6.1). Let A € m*, let m be the canonical E), I the annihilator of m in U(lo), and tpA the linear generator of M(-A
+
316 316
9.8 66 [CH.9.8 [CH.
HARISH-CHANDRA MODULES MODULES HARISH-CHANDRA
form on on O(b) O(b) which which isis the the composite composite of of the the mappings mappings form U(b)jj.. U(b)/Z+ U(b)/Z+ U(x-) U(zJ -+ -+ k. k. U(b)
Here the thefirst first arrow arrow iiss the the canonical canonical mapping,'the mapping,'the second second is is the the isomorphism isomorphism Here defined in 7.1.5, and the third is the mapping which relates every element oof defined in 7.1.5, and the third is the mapping which relates every element f to its its constant constant term.) term.) Then Then UU((LL)) to
KerqA= x-U(b)
+ U(a)x + C. U(b)(h+ A)(h)) hEm
+
We know that Z = U(a)x U(D)= Z @ U(x-) (7.1 S), hence
ZhC,,,
+ A(h)) (7.2.7 (i))
U(b)(h
and that
Ker On the other hand, whence we have the first equality of the lemma. Lastly, if h C m, then
hence and we prove the opposite inclusion in the same way, whence we have the second equality of the lemma. 9.6.4. LEMMA(notation as in 9.6.1, 9.6.3). Let be the linear form on U(g) = u(b) C3 U(b) such that fa(a@ b) = va(baT) for a,b E U(b). Then
h E m, 4 Let :
and hence
X(A, --A). On the other hand,
whence the lemma follows.
Then
CH. 9.8 61
317
SPHERICAL MODULES IN THE DIAGONAL CASE
9.6.5. LEMMA (notation as in 9.6.1,9.6.3,9.6.4). (i) For x,y E b and u E U(b), we set (x,y)u = yu - ux. This de$nes a g-module structure on U(b). (ii) For every u E U(b), let @,(u) be the linearform on U(g) = U(a) @ U(b) such that @,@)(a 63 b) = y,(buaT) for a,b E U(a). Then @, is a g-homomorphism of U(b) onto X(A,-A) such that @,(I) =h. (iii) @,(U(b)) = U(g)& and Ker @, is the annihilator of L(-A E).
+
Assertion (i) is obvious. Let x,y E b and u,a,b E U(a). Then %((x,y)u)(a @ 4 = @,bu - M a @ b) = qJ,(byuaT) - qJ,(buxaT)
+ @ M a x 8b) @ M ( a@ b)(x @ 1 + 1 8YN
= @ M a @ by) =
= ((xYY)@,(u>)(a @
and hence @, is a g-homomorphism. Clearly, @,(1)
.
@,(U(b)) = @,(U(g) 1) = U(g)h c X(-l,--l)
+
=fn,
and hence
from 9.6.4.
Let K be the largest sub-b-module of M(--l E ) which is distinct from M(-A E). Let A be the annihilator of L(--l E ) = M(-A E)/K.If we identify M(--l E ) with U(gJ canonically, then K c g-U(g-) from 7.1.11 (i). Let u E A, and a,b E U(b). Then buaTE A, and hence buaT transforms the canonical generator of M(--l E ) into an element of K and hence of g-U(g-). From 9.6.3we deduce that qJ,(buaT) = 0, whence @,(u)=O and A c Ker @* Finally, Ker @, is a sub-g-module of U(a), i.e., a two-sided ideal of U(a). Hence (Ker@,)- M(--l E) is a sub-a-module of M(--l E). If u E Ker @, and a E U(b), then
+
+
+
+
+
+
+
y,(ua) = @,(u)(aT 8 1)
+
+
= 0;
hence (Ker @), M(-A E ) $. M(--l E ) ; consequently, (Ker @bM(--l E ) c K, whence Ker @, c A.
+
9.6.6. LEMMA (notation as in 9.6.1). Let A E tu*, and let J and J' be the E ) and L(-A E ) , respectively. The following conannihilators of M(-A ditions are equivalent: (i) J = J'; (i) X(-l,--l)' generates the g-module X(A,--l). If these conditions are satisfied, X(A,-A) is isomorphic to the g-module U(b)/J (the g-module structure of U(b) being defined as in 9.6.5 (i)).
+
+
318
HARISH-CHANDRA MODULES
[CH. 9,8 6
From 9.6.5, whose notation we are using, we have a diagram of g-homomorphisms U(b)/J v!-+ U(b)/J' U(g)fA X(A, -A) , (1)
*+
where y1 is surjective, y2 is bijective, and y 3 is injective. Let t". From 9.6.2 (i), mtp(u,X(A,-A)) is the multiplicity of the weight 0 in u. On the other hand, the g-module structure on U(b) defines, by restriction to t, a representation o f t in U(b)which can be identified with the adjoint representation of b in U(b). From 8.4.3 and 8.3.9, mtp(u,U(o)/J) is the multiplicity of the weight 0 in u. Hence (2) mtp(u,U(b)/J) = mtp(u,X(A,-A)). Given this, we have J = J' e y1 injective H y 3 0 y z y 1 injective 0
w y 3 yz yl bijective 0
H y 3 0 y2 0 y1 Hy3
(from (2))
0
surjective
(from (2))
surjective
@ U(elh = m-4. The last assertion then follows at once.
9.6.7. PROPOSITION (notation as in 9.6.1). Let A E to*.
( A - e)(H,)G {-l,-2, then X(A,-A)'
If
...,} for all a E S,,
generates the g-module X(A,-A).
This follows from 7.6.24 and 9.6.6. 9.6.8. LEMMA(notation as in 9.6.1). Let eo be the one-dimensional trivial representation of t. Then X(2~,-2.5)~ is a sub-g-module of X(2&,-2.5).
zeEt-,e+eo
For all A € tu*, let us denote the annihilator of M(A) by JA.If A = 2.5, the conditions of 9.6.6 are satisfied from 7.6.24. Hence the g-modules X(2q-2~) and U(b)/J-= are isomorphic. On the other hand, J, = J-, from 8.4.4, and M(E)has a one-dimensional b-module as a quotient (7.2.6). Hence J-, is contained in a two-sided ideal of codimension 1 of U(o). Consequently, X(2.5,-2~) has a subg-module W of codimension 1. Then W is a sub-€-moduleof X(2&,--2.5), and X(2e7-2e)/ W is a trivial g-module, hence X(2~,-2e) is a complement of W in X(2~,-2.5), and
w = CX(2&,-2&). €erA,e+Qo
CH. 9, J1 61
319
SPHERICAL MODULES IN THE DIAGONAL CASE
9.6.9. LEMMA (notation as in 9.6.1). Let 1E tu*. There exists a g-invariant 2&,A - 2 ~ such ) that: bilinear form y on ~ ( 1 , - 1 x) X(-1 (a) for every e E I”, the restriction of y to X(1,-1& x X(-1 2&,A- 24,. is non-degenerate; (b) i f @ , @ ’ €€”and e’ =I=e*, then X(A,-l)Q and X(-1 243, - 2 ~ ) are ~’ orthogonal to each other with respect to y.
+
+
+
(a) We recall that we defined a multiplication on U(g)* (2.7.4). We have
M’(-A,A)*
- M’(A - 2&,-A + 2&)*c M’(-2&,2&)*.
Indeed, if f E M’(-Ail.)*,
z E n, we have, from 2.7.7,
g E M’(A - 2 ~ -1 ,
‘~0t)Cfg) = ( t u Y ) f lg
+f (‘&)
= (il.,-il.)b>f * g +f(-1
+ 2~)*, x E rn,
y E a and
+ 2&, - 24(y)g ?b
= (2&,-2&)0cfg) and similarly ‘L(x)cfg)= ‘L(z)(fg)= 0 . Again from 2.7.7, we have, for all rE 0, r u g ) = (rfk f(rg) hence X(il.,-il.)X(-A 2&,1 - 2&)c X(2&,-2&).
+
+
(b) Let !P be the sum of the finite-dimensional simple sub-t-modules of U(f)* for the left or right coregular representation (2.7.12). In 2.7.13 we defined the fundamental linear form cpo and the fundamental bilinear form Po on Y and P x Y respectively. Let T be the set of the gE U(€) such that g(rnU(t)) = 0. Let T‘ = T A Y. Then the mapping f w f IU(€), where f takes all values in MI(-&I)*, is a t-isomorphism of M’(-A,A)* onto T endowed with the right coregular representation (5.5.8), and hence defines by restriction a €-isomorphism of X(1,--il.) onto T’. We identify X(A,-1) with T‘ under this isomorphism. Then the multiplication
X(A,-1) x X(-1
+ 2&,il. - 2.5) -+ X(2&,-2&)
can be identified with the multiplication in T’ (2.7.4). The form cpolT’ can be identified with a linear form on X(2.5-2~) which, from 9.6.8, is ginvariant. (c) The bilinear form y :U , g )H cpocfg) on
X(1,-1) x X(-A
+ 2&,1 - 2&)
is hence g-invariant. It can be identified with Po, and hence, from 2.7.15 and 2.7.17, satisfies properties (a) and (b) of the lemma.
320
[CH.9,D 6
HARISH-CHANDRA MODULES
9.6.10. PROPOSITION (notation as in 9.6.1). Let A E m*. If (A-e)(Ha)e{l,2,
...} forall ores,,
then every non-null sub-g-module of X(A, -A) contains X(A,-A)'; in particular, the sub-g-module of X(A,-A) generated by X(A,-A)* is simple. From 9.6.7, X(-A
+ 2 ~il, - 2.5)' X(-A
generates the g-module
+ 2 5 2 -24.
Let W be a sub-g-module of X(A,-A). Then W = W ,? X(A,-A)Q. Let y be as in 9.6.9, and W' the orthogonal subspace of Win X(-A 2 ~A,- 2.5). From 9.6.9, W is the orthogonal subspace of W' in X(A,-A). If X(A,-A)* is not contained in W, then X(-A 2&,1- 2e)' is contained in W,' hence W' = X(-A 2 ~A, - 2 4 , and W = 0.
+
+
+
m*. If ( A - ~)(H,)cjZ - (0) for all a E S,
9.6.11. THEOREM (notation as in 9.6.1). Let A E then X(A,-A)
is simple.
This follows from 9.6.7 and 9.6.10. 9.6.12. THEOREM (k algebraically closed, g semi-simple). Let 0 be a Cartan subalgebra of g, B a basis for R = R(g,Ij), A E $*, whence we have a central character xn, and Jn = U(g) Ker xn. We assume that A(H,) cj Z - {0}for all a C R. Then Jnis the only two-sided ideal J of U(g)such that J A Z(g) = KerXr. In particular, JA is a maximal two-sided ideal of U(g). Indeed, the (g x g)-module U(g)/Jnis simple from 8.4.3, 9.6.6 and 9.6.11. 9.6.13. THEOREM (notation as in 9.6.1). Let x be a homomorphism of Z(b) into k , and let us identifr Z(g) with Z(b) @ Z(b). There exists one, and up to isomorphism only one, spherical simple g-module whose central character x' satisfies x'lZ(g) @ 1 = x. We set G = U(g), and bl = b x 0 c x b = g. Since g = bl Q f, we have G = Gt Q U(a,). For every xE by the inner derivation of G defined by (x,x) leaves Gt and U(b,) stable, and in U(bl) induces the inner derivation defined by x. Hence G* = (G' A GT) @ Z(bl). Let @ be the set of homomorphisms of G' into k which are zero on G* A Gt. Let @' be the set of homomorphisms of Z(bl) into k. Then q~I+ ylZ(b,) is a bijection of @ onto a'. Given this, the theorem follows from 9.1.12. References: [371, (791, [1001.
CH. 9.8 71
SUPPLEMENTARY REMARKS
321
9.7. Supplementary remarks
9.7.1. The representations of the principal series were first considered by Gelfand and Naimark for the classical complex semi-simple groups. For the case of general real semi-simple groups, the theorems are, roughly, of the following type: (1) a representation of the principal series is most often simple; (2) two representations of the principal series are in general equivalent if the initial data are conjugate under the Weyl group (intertwining operators); (3) we give conditions for a representation of the principal series to be unitary. In spite of the number and significance of the papers dedicated to these questions, the situation is not yet very clear. Some special cases (e.g. complex groups, groups with symmetric rank 1, spherical representations, unitary representations) are closer to a complete solution. In particular, we may cite [79], [loo], [131], [133]. In this chapter, we have only given those theorems which can be presented in an entirely algebraic form. The essence is due to Harish-Chandra [64], [65]. We have followed Lepowsky and McCollum [89] in 9.1, and Lepowsky [87] and Rader [lo31 in 9.2, 9.3, 9.4 and 9.5. The canonical mapping of G' into A @ K"' was introduced by Lepowsky and Rader. Some important special cases have been considered previously; for example, the canonical homomorphism of Gf into A was introduced by Harish-Chandra. Theorems 9.6.11 and 9.6.13 are due to Parthasarathy, Ranga Rao and Varadarajan [loo]; the method of proof of 9.6.1 1 given here was indicated to me orally by Duflo. 9.7.2. (a) (b) (c) (d)
We adopt the notation of 9.0, with I reductive in g. Gg,"is the set of the u E G such that u(Vu)c V, for every g-modulz V. Ge+'G",' c Ge9' for all e,o,tE I". Ge is the largest subalgebra of G containing GIg as a two-sided ideal. Let V be a g-module, and S c V,. Then (GS) A V, = G'*g. [89]
9.7.3. (notation as in 9.2.1). Let I' = [€,I], and let f'" be the set of the having the following property: there exists nE g" such that n(f' has a subrepresentation equivalent to 5. Let L be the left ideal of U(€') such that the left regular representation of €' in U(€')/Lbelongs to €'". Then U(g)Lis the intersection of the maximal left ideals of finite codimension of U(g) which contain U(g)L. [65]
[ € €"'
9.7.4. (notation as in 7.0). Assume that k = C and that g is s.imple. Let lj* and let € be a symmetrizing subalgebra of g containing 0. Let c be the centre of f . Then dim c = 0 or 1.
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(a) If dim c = 0, then L(1) is only a Harish-Chandra module relative to t if L(A) is finite-dimensional, i.e., if 1 - 6 E P++. (b) If dim c = 1, the following conditions are equivalent: (i) L(A) is a Harish-Chandra module relative to t; (ii) ( A - d)(H,)E N for every positive root 01 such that ga c t. [66] 9.7.5 (notation as in 9.2.1). Assume that k = C. Let 8 be the automorphism of g corresponding to f , Wthe Weyl group of (g,O,a) (1.14.14), 6 = $- C aER+ 01, and 6' = 61a. If w E W, let w* be the affine automorphism 1 H w(1 - 6') b' of the dual of a, whence there is an automorphism of A, which is also denoted by w * . Let AW* be the set of the elements of A which are invariant under w* for all w E W. (a) Let u E t", let t be the linear form u H tr u(u) on K, 6 the canonical antihomomorphism of G' into A @I K , and 6 ' ' the linear mapping (1 @ t) 6 of G' into A @I k = A. Then GU(G')= AW*. If B is the canonical mapping of S(g) into U(g), then 6,"IP(S(p)') is a bijection of B(S(p)') onto AW*. (b) In particular, the canonical homomorphism of G* into A has AW* as its image. Then 9.5.6 defines a canonical bijective correspondence between classes of spherical simple g-modules and W,-orbits in a*. ([87], [88])
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9.7.6 (notation as in 9.2.1). Let V be a spherical simple g-module. Then V" has dimension 0 or 1, and m V" = 0. [79] 9.7.7 (notation as in 9.2.1). Assume that k = C. Let L be a left ideal of K . Assume that K / L is finite-dimensional and that the left regular representation of f in K / L is semi-simple. (a) The g-module V = G/GL is a Harish-Chandra module. (b) Let o E t". Then V, is a module of finite type over Z(g). [64] (c) Let @,aE t". Then Ge9a/GPis a module of finite type over Z(g). (This is a special case of (b), as Lepowsky pointed out to me.) 9.7.8 (notation as in 9.2.1). Assume that k = C.Let V be a Harish-Chandra g-module having a central character. Consider the following conditions : (i) for all 5 E f", mtp(t, V ) < +oo; (ii) there exists a constant C such that, for all [ E t", we have mtp(l, V)5 C dim 5 ; (iii) V is a U(g)-module of finite type; (iv) V has a Jordan-Holder series. (a) From 9.5.1 and 9.7.7, we have (iv) + (ii) 3 (i), and (iv) *.(iii) * (i). From 7.8.16, (ii) does not imply (iii) and (i) does not imply (ii). (b) Let c be the centre of f . The weights of the adjoint representation
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of c in g generate a subgroup 0 in c * ; this subgroup is free and of rank dim c. For every integer n > 0, let €,^ be the set of the 6~ t" such that the linear form with which 51c can be identified belongs to (l/n)0. If there exists an n such that V = @w;Vt (a property which is automatically satisfied if f is semi-simple), then conditions (i) to (iv) are equivalent. Cf. problem 34. ([87], [90])
9.7.9 (notation as in 9.2.1). Assume that k = C.Let x be a homomorphism of Z(g) into C, and n a positive integer. Up to isomorphism, there only exist a finite number of simple Harish-Chandra g-modules V such that: (1) the central character of V is x ; (2) with the notation of 9.7.8 (b), V = @eu; V,. Cf. problem 34. ([87], [90]) 9.7.10. We now abandon the notation of 9.0. Let G be a simply connected semi-simple real Lie group, and G = KAN an Iwasawa decomposition of G. Let gR, f,, a,, ,TI be the Lie algebras of G, K , A, N, and g, f, a, 11 the complexifications of gR, f,, a,, .,t Let mR be the commutant of a, in fR, and m the complexification of mR.Let Mo be the connected Lie subgroup of G with Lie algebra m,, let Z be the centre of G, which is contained in K, let Ml = MoZ, and let M, be the centralizer of a in K; then Mo c Ml c M,, and these three groups have mR as their Lie algebra. For i = 0,1,2, let Qi = MiAN, which is a Lie subgroup of G with Lie algebra ~TT, @ a,@ ,t A finite-dimensional irreducible representation e of Qi is trivial over N and hence can be identified with a pair (A,@), where A € a* and oE M:. Let IC be the representation of G induced by e. We shall leave the case where i = 0 aside. (It would then seem reasonable to limit ourselves to the case where e is unitary, taking IC in the Mackey sense. Even in this case, the relations with the algebraic coinduced representation are not satisfactory.) If i = 1, then G/Ql is compact, and we can define IC (which is in general not unitary) in Lz(K/M,). The corresponding representation of U(g) in the space of K-finite vectors gives us a Harish-Chandra module which is one of the modules X(e,y) of 9.3.1. Since we often have M1 M,, the representation IC is often reducible, and consequently the module X(e,y) is also reducible. If i = 2, then IC belongs to what is classically termed the principal series. Let X = X"" be the corresponding Harish-Chandra module. Here are certain results concerning the A'*" (which unfortunately do not apply to
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the modules X(e,y) in the text except in certain cases). We shall restrict ourselves to the case where (Ker 0)A 2 has finite index in 2. (i) There exists a subset A of a*, whose complement is the denumerable union of algebraic manifolds distinct from a*, such that, if A E A, then X"" is simple. (Kostant, Wallach) (ii) Let us assume that u is trivial and one-dimensional. Let us define 6' as in 9.7.5. Let D be the set of the linear forms on a which are real-valued on aR and whose scalar product with the elements of R'+ (with the notation of 9.2.1) is non-negative. If il - 6'E D ia:, then X"" is spherical and (XA3b)'generates X'*" as a U(g)-module. [79] (iii) There exists a non-degenerate bilinear form which is canonically defined on XA*a~X26'-A+'*,is g-invariant, and is such that every t-stable vector subspace of XI*"is equal to its bi-orthogonal subspace. [79] (iv) We adopt the notation of 9.7.5. If M' is the normalizer of a in K, then W can be identified with M'IM,, whence we have an action of W in M:. Then, if W E W, the Jordan-Holder series of X"." and of Xw*A*wn have the same quotients with the same multiplicities. [87] (v) If E E g", then there exist A and 0 such that mtp(S,X".") > 0; we then have mtp(E,Xa,") = 1. [90] (vi) Let G, be the simply connected complex Lie group with Lie algebra g. Then there is a canonical homomorphism r of G into G c . If Ker u 2 Ker r , then XI" is generated as a U(g)-module by a single element. [90]
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9.7.1 1. For the Harish-Chandra modules' over 31(2,C) (with t = Ch), cf. 7.8.16. For the Harish-Chandra modules over Sl(2,C) x 31(2,C) (with t equal to the diagonal subalgebra), cf. [57]. Cf. also [186].