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Asymptotics of Elementary Spherical Functions
A N A L Y T I C A L AND NUMERICAL APPROACHES T O ASYMPTOTIC PROBLEMS 1 N A N A L Y S I S S . A x e l s s o n , L . S . Frank, A . van d e r Sluis [ r r i s . ) @ N o r t h - H o l l a n d Pub1 i s h i n q C o m p a n y , 1981
ASYMPTOTICS OF ELEMENTARY SPHERICAL FUNCTIONS by Hans Duistermaat ldathematisch Instituut Rijksuniversiteit Utrecht Utrecht, The Netherlands
This is a report on work in progress together with J.A.C. Kolk and V.S. Varadarajan, in continuation of [ l ] . 1. Symmetric spaces of negative curvature.
The most fruitful approach to the analysis of symmetric spaces is via group theory. S o for us a negatively curved symmetric space S is a space on which a noncompact connected semisimple Lie group G acts transitively and such that the stabilizer of a point of S is a maximal compact subgroup K of G. That is, S may be identified with K\G and the space of functions (resp. distributions) on S with the space of left K-invariant functions (resp. distributions) on G. A continuous linear operator 4 : Cm(S) D' ( S ) which commutes with the (right-)action of G on S is always of the form +
c"(K\G) 3 f
(1.1)
;a *
f E c"(K\G)
where is a uniquely determined compactly supported distribution on G which is both left and right K-invariant. A left and right Kinvariant distribution on G is called spherical and the space of compactly supported spherical distributions on G will be denoted by E'(K\G/K). Note that 4 actually is continuous linear: Cm(S) Cm(S), C:(S) C:(S) and extends to continuous linear mappings: D' ( S ) D ' ( S ) , E' ( S ) E ' ( S ) . Also that E ' (K\G/K) is an algebra with respect to convolution, the convolution corresponding to the composition of the corresponding operators on S. +
+
+
If
+
= TeG, resp.%
OJ
=
TeK denote the Lie algebra's of G, resp. K,
then let 4 be the orthogonal complement of
k
in
bd
with respect to
the Killing form (1.2)
:
K
Because on which
(X,Y)
+
Tr(ad X
0
ad Y ) .
is non-degenerate and k is a maximal linear subspace of
K K
is negative definite,g = 35
k 86 and
K
is positive o n h .
,
36
H. DUISTERMAAT
Furthermore the map (x,X) x.exp X is a diffeomorphism: K x /3 G and w : x.exp XI-+ x.exp -X ( x E K, X E n ) defines an automorphism -1 for any of G called the Cartan involution. Writing x' = v(x) x E G, it follows that +
(x,y)
+
xyx'
:
G
x
expb
+
exp4
defines an action of G on e x p h allowing to identify exp.3 with the symmetric space S, having K as a Riemannian metric for which G acts by isometries. L e t a b e a maximal linear subspace o f 4 which at the same time is a commutative sub Lie algebra of 9. Correspondingly, A = exp M is a maximal flat subspace of S through e, r = dim A is called the rank of the symmetric space S. Because the ad X, X E 4 form a commuting set of symmetric (with respect to K @ - KI ) linear operators-in g, one has a common I6 I* decomposition of into eigenspaces:
7
(1.3)
y
=
Z' aEA
, 9,
91,
non-zero linear subspace of
43
such that (1.4) (ad X) (Y) = [X,Y1 = a(Y).Y
for Y E (aa.
Of course the eigenvalues a(X) are real and depend linearly on X E bt, so the a E A actually are elements o f m * , called the roots. if we write m = k n Also N C @do, more precisely q o = m @3
vo.
The null spaces in otof the a E A,a f 0 are called the root hyperplanes. Thier common complement has connected components which are convex open cones with piecewise flat boundaries, called the Weyl chambers. If k E K normalizes tn, that is Ad k ( W c M, then (because Ad k is a Lie algebra homomorphism) Ad k permutes the root hyperplanes and therefore also the Weyl chambers. Furthermore each orthogonal reflection in a root hyperplane arises in this way. In fact the Weyl group W, defined as the group generated by the orthogonal reflections in the root hyperplanes, is equal to the group (1.5) {Ad k l a ; k
z
E
MI
-
,M
=
normalizer of &in
K. In other words,
W = M/M if (1.6) M = {k E K; Ad k(X) = X for each X E
-
denotes the centralizer of &in K. The Lie algebras of and M are both equal t o m , so M is open in M. On the other hand M is compact 5
31
ELEMENTARY S P H E R I C A L FUNCTIONS
as a closed subgroup of K and therefore W is finite (a fact which puts very severe restrictions on the system of root hyperplanes). It is easy to see that W acts transitively on the set of Weyl chambers, it turns out that this action is simple as well, so in fact#(W) is equal to the number of Weyl chambers. Now choose a Weyl chamber, which will be called the positive + Weyl chamber OL from now on. (Because of the above it does not matter which one we choose.) The non-zero roots do not change sign + , write on (1.7)
A+ = {a E A ;
a(X)
>
0 for some (all) X E
m+)
for the corresponding set of positive roots. Then A = ( - A + ) U ( 0 1 u A + and writing (1.8) Z = {a E A+ ; not a = B+y for some 0,y
E
A+]
for the set of simple roots, it turns out that is a basis of m* and that each positive root is a linear combination of simple roots with non-negative integral coefficients. As,a consequence the walls of &+ are contained in the null spaces of the simple roots. Because [ ga~QdBl C (1.9)
Z
?t =
aEA+
qa+Bl the
sum
%a
is a nilpotent sub Lie algebra of OJ and N sub Lie group of G. The map (k,X,Y)
+
=
exp N is a nilpotent
k.exp X.exp Y
is a diffeomorphism: K x O'L x U + G and as a consequence each x E G can be written as x = k.a.n with uniquely determined k E K , a E A, n E N, depending smoothly on x (Iwasawa decomposition). For any 41
E C:(K\G/K)
(1.10) ( A $ ) (X) = e-p'x)
here (1.11)
p =
5
Z
write
I N
@(exp X.n)ds, X
E
n,
dim g,.a.
U€A+
A is a continuous linear map: C I ( K \ G / K )
,
extending to a E' (a).More importantly A is a continuous linear map: E' ( K \ G / K ) homomorphism with respect to convolution and it maps to W-invariant distributions on oi. The deepest result however, on the proof of which we shall comment later,is that A actually is an isomorphism: +
+
Cz(W)
30
H. DUISTERMAAT
resp. (.8(K\G/K), * ) ( d ( M ) W , * ) . For Schwartz functions the result is due to Harish-Chandra and for compactly supported functions to Helgason ( in some special cases) and Gangolli 1 2 1 . In particular the algebra of G-invariant linear operators on S ( € I (K\G/K), * ) is commutative. For example, the G-invariant differential operators D on S are, since they are local operators on K\G, given by convolution with @ E E'(K\G/K) satisfying supp @ C K. But then supp A @ c {O}, that is A @ = Dgt6 for some uniquely determined differential operator D D&is an isomorphism from with constant coefficients on&, and D the algebra Diff (S)G of G-invariant differential operators on S onto the algebra Dif f (a) "61 of W-invariant differential operators with constant coefficients o n m . NOW, if u E D ' ( S ) is a common eigendistribution of all G-invariant differentialoperators D on S, that is (1.12) Du = A ( D ) . u for all D E Diff(S)G , +
+
A(D) is a homomorphism : Diff(S)G then D identification with Diff (a)"& is given by +
+
C, which via the
for some X EdC;f: which is uniquely determined modulo the action of the Weyl group. The space of common eigendistributions for a given X consists of analytic functions. It contains a spherical one, unique up to a factor, which is given by
Here H(y) is the element of ot defined by (1.15) y E K.exp H(y).N.
In view of the Iwasawa decomposition, H is a smooth mapping: G called the Iwasawa projection. In fact @ A is equal to the image of the exponential function X e(xrx' under the transpose: C"(0t) Cm(K\G/K) of the continuous linear map A : E' (K\G/K) E l (OL) , the integration over K is needed in order to make the functionright K-invariant as well. It may also be remarked that @ A is a common eigenfunction for all G-invariant C m ( S ) and that Fourier continuous linear operators : Cm(S) +
+
+
+
+
ELEMENTARY SPHERICAL FUNCTIONS
39
decomposition in ot shows that every element of E ' (K\G/K) can be expanded in @ p E a*.So the name elementary spherical function ip for the $ A is fully deserved. 2. Harish-Chandra's assptotics. The map kl.exp X.k2
-+
X (kl,k2 E K,X E m+) defines a fibration a
+
from an open dense subset U of G to OL I if f E C"(ot+) then a*f is a spherical function on U. For any differential operator D on G I
+
D(a*f)I + is a differential operator onot , called the radial M part Drad of D. On the other hand one has the integration formula f
-+
(2.1) J f(x)dx = J A(X). I f(kl.exp X.k2)dkldk2 dX G KxK R+ where dim q, (2.2) A(X) = const. II (f(ea (X)-e-a (X)) ) c i a +
and following Gangolli [2] it turns out to be somewhat more convenient to work with
Then D
+
6
is a homomorphism from Diff (S)G to the algebra of
+
differential operators on m with smooth coefficients, the conjugation with the factor A' makes that also D* = ( 6 ) * so in particular symmetric operators get mapped to symmetric operators. = A f $ / + are common eigenfunctions of all 6, Obviously the hot
D E Diff(S)G. If the 6 would have constant coefficients then it would follow that the $ A are sums of W(W) many exponential functions which are easy to analyse. This is not true,but asymptotically when + (away from the walls) then D approaches a constant X m in ot coefficient operator in an exponential fashion. In order to describe this, write -+
Then (2.5)
+
01.
=
[z EIR'
: 0
<
z,
1 for
ci
E
XIl
z a = 0 corresponding to infinity and z a = 1 to a wall of the Weyl + i chamber bl . Then 6 is an operator in OL with coefficients which can
40
H . DUISTERMAAT
L
be written as analytic functions of z E Q: , having only poles if z = 1 for some a E Z. The highest order part of 6 has constant caefficients, and also the constant coefficient part of 6, obtained by taking the values of the coefficients at z=O turns out to be equal to the operator DR E Diff (Ot)wrR mentioned in section 1. The common eigenfunctions of the DOLare linear comb nations of the exponential functions
and because Qw*h 1Y (2.7) Q A ( X )
= Qh
for all w
- c(h).wEW C e
E
W, it is likely that asymptotical-
(w(XI)
for some c(A). From this Harish-Chandra [ 3 1 deduces that if F denotes the Fourier transform mapping functions on ot. to functions on oL*, then tAo F - l is a unitary embedding : L2 (m*/W,B) L 2 ( G ) if onot*/W we take the measure +
(2.8)
6
=
Ic(ip)l-2 dp
(li E
m*)
called the Pencherel measure for spherical functions. This is the main step in the proof of the isomorphism property of A mentioned in the previous section, the remarkable fact being that the Plancherel measure is expressed in terms of the coefficient in the leading term of the asymptotic expansion of the elementary spherical m in &+. functions 4 (x) as x ili The most elegant proof of the Harish-Chandra asymptotics can be obtained by observing (like is done by Kashiwara e.a. [ 5 ] ) that in the za-coordinates the satisfy a "holonomic system with regular singularities at z a = 0 " . Because the only other singularities occur at z a = 1 this approach immediately leads to convergent power series expansions with radius of convergence equal to 1. Also this approach leads to a better understanding of the behaviour if A = ip, p approaching a root hyperplane, in which case the Harish-Chandra asymptotics breaks down. If the symmetric space has rank 1 , that is dim or = 1 , then the 6 are ordinary differential operators and one can apply the classical theory of such operators with regular singularities. However, the point is that if dimor > 1 the system of partial differential equations actually leads to a system of ordinary differential equations with regular singularities along +
+,
41
ELEMENTARY SPHERICAL FUNCTIONS
each curve in the z-space, to which then the same asymptotic theory can be applied.
as p
3. Asymptotics of I$~"(X)
+
m
in in*.
In [ 1 1 we studied the asymptotic behaviour of the common spectrum of the G-invariant operators on S, pushed down to a compact quotient S/T of S in order to make the spectrum discrete. It is not surprising that for more detailed information we need the asymptotic
1x1
(keeping Re A bounded) which is complementary to the asymptotics of Harish-Chandra.
behaviour of I$,(X) as
+
m
Here the approach is to incorporate Re X in p in (1.14) and read (1.14) as an oscillatory integral
i( p,H(xk)) g (H(xk)) dk e K with phase function f : k (p,H(xk)) and amplitude k g(H(xk)) . UfX + In view of the decomposition G = K . e x p 4 .K we may take x = exp X, + X E OL and then H(xkm) = H(xk) for m ' E M = dentralizer of M. in K (see (1.6)) , so (3.1) can actually be seen as an integral over the flag variety K/M rather than over K. (The name is because for the classical groups, M is the stabilizer group for a natural action of K on a set of flags.) The first step is to observe that the asymptotic expansion of (3.1) for p = w.v, w E I R , w m is determined by the behaviour of the amplitude near the stationary point of the phase function fv,X. (3.1)
I
+
+
+
3.1. Theorem. The set of stationary points of f
VfX
(3.2)
Cv,x =
is equal to
K'GK',
which is a smooth submanifold of K, and the rank of the Hessian of is equal to the codimension of C in K. f v f X at each point of C vtx V,X (Clean stationary point set in the sense of Bott.) Obviously f is constant on the connected components of C /PI v,x v,x and because W acts transitively on the set of these connected components, an application of the method of stationary phase immediately leeds to the asymptotic expansion
as p = o.v, w
+
m.
42
H . DUISTERMAAT
An explicit calculation of the Hessian of f at c gave us an v,x v,x explicit formula for the leading coefficient cw,ot if both v and X are regular (not in a root hyperplane), then
Although the proof is quite different, the result is very similar to the asymptotics of Harish Chandra. In fact I believe that in the same way as Harish Chandra's asymptotics implied that tA*°F-l is a unitary embedding: L 2 ( O t * / W , B ) L 2 ( G ) , our asymptotics implies that the adjoint F o A is a unitary embedding:-L2 ( K \ G / K ) --* L2( */W,B). As a consequence A is injective, or equivalently A*OF-' has a dense range, which was the missing bit of information in the proof that A is an isomorphism. With a lot more work we are able also to get uniform asymptotic estimates when 1 ~ --*1 m , allowing u to pass through the root hyperplanes. (There the behaviour is quite' singular because the dimension of C then suddenly increases, leading to a jump in the order of v,x the asymptotic expansion ( 3 . 3 ) . This is an example of the Stokes phenomenon.) Here X is kept in compact subsets: we are also working on the still harder problem to understand what happens if both p and X run to infinity, that is to unify our asymptotics with the asymptotics of Harish-Chandra. I would like to close with some comments on the remarkable properties of the Iwasawa projection +
(3.4)
k
H(xk)
:
K/M
+
OL
which are a consequence of the description of the set of stationary points of fv,x. (fv,x is nothing else as testing the image with the linear function v.) The remarkable fact is that the set of stationary points in K/M does not move continuously with v : K v only depends on the set of roots orthogonal to v and this varies within a finite set, the dimension only going up when v enters the intersection of more root hyperplanes. Assuming for convenience that X is regular + ( = not in a wall of O t ) then if v is regular as well, /M = W all v,x the time, so for instance the maximum value of fv,.j( is equal to ( v, (w-l(X)) for some w E W, all the time. It immediately follows
ELEMENTARY SPHERICAL FUNCTIONS
43
that the image of K/M under the Iwasawa projection is contained in the convex hull of the finite set {w-l(X) ; w E W) and expanding this argument a little further one finds back the 3.2.
Theorem (Kostant [ 6 1 ) . The image of K/M under the Iwasawa projection ( 3 . 4 ) is equal to the convex hull of the Weyl group orbit of X in 4. This result is very remarkable indeed because K/M is a beautiful smooth compact manifold without boundary, the Iwasawa projection is analytic, nevertheless the image is a polyeder, full of faces, edges and corners. Of course for us the surprise came already at an earlier stage, namely that the asymptotics of ( 3 . 1 ) could always be obtained by just applying the methods of stationary phase on a set of stationary points depending in a discrete fashion on the parameter V.
For more convexity results, generalizing Kostant's theorem, see the thesis of Heckman [ 4 ] , defended yesterday in Leiden.
REFERENCES Duistermaat, J.J., Kolk, J.A.C. and Varadarajan, V.S., Spectra of compact locally symmetric manifolds of negative curvature, Inv. Math. 2 ( 1 9 7 9 ) , 2 7 - 9 3 . Gangolli, R . , On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Liegroups, Ann. of Math.
93
( 1 9 7 1 ) , 150-165.
Harish-Chandra, Spherical functions on a semi-simple Liegroup I,II, Amer. J. Math. 80 ( 1 9 5 8 ) , 2 4 1 - 3 1 0 , 5 5 3 - 6 1 3 . Heckman, G.J., Projections of orbits and asymptotic behaviour of multiplicities for compact Lie groups, Thesis, Leiden, 1 9 8 0 . Kashiwara, M., Kowata, A., Minemura, K., Okamoto, K., Oshima, T. and Tanaka, M., Eigenfunctions of invariant differential operators on a symmetric space, Annals of Math., 107 ( 1 9 7 8 ) , 1-39.
Kostant, B . , On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ec. Norm. Sup. 6 ( 1 9 7 3 ) , 4 1 3 - 4 5 5 .
ASYMPTOTICS OF ELEMENTARY SPHERICAL FUNCTIONS by Hans Duistermaat ldathematisch Instituut Rijksuniversiteit Utrecht Utrecht, The Netherlands
This is a report on work in progress together with J.A.C. Kolk and V.S. Varadarajan, in continuation of [ l ] . 1. Symmetric spaces of negative curvature.
The most fruitful approach to the analysis of symmetric spaces is via group theory. S o for us a negatively curved symmetric space S is a space on which a noncompact connected semisimple Lie group G acts transitively and such that the stabilizer of a point of S is a maximal compact subgroup K of G. That is, S may be identified with K\G and the space of functions (resp. distributions) on S with the space of left K-invariant functions (resp. distributions) on G. A continuous linear operator 4 : Cm(S) D' ( S ) which commutes with the (right-)action of G on S is always of the form +
c"(K\G) 3 f
(1.1)
;a *
f E c"(K\G)
where is a uniquely determined compactly supported distribution on G which is both left and right K-invariant. A left and right Kinvariant distribution on G is called spherical and the space of compactly supported spherical distributions on G will be denoted by E'(K\G/K). Note that 4 actually is continuous linear: Cm(S) Cm(S), C:(S) C:(S) and extends to continuous linear mappings: D' ( S ) D ' ( S ) , E' ( S ) E ' ( S ) . Also that E ' (K\G/K) is an algebra with respect to convolution, the convolution corresponding to the composition of the corresponding operators on S. +
+
+
If
+
= TeG, resp.%
OJ
=
TeK denote the Lie algebra's of G, resp. K,
then let 4 be the orthogonal complement of
k
in
bd
with respect to
the Killing form (1.2)
:
K
Because on which
(X,Y)
+
Tr(ad X
0
ad Y ) .
is non-degenerate and k is a maximal linear subspace of
K K
is negative definite,g = 35
k 86 and
K
is positive o n h .
,
36
H. DUISTERMAAT
Furthermore the map (x,X) x.exp X is a diffeomorphism: K x /3 G and w : x.exp XI-+ x.exp -X ( x E K, X E n ) defines an automorphism -1 for any of G called the Cartan involution. Writing x' = v(x) x E G, it follows that +
(x,y)
+
xyx'
:
G
x
expb
+
exp4
defines an action of G on e x p h allowing to identify exp.3 with the symmetric space S, having K as a Riemannian metric for which G acts by isometries. L e t a b e a maximal linear subspace o f 4 which at the same time is a commutative sub Lie algebra of 9. Correspondingly, A = exp M is a maximal flat subspace of S through e, r = dim A is called the rank of the symmetric space S. Because the ad X, X E 4 form a commuting set of symmetric (with respect to K @ - KI ) linear operators-in g, one has a common I6 I* decomposition of into eigenspaces:
7
(1.3)
y
=
Z' aEA
, 9,
91,
non-zero linear subspace of
43
such that (1.4) (ad X) (Y) = [X,Y1 = a(Y).Y
for Y E (aa.
Of course the eigenvalues a(X) are real and depend linearly on X E bt, so the a E A actually are elements o f m * , called the roots. if we write m = k n Also N C @do, more precisely q o = m @3
vo.
The null spaces in otof the a E A,a f 0 are called the root hyperplanes. Thier common complement has connected components which are convex open cones with piecewise flat boundaries, called the Weyl chambers. If k E K normalizes tn, that is Ad k ( W c M, then (because Ad k is a Lie algebra homomorphism) Ad k permutes the root hyperplanes and therefore also the Weyl chambers. Furthermore each orthogonal reflection in a root hyperplane arises in this way. In fact the Weyl group W, defined as the group generated by the orthogonal reflections in the root hyperplanes, is equal to the group (1.5) {Ad k l a ; k
z
E
MI
-
,M
=
normalizer of &in
K. In other words,
W = M/M if (1.6) M = {k E K; Ad k(X) = X for each X E
-
denotes the centralizer of &in K. The Lie algebras of and M are both equal t o m , so M is open in M. On the other hand M is compact 5
31
ELEMENTARY S P H E R I C A L FUNCTIONS
as a closed subgroup of K and therefore W is finite (a fact which puts very severe restrictions on the system of root hyperplanes). It is easy to see that W acts transitively on the set of Weyl chambers, it turns out that this action is simple as well, so in fact#(W) is equal to the number of Weyl chambers. Now choose a Weyl chamber, which will be called the positive + Weyl chamber OL from now on. (Because of the above it does not matter which one we choose.) The non-zero roots do not change sign + , write on (1.7)
A+ = {a E A ;
a(X)
>
0 for some (all) X E
m+)
for the corresponding set of positive roots. Then A = ( - A + ) U ( 0 1 u A + and writing (1.8) Z = {a E A+ ; not a = B+y for some 0,y
E
A+]
for the set of simple roots, it turns out that is a basis of m* and that each positive root is a linear combination of simple roots with non-negative integral coefficients. As,a consequence the walls of &+ are contained in the null spaces of the simple roots. Because [ ga~QdBl C (1.9)
Z
?t =
aEA+
qa+Bl the
sum
%a
is a nilpotent sub Lie algebra of OJ and N sub Lie group of G. The map (k,X,Y)
+
=
exp N is a nilpotent
k.exp X.exp Y
is a diffeomorphism: K x O'L x U + G and as a consequence each x E G can be written as x = k.a.n with uniquely determined k E K , a E A, n E N, depending smoothly on x (Iwasawa decomposition). For any 41
E C:(K\G/K)
(1.10) ( A $ ) (X) = e-p'x)
here (1.11)
p =
5
Z
write
I N
@(exp X.n)ds, X
E
n,
dim g,.a.
U€A+
A is a continuous linear map: C I ( K \ G / K )
,
extending to a E' (a).More importantly A is a continuous linear map: E' ( K \ G / K ) homomorphism with respect to convolution and it maps to W-invariant distributions on oi. The deepest result however, on the proof of which we shall comment later,is that A actually is an isomorphism: +
+
Cz(W)
30
H. DUISTERMAAT
resp. (.8(K\G/K), * ) ( d ( M ) W , * ) . For Schwartz functions the result is due to Harish-Chandra and for compactly supported functions to Helgason ( in some special cases) and Gangolli 1 2 1 . In particular the algebra of G-invariant linear operators on S ( € I (K\G/K), * ) is commutative. For example, the G-invariant differential operators D on S are, since they are local operators on K\G, given by convolution with @ E E'(K\G/K) satisfying supp @ C K. But then supp A @ c {O}, that is A @ = Dgt6 for some uniquely determined differential operator D D&is an isomorphism from with constant coefficients on&, and D the algebra Diff (S)G of G-invariant differential operators on S onto the algebra Dif f (a) "61 of W-invariant differential operators with constant coefficients o n m . NOW, if u E D ' ( S ) is a common eigendistribution of all G-invariant differentialoperators D on S, that is (1.12) Du = A ( D ) . u for all D E Diff(S)G , +
+
A(D) is a homomorphism : Diff(S)G then D identification with Diff (a)"& is given by +
+
C, which via the
for some X EdC;f: which is uniquely determined modulo the action of the Weyl group. The space of common eigendistributions for a given X consists of analytic functions. It contains a spherical one, unique up to a factor, which is given by
Here H(y) is the element of ot defined by (1.15) y E K.exp H(y).N.
In view of the Iwasawa decomposition, H is a smooth mapping: G called the Iwasawa projection. In fact @ A is equal to the image of the exponential function X e(xrx' under the transpose: C"(0t) Cm(K\G/K) of the continuous linear map A : E' (K\G/K) E l (OL) , the integration over K is needed in order to make the functionright K-invariant as well. It may also be remarked that @ A is a common eigenfunction for all G-invariant C m ( S ) and that Fourier continuous linear operators : Cm(S) +
+
+
+
+
ELEMENTARY SPHERICAL FUNCTIONS
39
decomposition in ot shows that every element of E ' (K\G/K) can be expanded in @ p E a*.So the name elementary spherical function ip for the $ A is fully deserved. 2. Harish-Chandra's assptotics. The map kl.exp X.k2
-+
X (kl,k2 E K,X E m+) defines a fibration a
+
from an open dense subset U of G to OL I if f E C"(ot+) then a*f is a spherical function on U. For any differential operator D on G I
+
D(a*f)I + is a differential operator onot , called the radial M part Drad of D. On the other hand one has the integration formula f
-+
(2.1) J f(x)dx = J A(X). I f(kl.exp X.k2)dkldk2 dX G KxK R+ where dim q, (2.2) A(X) = const. II (f(ea (X)-e-a (X)) ) c i a +
and following Gangolli [2] it turns out to be somewhat more convenient to work with
Then D
+
6
is a homomorphism from Diff (S)G to the algebra of
+
differential operators on m with smooth coefficients, the conjugation with the factor A' makes that also D* = ( 6 ) * so in particular symmetric operators get mapped to symmetric operators. = A f $ / + are common eigenfunctions of all 6, Obviously the hot
D E Diff(S)G. If the 6 would have constant coefficients then it would follow that the $ A are sums of W(W) many exponential functions which are easy to analyse. This is not true,but asymptotically when + (away from the walls) then D approaches a constant X m in ot coefficient operator in an exponential fashion. In order to describe this, write -+
Then (2.5)
+
01.
=
[z EIR'
: 0
<
z,
1 for
ci
E
XIl
z a = 0 corresponding to infinity and z a = 1 to a wall of the Weyl + i chamber bl . Then 6 is an operator in OL with coefficients which can
40
H . DUISTERMAAT
L
be written as analytic functions of z E Q: , having only poles if z = 1 for some a E Z. The highest order part of 6 has constant caefficients, and also the constant coefficient part of 6, obtained by taking the values of the coefficients at z=O turns out to be equal to the operator DR E Diff (Ot)wrR mentioned in section 1. The common eigenfunctions of the DOLare linear comb nations of the exponential functions
and because Qw*h 1Y (2.7) Q A ( X )
= Qh
for all w
- c(h).wEW C e
E
W, it is likely that asymptotical-
(w(XI)
for some c(A). From this Harish-Chandra [ 3 1 deduces that if F denotes the Fourier transform mapping functions on ot. to functions on oL*, then tAo F - l is a unitary embedding : L2 (m*/W,B) L 2 ( G ) if onot*/W we take the measure +
(2.8)
6
=
Ic(ip)l-2 dp
(li E
m*)
called the Pencherel measure for spherical functions. This is the main step in the proof of the isomorphism property of A mentioned in the previous section, the remarkable fact being that the Plancherel measure is expressed in terms of the coefficient in the leading term of the asymptotic expansion of the elementary spherical m in &+. functions 4 (x) as x ili The most elegant proof of the Harish-Chandra asymptotics can be obtained by observing (like is done by Kashiwara e.a. [ 5 ] ) that in the za-coordinates the satisfy a "holonomic system with regular singularities at z a = 0 " . Because the only other singularities occur at z a = 1 this approach immediately leads to convergent power series expansions with radius of convergence equal to 1. Also this approach leads to a better understanding of the behaviour if A = ip, p approaching a root hyperplane, in which case the Harish-Chandra asymptotics breaks down. If the symmetric space has rank 1 , that is dim or = 1 , then the 6 are ordinary differential operators and one can apply the classical theory of such operators with regular singularities. However, the point is that if dimor > 1 the system of partial differential equations actually leads to a system of ordinary differential equations with regular singularities along +
+,
41
ELEMENTARY SPHERICAL FUNCTIONS
each curve in the z-space, to which then the same asymptotic theory can be applied.
as p
3. Asymptotics of I$~"(X)
+
m
in in*.
In [ 1 1 we studied the asymptotic behaviour of the common spectrum of the G-invariant operators on S, pushed down to a compact quotient S/T of S in order to make the spectrum discrete. It is not surprising that for more detailed information we need the asymptotic
1x1
(keeping Re A bounded) which is complementary to the asymptotics of Harish-Chandra.
behaviour of I$,(X) as
+
m
Here the approach is to incorporate Re X in p in (1.14) and read (1.14) as an oscillatory integral
i( p,H(xk)) g (H(xk)) dk e K with phase function f : k (p,H(xk)) and amplitude k g(H(xk)) . UfX + In view of the decomposition G = K . e x p 4 .K we may take x = exp X, + X E OL and then H(xkm) = H(xk) for m ' E M = dentralizer of M. in K (see (1.6)) , so (3.1) can actually be seen as an integral over the flag variety K/M rather than over K. (The name is because for the classical groups, M is the stabilizer group for a natural action of K on a set of flags.) The first step is to observe that the asymptotic expansion of (3.1) for p = w.v, w E I R , w m is determined by the behaviour of the amplitude near the stationary point of the phase function fv,X. (3.1)
I
+
+
+
3.1. Theorem. The set of stationary points of f
VfX
(3.2)
Cv,x =
is equal to
K'GK',
which is a smooth submanifold of K, and the rank of the Hessian of is equal to the codimension of C in K. f v f X at each point of C vtx V,X (Clean stationary point set in the sense of Bott.) Obviously f is constant on the connected components of C /PI v,x v,x and because W acts transitively on the set of these connected components, an application of the method of stationary phase immediately leeds to the asymptotic expansion
as p = o.v, w
+
m.
42
H . DUISTERMAAT
An explicit calculation of the Hessian of f at c gave us an v,x v,x explicit formula for the leading coefficient cw,ot if both v and X are regular (not in a root hyperplane), then
Although the proof is quite different, the result is very similar to the asymptotics of Harish Chandra. In fact I believe that in the same way as Harish Chandra's asymptotics implied that tA*°F-l is a unitary embedding: L 2 ( O t * / W , B ) L 2 ( G ) , our asymptotics implies that the adjoint F o A is a unitary embedding:-L2 ( K \ G / K ) --* L2( */W,B). As a consequence A is injective, or equivalently A*OF-' has a dense range, which was the missing bit of information in the proof that A is an isomorphism. With a lot more work we are able also to get uniform asymptotic estimates when 1 ~ --*1 m , allowing u to pass through the root hyperplanes. (There the behaviour is quite' singular because the dimension of C then suddenly increases, leading to a jump in the order of v,x the asymptotic expansion ( 3 . 3 ) . This is an example of the Stokes phenomenon.) Here X is kept in compact subsets: we are also working on the still harder problem to understand what happens if both p and X run to infinity, that is to unify our asymptotics with the asymptotics of Harish-Chandra. I would like to close with some comments on the remarkable properties of the Iwasawa projection +
(3.4)
k
H(xk)
:
K/M
+
OL
which are a consequence of the description of the set of stationary points of fv,x. (fv,x is nothing else as testing the image with the linear function v.) The remarkable fact is that the set of stationary points in K/M does not move continuously with v : K v only depends on the set of roots orthogonal to v and this varies within a finite set, the dimension only going up when v enters the intersection of more root hyperplanes. Assuming for convenience that X is regular + ( = not in a wall of O t ) then if v is regular as well, /M = W all v,x the time, so for instance the maximum value of fv,.j( is equal to ( v, (w-l(X)) for some w E W, all the time. It immediately follows
ELEMENTARY SPHERICAL FUNCTIONS
43
that the image of K/M under the Iwasawa projection is contained in the convex hull of the finite set {w-l(X) ; w E W) and expanding this argument a little further one finds back the 3.2.
Theorem (Kostant [ 6 1 ) . The image of K/M under the Iwasawa projection ( 3 . 4 ) is equal to the convex hull of the Weyl group orbit of X in 4. This result is very remarkable indeed because K/M is a beautiful smooth compact manifold without boundary, the Iwasawa projection is analytic, nevertheless the image is a polyeder, full of faces, edges and corners. Of course for us the surprise came already at an earlier stage, namely that the asymptotics of ( 3 . 1 ) could always be obtained by just applying the methods of stationary phase on a set of stationary points depending in a discrete fashion on the parameter V.
For more convexity results, generalizing Kostant's theorem, see the thesis of Heckman [ 4 ] , defended yesterday in Leiden.
REFERENCES Duistermaat, J.J., Kolk, J.A.C. and Varadarajan, V.S., Spectra of compact locally symmetric manifolds of negative curvature, Inv. Math. 2 ( 1 9 7 9 ) , 2 7 - 9 3 . Gangolli, R . , On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Liegroups, Ann. of Math.
93
( 1 9 7 1 ) , 150-165.
Harish-Chandra, Spherical functions on a semi-simple Liegroup I,II, Amer. J. Math. 80 ( 1 9 5 8 ) , 2 4 1 - 3 1 0 , 5 5 3 - 6 1 3 . Heckman, G.J., Projections of orbits and asymptotic behaviour of multiplicities for compact Lie groups, Thesis, Leiden, 1 9 8 0 . Kashiwara, M., Kowata, A., Minemura, K., Okamoto, K., Oshima, T. and Tanaka, M., Eigenfunctions of invariant differential operators on a symmetric space, Annals of Math., 107 ( 1 9 7 8 ) , 1-39.
Kostant, B . , On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ec. Norm. Sup. 6 ( 1 9 7 3 ) , 4 1 3 - 4 5 5 .