Harmonic Analysis on Real Reductive Groups (Lecture Notes in Mathematics 576)

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

576 V. S. Varadarajan

Harmonic Analysis on Real Reductive Groups

Springer-Verlag Berlin. Heidelberg-New York 1977

Author V. S. Varadarajan Department of Mathematics University of California at Los Angeles Los Angeles, C A 9 0 0 2 4 / U S A

Library of Congress Cataloging ila Publication Data

Varadarajan,

V S Harmonic analysis on real reductive groups.

(Lecture notes in mathematics ; 576) Includes bibliographical references. 1. Lie groups. 2. Lie algebras. 3. Harmonic analysis. I. Title. II. Title : Real reductive oups. III. series: Lecture notes in mathematics erlin) ; 576. 0A3.L28 no. 576 [QA387] 510'.8s [53-2'.55] 77-22]-6

~B

AMS Subject Classifications (1970): 22 E30, 2,.r ) E45

ISBN 3-540-08135-6 ISBN 0-387-08135-6

Springer-Verlag Berlin- Heidelberg. New York Springer-Verlag New York • Heidelberg • Berlin

This .work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin " Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

PREFACE

The contents of these notes are essentially the same as those of a Seminar on semisimple groups that I conducted during 1969-1975 at the University of California at Los Angeles.

I am very grateful to Professors Gangolli and Eckmann

for suggesting that this material appear in the Springer Lecture Notes Series and encouraging me to prepare them for publication. My aim here has been to give a more or less self-contained exposition of Harish-Chandra's work on harmonic analysis on real reductive groups, leading to the complete determination of the discrete series.

I have kept quite close to

his view of the subject although the informed reader may perceive departures in detail here and there. These notes are in two parts.

Part one deals with the problems of invariant

analysis on a real reductive Lie algebra.

It contains a full treatment of regular

orbital integrals and their Fourier transforms; the theorem that invariant eigendistributions

it presents a detailed proof of

are locally integrable functions;

and concludes with the proof of the theorem that an analytic invariant differential operator that kills all invariant distributions.

C~

functions, kills all invariant

Part two treats the theory on the group, with descent to Lie al-

gebra playing a key role in many proofs.

Here I have proved that invariant

eigendistributions on real reductive groups are locally integrable functions~ given the explicit construction of the characters of the discrete series~ and treated all the aspects of Schwartz space and tempered distributions that are needed to reach the goals I set out with. Due to obvious limitations I have not made any attempt to discuss other contributions to this subjeet~ such as orbital integrals of nilpotents~ analysis over local fields, to mention a f e ~

invmri~nt

The subject is in a very active

phase of development and many recent contributions

suggest a real possibility of

a significantly different way of treating some of these questions.

However~ I

feel that an exposition that attempts to maintain the original and pioneering perspective of Harish-Chandra deserves a place in the literature. I wish to thank all my friends with whom I ha~e discussed this subject over the past several years.

In addition, I would like to thank Peter Trombi, King

Lai~ Mohsen Pazirandeh and Thomas Er~right for encouraging me to continue the seminar during the period it was being run, and for help in checking the manuscript.

Without this help these notes would not have appeared.

I am above all

deeply grateful to Harish-Chandra for giving me his time and ideas so generously

durir~ my various visits to Princeton and for helping me to understand his view of the subject. Chaa~lotte Johnson typed these notes with great skill, patience~

and speed.

I am very grateful to her for putting up with all my demands and carrying out the many and often confusing changes I wanted.

Alice Hume typed an early draft

of a section of these notes and Elaine Barth helped me in preparing these notes at all stages.

To both of them my gratitude.

Finally, I wish to acknowledge my indebtedness

to Various institutions

Foundations that supported me during the many stages of the preparation: Alfred P. Sloan Foundation;

and

to the

to the National Science Foundation for the grant

that has supported me over the past several years; to the I.H.E.S. at Bures/ Yvette, the Mathematics Institute for Advanced

Institute

of the Rijks University of Utrecht and the

Study at Princeton for their hospitality during 1975;

and, to the lively and diversified

group of young men and women at the Huize

Fatimah in Zeist, Holland for providing me with a most unusual working atmosphere during the Fall of 1975 when I wrote these notes in their present form.

Pacific Palisades,

1976

V.S.

Varadarajan

CONTENTS

PART

I

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

0.

Summary

3

i.

Orbit structure

2.

Transfer of d i s t r i b u t i o n s

3.

~le i n v a r i a n t

4.

Local structure

. . . . . . . . . .

behaviour 5.

of the adjoint r e p r e s e n t a t i o n

integral

6.

Local structure

7.

Tempered

Subject

on ~:

f(0)= ~(8(~b)~f,b)(0 ) operators

78

on ~:

singular p o i n t s . . . . . . .

. . . . . . . . . . . . . .

that annihilate

96 . .

|05 123

all i n v a r i a n t

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Index of symbols

58

. . . . . . . . . . . . . . . . . . . . . . eigendistributions

23 36

. . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

index

9 . .

on 9:

points

eigendistributions o n a reductive Lie algebra.

differential

Appendix

eigendistributions

of the f u n c t i o n F around

limit f o r m u l a

References

f r o m ~ to ~.

. . . . . . . . . . . . . . . . . . . . .

of invariant

distributions i0.

theorem

invariant

The

. . . . . . . . . . . .

operators

of i n v a r i a n t e i g e n d i s t r i b u t i o n s

the b e h a v i o u r

Invariant

on ~

of invariant

the f u n d a m e n t a l

9.

and d i f f e r e n t i a l

around r e g u l a r and s e m i r e g u l a r

Local structure

8.

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

132 151 166 168 173

PART II

Contents

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.

Summary

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i.

Groups of class

]~

2.

Orbit

3.

Descent

structure

4.

Local

5.

The d i s t r i b u t i o n s

6.

Parabolic

from

in

G

G

to

structure

%,

subgroups

7.

Some r e p r e s e n t a t i o n The functions

9-

Schwartz

. . . . . . . . . . . . . . . . . . . . . . . . Z

and

. . . . . . . . . . . . . . . . . . .

~

distributions

. . . . . . . . .

192 202 221 233

. . . . . . . . . . . . . . . . . . . . . . . .

243

. . . . . . . . . . . . . . . . . . . . . . . . .

279

theory

. . . . . . . . . . . . . . . . . . . . .

302

. . . . . . . . . . . . . . . . . . . . . .

320

and

~

space and t e m p e r e d

The invariant

~

of invariant 8 - f i n i t e

8.

10.

. . . . . . . . . . . . . . . . . . . . . . . . .

! 76 177

integral

on

distributions C~(G) e- -

. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

341 363

Pa~e ]1.

A fundamental estimate

. . . . . . . . . . . . . . . . . . . . . . .

12.

The invariant integral on

13.

Tempered invariant eigendistributions

14.

Asymptotic behaviour of eigenfunctions

15.

The discrete

. . . . . . . . . . . . . . . . . . . . .

435

16.

The space of cusp forms

. . . . . . . . . . . . . . . . . . . . . . .

459

17.

Determination of

. . . . . . . . . . . . . . . . . . . . . . .

478

series for

c(G)

C(G)

G

. . . . . . . . . . . . . . . . . . .

374 386

. . . . . . . . . . . . . . . .

401

. . . . . . . . . . . . . . .

410

18.

Appendix

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

485

19.

Appendix

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

500

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

508

References

Subject index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Index of symbols

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

510 518

PART ONE

INVARIANT ANALYSIS ON A REAL REDUCTIVE LIE ALGEBRA

PART I CONTENTS Pa~e 0.

Summary

i.

Orbit structure

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.

Transfer of distributions

3.

The invariant

4.

Local structure

5-

Local structure

6.

Local structure of invariant

behaviour

of the adjoint r e p r e s e n t a t i o n

of invariant eigendistributions theorem

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . eigendistributions

7.

Tempered The

invariant

9-

Invariant differential

i0.

Appendix

limit formula

eigendistributions

operators

21 34

55

75

on Z:

on a reductive

f ( O ) = E(~(~b)gf,b)(O )

7

on ~:

of the function F around singular points . . . . . . . . . .

8.

distributions

f r o m 9 to ~ . . . . . .

i

on ~:

around regular and semiregular points

the behaviour

Subject

operators

. . . . . . . . . . . . . . . . . . . . . . . .

of invariant eigendistributions

the fundamental

References

and differential

integral on ~

. . . . . . . . . . . . . . .

92

Lie algebra . . . . . .

i01

. . . . . . . . . . . . . . . . .

ll8

that annihilate

all invariant

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

160

index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Index of symbols

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

162 167

0.

Summary

In this part we shall be concerned with questions real reductive Lie algebra 9. G-invariant

elements

Let

G

of the symmetric

aAgebra over

~c ~

Then our main objects of study are the G-invariant transforms

on

~

I

~(u)

(u c I).

the algebra of

the complexification

distributions

especially those that are eigendistributions

erential operators the references

of invariant analysis on a

be its adjoint group and

of

and their Fourier for each of the diff-

The entire theory is due to Harish-Chandra

cited at the end) and is th~ foundation

9.

(cf.

on which the harmonic analysis

of real reductive groups will be erected later on in Part Two. Basic to all the considerations of

G

on

9.

and its corollaries. neighborhoods ~

its centralizer

in

~

and

Z

open neighborhoods of

with fiber

X

U;

in

9

moreover, X.

algebra (CSA) containing

X,

then

to

9;

~,

of

X

in

the neighborhoods

of

This theorem permits

U

X e 9.

If

X

G.

~

G • U= V

and

Z

X

be such a point;

Then~ for sufficiently is a G-invariant

over the orbit

V

G" X

~

V

become trivial,

(CSG) L V:(G/L) ×U.

one to transfer many problems of invariant analysis from

especially local questions

involving

X

is the unique Cartan sub-

is the Cartan subgroup

in this case the fiber bundles

of

form a basis for the family of

is regular and

~= ~

Let

in

which is a fiber-bundle

G-invariant neighborhoods

corresponding

(s.s.) point

its centralizer

small Z-invariant

9

The main results are Theorem 1.20

This theorem gives a detailed description of the invariant open

of an arbitrary semisimple

open neighborhood in

here is the study of the geometry of the action

This is carried out in Section i.

the structure

9

to

of invariant distributions.

This is the so-called method of descent ; it is one of our main tools. In order to be able to use systematically to have a detailed knowledge tions from

~

tributions

~.

of the transfer of distributions

and differential

These details are worked out in Section 2.

In particular, A

(D ~

if

(D))

X

is a s.s. point of

and

~ ( T ~ ~T)

~

and

U, V

are as above~ we

with the following properties:

is the canonical transfer map that carries the vector space of G-invariant tions on

V

injectively into the space of Z-invariant

distributions

D ~ A (D)

is a map that carries the space of G-invariant

operators

D

U;

is called a radial component of

A (D)

invariant C =

on

V

into the space of Z-invariant

functions

f

invariant distribution D

on

V,

~DT =A

D;

and

A~

on

T

(D)o T.

one radial component operator

equa-

For invariant dis-

Theorem 2.3 does this, while Theorem 2.14 handles invariant differential

operators. have maps

to

the method of descent it is necessary

V

on If

A~(D)

and all

V X

D D,

analytic differential

Dflu=A

(D) .flu;

while

operators

on

and for any G-

analytic differentia& ~

for

to any G-invariant

is a homomorphism. S

distribu-

U,

in view of the fact that for all G-

and any G-invariant

corresponding

T

analytic differential

is regular and we write

(D !--A~(D))

on

T~

If

X

~,

operator

there is only

analytic differential is not regular

(but

still s.s), As(D ) however, A (D)

is in general not unique although canonical choices can be made;

is uniquely determined as an endomorphism of the space of Z-invar-

iant distributions on

U.

Theorems 2.15, 2.21 and 2.22 give the determination of

the endomorphisms defined by the

& (~(p))

when

p ¢ i.

In Section 3 we make a detailed study of the invariant integral on for any CSA 9 tion of

f f

on

we examine the map ~

the function

f ~ ~f,9

~f,9

on

of the (suitably normalized) mean values

9'.

Since these orbits are unbounded in

general there are convergence problems which force one to restrict Schwartz space

S(~)

of

~.

namely,

that associates with a continuous func-

9'

on the orbits of the points of

~;

f

to be in the

Here it is useful to keep in mind the analogy with the

classical theory of spherical and hyperbolic means on

~3,

the spherical means being

the mean values over the spheres having the origin as center, while the hyperbolic means refer to the hyperboloids having the origin as center and their principal axes along the coordinate axes. on

~'

whenever

f

It is easy to see that

is compactly supported.

the appendix) reminiscent of defined for all

f e S(~)

further that for any

the

and

~f,9

is well-defined and

Sobolev estimates we prove that

~f,9

C~

Using some estimates (cf. Section i0, @f,9

is well

is an element of the Schwartz space of

u e I, ~ ( u ) f , 9 = ~ ( u g ) ~ f , ~ ~ u 9

~', and

being the '~rojection" of

u

on S(9)(Theorem 3-9)In general, the case when

G

~f,~

does not extend continuously to all of

is the adjoint group

of SL(2,~) .

However,

9;

this is already

~f,~

and its de-

rivatives have only discontinuities of the first kind, i.e., for any differential operator component

P

of

D

~'

on

9

H 0 e 9,

any

with polynomial coefficients, and any connected

in whose closure lies

HO,

the limits

lim (D~f,~)(H) F~H~H 0 exist for all course on

f e S(~).

Now

this limit will depend on the choice of

the nature of this dependence on 3.23, 3.26 and 3.30. of any point S(Ho)

HOe 9

F.

%f~9

has a

C~

extension in the neighborhood

at which no singular imaginary root vanishes;

of singular imaginary roots vanishing at H0

H0

that~ in the case of a semiregular ~'

containing

H0

H0

that if the set

is nonempty~ then~

for all differential operators

skew symmetric with respect to all the Weyl reflexions

of

(and of

The main results are contained in Theorems

They show that

tends continuously around

F~

F

f), and for our applications it is necessary to make a detailed study of

D~f,~

D e Diff(~c)

sB, 8 e S(Ho) ;

ex-

that are

and finally

when there are only two connected components

in their closure, the 'jump'

(D~f,9)+(~O) - (D~f,9)-(~0) is, up to a nonzero multiplying constant, HO

not conjugate to

9

and

D~

(DV~f,~)(HO)

where

~

is the differential operator on

is a CSA through ae

that is the

image of

D

under a canonical isomorphism.

At this stage we begin the study of G-invariant ~(!)T

is a finite dimensional

distributions

T

such that

space - the invariant l-finite distributions.

basic theorem is Theorem 5.28, which asserts that such distributions tegrable functions which are analytic at regular points.

The

are locally in-

The proof of this theorem

is quite long~ and is carried out in Sections 4 and 5; it depends in an indispensable manner on the results of Sections i through ~. this theorem in outline, all of

~,

If

confining our attention to a distribution

X c~

neighborhood

is a regular point and

of

X

in

~;

belong to some positive tion in a neighborhood

~

FT

on

@'

here

system. of

X

7~

is the CSA that contains

such that

~. T=F T

4.17).

FT

on

it can be proved that

(The o r e m

Although

p = ~,

for all

p ~I

FT

being the open dense set of

is locally integrable

T = T'.

is 1-finite.

s.s.

the proof that

X,

~

finiteness

(Lemma 5.5).

~.

is zero around

X

S

S

must be contained

operators

~(~),

and the Euler vector vector field isomorphic

to

is an m-module. ~(~)-finite completing

~](2,C),

simple Lie algebra

is 1-finite,

in the set

element

S

(Section 5, Theorem 5.26).

5

This can be

then

S= 0

dim(B), •

(Theorem 5.27).

one sees at once

of nilpotents by

in

is any invariant dis-

of

~.

Now~

~, The Casimir polynomial~

span a three dimensional

the proof of the main theorem.

~.

We thus obtain the l-

Lie algebra

and the space of invariant distributions

A detailed study of this m-module

s.s.

~ ~I(2,]R),

The last step now consists

~ (= multiplication E,

or

is reduced to the proof of the

By the method of descent coupled with an induction on

the differenti~l

We then prove ( T h e o r e m

More generally we prove that if

tribution with singular support and if

that the support of

~(w)T' -

distribution with singular

(Section 4.5).

T' (Theorem 5.17 and its corollary). T - T ' = 0.

The

reduces this to the case

Applying the method of descent to a

assertion in the three dimensional

of

~.

is zero as soon as it is so in the neighborhood

done by an elementary explicit computation

proving that

on

This comes to proving that

and noting that in this case [~,~]~u(2, C)

~(~)T' - (~(~)T)'

T'

In Section 5 we study the distribution

semiregular point of

semiregular point

corresponding

on

on all of

The next step, and this is the

and a technical argument

support and having a very special structure

of each

Z'

very closely and prove that it is a G-invariant

5.16) that such a distribution

in a that

may become infinite when we approach the singu-

T'

the Casimir element.

(9,~)

coincides with an analytic func-

It therefore defines an invariant distribution

is to show that

~(p)T'= (~(p)T)'

T

@',

main theorem is of course the assertion that most difficult,

X, the method of

is an exponential polynomial

In this way we obtain an invariant analytic

Z.

~,

~OT

In particular,

in

regular points of

(~(~)T)'

defined on

is the product of the roots of

lar points of

when

T

invariant and I-finite.

descent developed in Section 2 shows that

function

We shall now describe the proof of

m

with supports c

shows that it cannot have any

Thus

S=O.

This proves

T = T',

Once we have established that

T

by the locally integrable function

coincides with the distribution defined on

FT,

the question arises as to how

in the neighborhood of the singular points of ~

• (FTI~,)

and

Y~= ~('~)~,

where

itive system of roots of (Zc,~c)~ algebra of ~

and

~c'

H~

the relations between

on

if

~

is the element

~'(R)

@~

~'(R),

in

@~ =~F and if

on

F

~.

~

~c"

as well as

(~,~)

vanishes,

there is an exponential polynomial

in particular,

@~

extends to an analytic

is of compact type, i.e., all roots of

imaginary, then there is a sir~le exponential polynomial on

~

We find (Theorems 6.3, 6.5) that

~'(R)

F O ~';

~= b

in a pos-

In Section 6 we investigate

where no real root of

of

}~ =

in the positive system that defines

~

for different

is the subset of

such that

function on

~

~

behaves

of the symmetric

in the neighborhood of an arbitrary point of

them, for each connected component ~F

is the product of the roots Z~H

FT

we write

~

being the canonical image of }~

For any CSA ~

~f~

the product being over all

the behaviour of

(i)

mud

~.

~

on

~

(~,b)

such that

are

~=

b' (ii)

if

H0 c ~

is such that the set

RHo

nonempty, then for any differential operator with respect to all the Weyl reflexions to a neighborhood of tion (also denoted by

H0;

on a&l of

D e Diff(~c)

s 8 (~ c RH0 ) .

in particular,

Y~)

of real roots vanishing at

~ = ~(Zg)@~

~,

if

~i,~2

which is skew symmetric D¢~

extends continuously

extends to a continuous func-

and this continuous function does not de-

pend on the choice of the positive system defining (iii)

are any 2 CSA's, ~ I - Y ~ 2 -

~. 91 N ~2"

on

-

Note that these properties can be verified as soon as one knows given an invariant distribution are necessary for tribution on

~'

to ensure that

T

T

on

~

to be 1-finite on

H 0 is

FT .

that is l-finite on ~.

We prove that if

~', T

If we are these conditions

is an eigen dis-

with regular eigenvaAues, then these conditions are also sufficient T

is an eigendistribution on

~

(Theorem 6.9).

In Section 7 we combine the results of Sections 4-6 with the theory of Fourier transforms to study the behaviour of invariant eigendistributions on also tempered.

~

which are

For regular eigenvalues these turn out to be linear combinations of

the Fourier transforms of the invariant measures that live on the regular orbits of (these measures are tempered in view of the results of Section 3).

A special case,

of very great importance for the construction of the discrete series of representations of

G, arises when

~

has a

CSA b

of compact type and the eigenvalues are

defined by the orbits of regular points of distribution

T

b'

We prove that a tempered invariant

corresponding to such an eigenvalue is completely determined by its

restriction to the regular elliptic set

(= (b')G),

and that on

a linear combination of exponentials (TheoremS 7.13 and 7.15).

b', 7b" (FTIb,) is Apart f~om the

theorem that invariant 1-finite distributions are locally integrable functions,

these are among the most important results in this Part. and uniqueness

The fundamental

existence

theorems of the discrete series ultimately depend on these theorems.

In describing

the results of Sections 4- 7 we have assumed that our distributions

are defined on all of

~.

Actually it is necessary to work with distributions

fined only on open subsets of

~,

de-

and this is what is done in the appropriate places.

In Section 8 we obtain a formula which gives an expression for the Dirac measure on

~

located at

admit a

CSA

0

in terms of the invariant

of compact type, say

maximal subalgebra of constant

e > O

~

b,

integral on

and let

of compact type (q

q=½

~.

More precisely~

dim(~/~)

is an integer).

where

~

let

is a

Then there exists a

such that f(O) = (-1)qc (~(~b)¢f,b)(O)

for all

f e S(~);

we have to remember here that as

respect to all the Weyl reflexions~ all of

B(~b)¢f,b

G;

it then constitutes G

(cf. Harish-Chandra

~

this lifting from ~

exp(U) = V

g

to

the map tion

exp.

G,

E

on

V

by transporting

v2

via t

t

exp.

on

operators

~(z)

and so the theory developed there is a canonical for all invariant

C

on

G,

~

~

v

be the positive invariant

t

on

V

exp;

determinant

functions

(Jacobian)

of

we associate the distribu-

invariant differential

oper-

Then one may hope to study the properties which are invariant eigendistributions ~.

However,

as

G

of

for

is noncommutative~

the

in general do not have constant coefficients

2(z - ~(z)) f

U

exp : U "

to any analytic invariant diff-

so far cannot be applied to it.

isomorphism

Let

and let

via

on which

G.

in

we associate the analytic

E

characters

operators on

0

is the functional

by looking at the distributions

differential

differential of

To any invariant distribution

the irreducible on

such that

obtained by transporting

erential operator E

U

Section 9 is de-

In order to explain what is done we require some nota-

is an analytic diffeomorphism,

~

ator

G.

small invariant neighborhood

analytic function on

map.

and proof of a crucial theorem which allows us to accomplish

be the algebra of biinvariant

be a sufficiently

[ 4 ] [12]~

is that one hopes to carry

most of these results over to G by means of the exponential voted to the formulation

Let

map, this

one of the princ-

[ i]).

The main reason for doing all of this work on

tion.

function on

Using the exponential

ipal steps in proving the Plancherel formula for Gel'fand and Graev

is skew symmetric with

extends to a continuous

b, by virtue of the results of Section 3.

relation can be carried over to the group

B(fgb)

on

of

~

onto

But it turns out that I

such that one has,

U~

~(z'-~)f = (v -I o ~(~(z)) o v)f (this formula is essentially formal and is substantially character formula). on

V,

tO = v - ~ ,

In other words, then

equivalent to the Wey!

if we write, for any invariant distribution

t

(~(z)t) ° : ~(~(~))t ° at least when

t

is a

distributions

t~

C~

function.

(~ ~ 8)

If one could establish this for all invariant

then it follows that

t

is ~-finite if and only if

finite~ so that we can apply all of our theory on

~

to the study of

result of Section 9 allows us to do this (Theorem 9.23).

to t.

is lThe main

It asserts that if

D

is

an analytic invaris~t differential operator on a completely invariant open subset of

~

variant

(in particular one can take f

in

C~(~),

then

DT= 0

~= U

above) such that

Df= 0

for all in-

for all invariant distributions

T

on

~.

This theorem is however difficult to prove and most of Section 9 is devoted to obtaining the estimates necessary for proving it. and the method of descent. down to the case when that

T

The proof is by induction on

dim(~

By arguments similar to the ones used earlier one comes

supp(DT)

is contained in the nilpotent set

~.

If we know

is te~pered~ then one can use Fourier transforms and combine it with the

fundamental theorem of Section 5 to conclude that

D T = 0.

However~

in general

T

is not tempered and so a method has to be devised by means of which one can reduce the proof to the tempered case. variant distribution~

Theorem 9.12 does this.

It shows that every in-

defined on an invariant neighborhood of the origin in

9, is

necessarily tempered on some (possibly) smaller invariant neighborhood of the origin in

9. Finally,

in Section i0 which is an appendix, we collect together for convenience

a few results that are used in various places of the exposition.

i.

i.

Orbit structure of the ad~oint representation

Preliminaries

~]uroughout what follows its eomplexification. ~c), ~.

and for S(~c)

G

XeGc,

~

is a

real reductive algebra

(resp. Gc)

X e ~c'

we write

Ad(x)(X) = X x = x -X.

is the symmetric algebra over

nomials on

~c;

Is(~c) = IS

and

~c(~ ~)

is the (connected) adjoint group of

~e

and

P(~c)

(resp. Ip(~c ) = Ip)

¢

is

~ (resp.

is the center of

is the algebra of poly-

is the subalgebra of

S(~c)

(resp. P(~c) ) of all elements invariant under G c. Let m = dimc(~e)• ~= rk(~e) • rk denoting rank. We use without comment the standard terminology of semisimple Lie algebre~ and Lie groups. For any indeterminate The

qie ~ ,

qm=l,

qs=0

T

and X 6 ~e' let det(T • i - ad X) = Z0 0

and

Let

m ~9, Xe~.

s.s. points of

and Then

Let

~

~,

If

then

~

Then

e2tXeG "XVt and

X s e~,

ad X

[X~,X~] = 0;

so

X s e CI(G .X)

then

cad X

g •Xe ~

Orbits in

Let

N

p(0) = 0.

Let

X

is

(resp.

9.

X e ~

c { i;

and

X = X' + X' s n adX' n

p(X)=P(Xs) V

in this case, ~

X

X

in

s ~.

in

9.

Then

Xe ~

cXeG

"X~

containing all

and so

By Lemma 2 ] H,Y e 8

conversely, if

cXeG

.X

as

X=X

+X is s n {H,Xn,Y] is

such that

t --~

for some

If

X

c # i,

is nilthen

have the same eigenvalues which must all be zero. geG~

so

If

X e ~.

Zc" be the set of nilpotents of

9~c

~c

9

(resp. 9c ).

is the set of common zeros of

~,

the set of all

p e

is Gc-stable and splits into finitely many orbits.

The first assertion is standard.

The second one is due to Kostant [i].

shall need it in the real case also and prove it in Theorem 15. 9c

X=X s +X n is s.s.

(CI = closure), and

for some

is a

ad X' is s.s., s X's = Xs' X'n = Xn.

is a G-invariant open subset of

for some

(resp. Nc)

Proposition >.

and

Then

(exp tH) • X = X s + e 2 t X n - X s

e~;

and

2.

with

~.

be the centralizer of

a standard triple.

X e~

relative to

~ c 9

9 = 9.

the Jordan decomposition of

potent,

X

is nilpotent ~ c X e G ' X 0 e CI(xG).

Xn e ~

and

in particular,

if and only if it is so relative to

be the Jordan decomposition of

If

We

As the result for

follows by applying Theorem 15 to the real Lie algebra underlying

9c,

we omit

the proof at this time. Let

PI'"" "'P£

Ip = C[Pl,...,p Z]

(3)

be algebraically independent homogeneous polynomials such that (Varadarajan [i], p. 335).

~(x) : (pl(x), ....p~(x)),

~ : ~-z( = ~ - l ( ~ ( B ) ) . in

(cf. Varadarajan

Then

~. s

Since is in the

lies in the Weyl group of

utxl~ = identity.

But then

utxeexp(~)mZc,

i.e.,

x~ Z . e COROLLARY ii. centralizer y •Y = Y

3.

of

e

X

in

Z.

be s.s.;

Let

Invariant

open sets in [c"

nQ

and

X e ~c

is the centralizer

Z~

the centralizer

~ = center(~).

Ye~

(4)

Then

X

X

in

det(Ad(y) Is) : i

G; ~, V

Ye Z

the and

y e Z.

The sets

Uw,V w.

is a fixed s.s. element; of

of

in

G . c

~c = centralizer

of

X

in

~c;

Define

"8c = [Y: Y~ 8c' det(adY)[c/~ c }/ O]

(here

(ad Y)~c/~c

Xe'~c, that

Xc [

for all

In this Z

Let

while

"~c

is the endomorphism of

is Ze-stable , dense and open in

Y e " ~c Ys ~" de"

With the usual

9c/~ c

Let

identificstions

@

be the map of 12

the

induced by

ad Y).

~c"

If

Y

(y3Y) ~ Y

of

Gc × " ~c

tangent

Then

e ~c' it is clear

spaces

into to

Ge,

~e"

ii

~c

and

~c'

we find t h a t V y £ G c ~

(X'• ~c, Y'• ~e ). (d~)(y,y) of

Let

As

U ~X

is open in m

any subset (5)

(ady) ~c/~c

is surjeetive.

Sc' Gc "U

X•~c.

As

is surjective,

Thus

~

~c"

Let

(resp. ~X)

~c_~ c

Y•'~c' (d~)(y,y)(X',Y')= [X',Y]Y+Y 'y

~c

be a CSA of ~c;

be the Weyl group of

U

it is a CSA of ~c and

(~c,~c)

(resp. (~c,~c)).

For

let Y

lies in

Zc

"~],

v

=~

c

"u.

is the centralizer of X in m we can find an open subset ~0 of ~c con-

raining

X

such that

(~)

~Sn%:~/ (~ro\~ o

x~%ff'~e'%:% (~%)'

It is clear from these definitions that if Y e U V

Y e Vco' Ys

also.

assume~ since Then

so

is submersive and so, for any open subset

{Y: Ye ~c' s.s. component of

=

~c+ramge(adY)=~c ;

We also observe that Ye V ~ M y C \ ~ .

M y = M Y s , that

x • Y s' = Y

for some

Y

x• G c . Let

~ consequently, if

To prove this we may

is s.s. and further, that

centralizer ~y of Y in ~c is c ~c" so Y " e U

, then Y s e U

x) "

YeU

N : x • Y'n' Y" = Y + N .

. Let

Since

So Ne ~c' giving Y"• ~c"

Ye

Y' eMy. ~e ~ the

But then Y: Y"s and

~ i.e.; Y' e V . Finally~ since ~ 0 C ' ~ c , U~0C'~c.

L~4MA 12. of M onto N.

Let M,N be analytic manifolds~

7. : M ~ N

a submersive analytic map

Then~ given any compact F~_N, ~ a compact E c M with

7F(E)= F.

This is clear since submersive maps are coordinate projections in local eoordinate s. Proposition i~. ~e"

For any w open in w0, U w is open in " ~c' while V

U W is invariant under Z

is open in

and contains~ along with any element, its s.s. com-

c If h is the class of all open subsets of c00, we have the following, valid

ponent.

kv'W,Wl,~ 2 • h: (7)

Y•ac, U

YY

in

gc CSA's

~Uw/~. are

y e Z e ; C l ( % ) S ~ l ~ C l ( V ~ 2 ) f f V ~ l"

Y

UwC-" ~c' V~

Then ~ s.s. Y w i t h

c ~c;

of

~c'

ZYne ~yyA Be c " ~e assume

/0~

is open by Lemma 9; since

~eh, y-U

also

(y.U)nU

moreover, showing that

and

Se~x, i.e., y - X = X .

~.

rk(~y) = rk(~c) ~y~c/~.

Then If

3

xl,x 2• Zc

So

with

z=x2YXl, then ~Z:~c,

Lemma i0 ~ y e

Bc"

Let

y e Gc,

. The centralizers By, ~yy

so that Zn,Z • U

to be regular.

both regular and in

is open in

Y,YYeu

so that ~ if

CSA's

of

~y

Zne ~y~] De' tending to n

is large.

H=Xl I •Y and

and

of Y, are Y;

We may thus x2Yx I 'H

s=Ad(z)l~ce~;

so by (6)

Z . For the second result in (7) let ~i'~2 6 h C

with

Cl(m2)C-Wl' Y n e U w 2 ' Y n e G c ' Y n ' Y n ~ Y c ~c"

13

We may assume

Yn

regular V

12

n.

So ~ H n { ~ 2 , tn ~ Zc with Y : t • H . Since P(Hn) ~ p(Y) for all n n n we may assume in view of Lemma 6 that lim H : H exists. Then H ~ ~ n- ~ n i YCMH, so that Y ~ V I. COROLLARY 14. sequences in Ixn • X]

U

If

m c~

and

Gc

is bounded in

We may assume F ~ V

x

"X

B ~ Gc for all

n,

4.

F ~ B "A. x

n

•X

~ Y

as

n ~ ~.

Y~V

Xn "Yn So

for all

n.

By Lemma 12, ~ compact sets

Xn "Xn = Yn "Yn' Yn {A' Yn [B"

~.

= 01 U ... U 0 s

Oi U .. • U 0 s

We may assume

H, X, Y

where the

0i

~

of

~,

~c

Since

~,/G will

and

(ii)

~i ..... ~£

~

of

g

has only finitely many mutually

H,

~(H) = 2.

Also

But if

X~Z

(gc,~c)

(gc,~c)

so that

the set of possible H.

For (i), let

cQ g~

where

H = ~ < i < ~ aiHi Q

~0

be

~

..... HZ} ,

where the

ai

is the set of positive roots

a i = ~i(H) > 0, representation theory of

~I (2,C)

the are all

~

with

shows that

[X,g_~. ] ~ 0 , ~ 3 ~ = ml~ I + ... + mz~ £ c Q with m i > 0 and so, as 2 = (~(H) = l mjaj~ we find a i = 1 or 2. For (ii), let gp be the set of all Y ~ g with [H~Y] = p Y Let

'g2

(p c Z~),

sentation of

GO,

'g2

~(H)

if

such that

is thus nonempty.

[ g 0 , B ] c ~p ~v'p

r,

Furthermore,

'g2

G

corresponding to

[ X " g 0 ] = g2"

in a vector space

for the eigenvalue

by a theorem of Whitney [i], other hand,

the analytic subgroup of

X ' ~ g2

C "H + C .X + C • Y

eigenveetors of X ~ 'g2"

and

be the set of all

W

and

~(X)(W0) = W 2. g2\'~2

If W

r

~

GO

X' ~ 'g2~ the differential of the map

leaves

is the subspaee of This implies that

is an algebraic set.

~2'

x ~ x .X'

14

as well as of

GO

gO"

is any repre-

has finitely many connected components.

and so

H

X

take real values; by Lemma 3, ÷ 3 a chamber ~0 with

be the corresponding simple roots;

dual basis of ~0~g~, the root spaces. Then _> 0.

More-

follow if one proves: (i) with X ~

for fixed

where all roots of

~(H)~2Z xv/ roots Let

G.

are disjoint orbits and for

as in Lemma 2, there are only finitely many possibilities for

the set of points of

integers

xn{ynZc ;

B • X.

splits into finitely many orbits under the stabilizer of

HcCI(~O).

, 0 so,

We begin by proving Kostant's theorem [i].

to be semisimple.

in a preassigned CSA

and

By (7)

A ~U

is a closed set containing 0 i as an open subsel; 0 s = [0].

nonconjugate CSA 's, the finiteness of

H c ~0

Zc ~

by (7) and so ~ a compact

splits into finitely many orbits under the action of

15.

fixed and

are

~.

over, we can write i < i < s~

[Xn] , Ixn}

is bounded in

~0

is the set of nilpotents of

~0~M

Then

lies in the compact set

Orbits in

and

[x • X ] n n

n

containing

0 with

CI(~) ~ ~0'

and

~c'

n set

is such that

respectively, such that

p c~,

into

Hence On the

'g2' invariant; '~2

is the map

13

Y m [Y;X'] mersion,

of

~0

into

showing that

each connected

~2

which is surjective.

G O • X'

component of

is open in '92

only finitely many Go-orbits

in

'~2 v X ' c

this case

Let

X e g.

(G c • X) n ~

closed G-orbit;

and

'Z2' proving

Then

X

in

g.

Let

We assert that xeG

, yeG, c (G c .X) N ~.

cisely

~e~

(G c "X) N B

can be closed only when Let X 0 e B

xZ ~ x

be

V

s.s.

and

~

Then

(G c -X) N ~,

.X.

So

G.X

and let

Z

B,

is finite for any CSA

of

n 8, v = G • U.

We claim that

M x N ~ which is closed in G. (X + n ) c M x n V . X

Suppose

and so x .Ys = x ~ .X

i.e.~ X = Y s .

5.

of

G/Z

onto

G -~

is locally compact

and

polynomials

on

Let

I

Ul~ ...,u~ !c"

Let

~

and E = G "

YeX

where

+n~

(ii)

splits

(X + u).

E

is

of

G-

~ e h,

xeG,

YeU.

showing that

=X

"Ys e G c

so that x " X

= x "Ys'

x • YeE.

~.

19

of rank

~

with adjoint

of the algebra of all invariant

Vl,...,v p (p = dim I) 15

So U = Um

Certainly

Then ( x - Y ) s = X

By (7), x -Ix' .X

generators

write

this will imply that E is open in

be a s.s. Lie algebra over

Further let

E

such that

be the centralizer

We select

of a s.s. point of

homogeneous

E c ~

splits into finitely many G-orbits.

G • (X + n ) = M X A V ;

X

of

G " X O.

and hence that E is locally compact.

Invariaat neighborhoods

L

G "~.

is locally compact.

x-YeM

Then

Using the same category argument as in

8, X ~ ( ~ ) s ,

for some x ' e G c .

In other words~

LEMMA 18. group

~

E

That

is a regularly imbedded analytic submanifold

while Theorem 15 ~ E

it remains only to show that

~ c ~.

X 0 in G.

is G-stable and locally compact in the relative topology

of

is open in

we conclude that

showing that they are pre-

be the centralizer

into finitely many G-orbits one of which is

stable and G • X 0 c E

~.

X'eU,

is s.s. is immediate from Lemma 4.

in its relative topology.

(XO)s, n the set of nilpotents

is a

is open in

for some

the discussion of the complex case it is enough to construct a set E

In

and that they are all closed also.

As in Proposition 8 this reduces to proving that

(i)

is s.s.

the centralizer

V

x' e (G c .X) N B,

.X 0 is an analytic diffeomorphism

second countable

X

each component

Y = x .X = y . X ' e V

(G c "X) N ~,

(G c • X) n ~ X

X

= x .X = y - X e G

is open in of

by Lemma 4.

X.

n 8, v = G • U.

For, if

-X = X~Y

the connected components

and

Let

U = U

Since this argument can be used

G . X 0 is open in its closure in B,

{0]

is closed if and only if

n°3.

and define

then (7) ~ y - l x

THEOREM 17.

Using the Baire category theorem

is the component containing

They are finite in number since G'X

(ii).

This implies at once that and hence that there are

has finitely many connected components;

G •X

V n (G c .X) = G .X.

each G-orbit in

GO

The closed orbit is

G" X

We use notation and results of of

'92"

is an orbit under

we complete the proof as in Proposition 7. THEOREM 16.

Hence the previous map is a sub-

be the polynomials

such

14

that ~ e t ( ~ - ~ - a d X ) = @ - v l ( X ) @ I(t) = [Y : Y e l~ I~I < t {Y:Yel,

lvj(Y)[

i 0 with U c U . By (7) we have: y ~ G , y • U N 7,T -- ~0 ;T The assertion (i) follows at once from this. To prove (ii) note

Z . C = Ul
_dim(~); X'

N.X.

are as in Theorem 1.20 and the intersection above is taken over open

neighborhoods ~ c

onto

is an analytic diffeomorphism of

G • (X + n) = ~

where

the

N

which maps ~y.X~ ×__\,~ onto - l ( ~ y . X )

(i0)

set

is an open neighborhood of

(y e N, Y e U7,~)

the G-invariant open neighborhoods of

is closed in

is a regularly imbedded analytic submani-

is an analytic diffeomorphism of

(~.x) ×_~,~ onto - i ( ~ . X ) each

N

X'~ '~

such that for any compact set

5

be a compact subset of

~, c U 7,~, '~

can serve instead of

we can find a neighborhood

5' c 7', G • (5' + n)

19

'~

and is Z-invariant. X 7'

is closed in

So

in the of ~.

X'

18

Applying this to each point of a compact set is closed in

~.

This proves (ii).

We take up next the assertion (i). in (i) and hence

G ° (X + m)

uppose

z~G

Y' "YI' = Y" for all

Y = y " Yll

As

y

YI : y

~' , T',

'~.

Clearly

all

as

Y[

" Y i; y

V , y

Suppose

,,

G • (¢ + n)

is closed in

ove.

for all

~

As

~.

~',T'

Y~sUT,

3

T,

and

as

and

y'~G,

with

e z an~ so YI

~T'

It remains to prove (iii) . X'{CI(G"

Yn { G

such that

n ~.

As Yn is the s.s. component of Y n + % ,

as n ~ .

we conclude that G " (X + n)

X + n c U7',T'

as a

i.e., YI ~ X + ~ .

relatively closed in

'~

is contained in the intersection on the right side of

for

such that

~ c

In particular,

Yn" (Yn + Nn) ~ X'.

({ + ~)).

Then

Let ~

{ m

'~

be

Yn ~ {' Nn{ ~ ' Y n S G

Then for any p s I p ( ~ e ) ; P ( Y n + N n ) - p ( X ' ) p(Yn)=p(%+Nn),

as

showing p ( Y n ) - P ( X ' )

As we can find a CSA of ~ (hence of ~) eontsining &, we conclude from Lemma

1.6 that the Y

remain within a compact subset of ~. We may assume that Y ~ Y as n n n - ~ for some Y ~ . As p ( Y ) = p ( X ' ) for all p ~ Ip(~e)~ Y and X's are conjugate under

G c.

In particular, dim(~x~ ) = dim(~y) ~dim(~),

dim(~x~ )= dim(~), then ~y= ~ so that Y e '~.

as ~ y m ~.

If we now assume that

But now there is a compact set 6 ~

taining Y and all the Yn' and so X ' e CI(G • (8 + m ) C G ( ~ + n), by (ii). X' = X ' " ( Y ' + N ' )

( Y ' e { , N ' e ,,x'eG),

then X $ = x ' "

con-

Of course if

Y' and s o d i m ( ~ X ~ ) = d i m ( ~ y , )

=

dim(~). COROLLARY 2~.

Let m(~) be as in Lemma 19.

If 0 < T I ~ T and 71 is any open

neighborhood of 0 in 7, ~ h , ~ l O a = U s s m ( a ) s "~i' the union being disjoint. Let YeUTI,TI, so that ~ = y • ~.

y£G

UTI,T I n 'a=71.

for

some

~,

so that

and y • Y c a .

So y ' ~ = ~ ,

As Y s '~ and y • Y e a ,

and y . Y s ~ N

'~= 'a.

we have ~ C ~ . y = y

Thus we are through but for the disjointness.

s ~(~),

and

y "X=X.

s=Ad(y)lm Thus

for

s -X=X,

• ~,

If s = A d ( y ) l a , then Y = s-l(y • Y)

some

ycN(~),

and as

m(~)

we h a v e

If s . ~ i ~ i

~

y .U~I,TlOUT1,T1 {

acts freely on

~,

s

must be

is the CSA containing

X,

Z

acts

the identity.

COROLLARY 24. trivially on (G/~X %,T

is closed in LEMMA 25. that X

~,

onto

Suppose and the map

VT, ~.

For, let

Then

~

(yZ~Y) ~ y • Y

is an analytic diffeomorphism of

Moreover, for any compact subset

~

of

~' = ~ n ~,, G .

~. Let

L c ~

G • (Y + n) n L = ~ V

is regular,

X c ~'

~= ~

and

be a compact set. Yc~\M, m=0,

~

so that

q0 be as in (3), and

M=~

Then

3

a compact set

M c ~

being the set of nilpotents of G- (~\M)

does not meet

L.

0 qt-l((p(L)), and use Lemma 6.

20

~.

such If

19

6.

Localization.

LEMMA 26. 0 c ~. I

Then

(ii)

Let

~

I

be a s.s. Lie algebra over

f e C~(1)

0 < f < i and

Let

p = dim ]~

such that (i) f = i

and

Let

I~

let

Let

g eC~(~P)

f(x) = g(vl(x ) ..... ~ ( x ) ) THEOREM 27 . Then X

~

f

(iii) If

Let

in

I

7,T

X e~

be s.s.,

C=(~)

with

(i)

with

0

let

f

~

s > 0

such that

~(s) ~ ~.

I~ : {(X 1 .... ~ p ) : X ie ~, IXi[ < a, on

I~/2~supp g c_ IP~5~/~/I.; take

an invarismt open neighborhood of

is G-invariant

Choose gl = i

g2 e C ($8)

$8

and

geC~(~)

(ii)

X.

0 < f < i~ f = i

with

7',T'

Let

(i)

0

C = supp g.

outside

Following Harish-Chandra, invariant if

Cl(~.~) ~

(i)

around

(ii)

Ye~,

g2

O "C f

C

c_ ~ n % ~ T . X

in

g2 = 1

Let

~,

and

in an open neighbor-

(Yco,

Z¢$~).

(i)

f

(Lemma

is G-invariant

is closed in

is closed in

Z

From (iv) of (ii)

~, Z-invariant~

~.

Extend

f

to

and ~

by

has the required properties.

we shall call an open subset

is invariant under

(i)

G

(ii)

if

K

Q

of

~

com~letely

is a compact set

The following statements on an invariant open set ~

is completely invariant •

(ii)

3 f eC~(g) such that (a) f is G-invariant

open neighborhood of (i) ~ ( i i )

Y

(e)

(c)

~ ~,

f ~ 1

Then

V

(ii) ~ (iii)

is open.

Y e Q,

(b)

then

~ c 9

are

Ys ~ .

0 < f < 1, f = 1

3

f e C~(~)

by Theorem 27.

with

in an open neighborhood of

COROLLARY 29 . Let U = U 7~ T completely invariant open set. U c '8~

If

(iii) in em

supp f ~ Q.

by Lemma 4, and

be a compact set.

supp f c n

X ~ eenter(~).

is invariant under

such that Then

So let

~.

THEOREM 28. equivaient:

~

G • C;

%',T'

0 J g2 ~ i,

g(Y + Z) = gl(Y)g2(Z)

So~ by (ii) of Theorem 20,

defining it to be

with

in a neighborhood of

supp g2 ~ $~(T') by

= g I ,U ,. _T

c_ U 7'~T,.

As

The

the result is immediate from Lemma 26.

0 j gl < i ,

in

Define

K ~ ~

3

0 _< g £ i~ g = i

Theorem 20 it follows that ~ f s C~(V ,,T,)~

If

supp f ~ ~.

(x ~ I).

be as in Theorem 20.

supp gl ~ ~';

f l ~u , _ _

(iii)

v. are real on I, inJ Q(s) = [X :X £ I, [Vs(X) l < s for

and if

given by

with

0

supp f ~ Q.

gl ¢ C~(°)

26).

be as in Lemma 18.

Let

X e center(~);

hood of

an invariant open set~ and

is invariant under all automorphisms of

then Lemma 18 shows that

be the open cube in ~P

l < i < p].

~

in an open neighborhood of

v. J variant under all automorphisms of i < s < p] (a > 0),

f

IR,

be as in (9).

Moreover

V 21

(a)

f

K

in

If

satisfies

Assume (iii) and let

is G-invariant g.

U c '~,

Then then

(b)

C =

CI(G • K ) ~ C ~ . V=G'U

(ii) of Theorem 28.

is a

20

COROLLARY ~0.

Let

invariant subset of

~

~ c ~

be a completely invariant open set and

containing all

Let

Xc~.

Then

for some

g~G.

But then

Xs c C I ( G ' X ) X~I.

s.s.

points of

by Lemma 4. So

So

~.

Xs~

Then

~

, ~XsC~l

~i

an open

= ~. ~g

.Xc~ I

~i = g~"

There are many papers dealing with slices and their properties for transformation groups; see for instance R. S. Palais, Annals of Mathematics 73 (1961) 295-323.

22

2.

Transfer of distributions and differential operators from

The main technique of invariant analysis on

Z

~

to

~.

is the method of descent.

In

this section we shall examine the question of transferring distributions and differential equations from

~

to

~.

i.

Transfer of distributions.

Let

M

C~

be a

manifold .

The map

For any

T ~ a T.

r

with

the vector space of complex valued functions on < r

and have compact supports;

we write

M

Cc(M )

we denote by C ~r)( M)

0 < r < ~,

which have derivatives of order instead of

c(O)(M).

--

These

C

spaces are topologized in the usual manner (cf. Schwartz [i ]). M

is a continuous linear function

T : C~(M) ~ £; it is of order

(necessarily uniquely) to a continuous linearC- function dim(M) = m

and

~

is an m-form on

positive Borel measure

~

A distribution on

M

in a natural manner; given F

on

M

if it extends

T' :c~r)(M)- ~ £.

that vanishes nowhere,

identify any locally integrable function

r

~,

~

If

gives rise to a

it is customary to

with the distribution

co

TF : ~

fM F ~ d ~

(~ e Cc(M ));

Furthermore, associated of the algebra of SM (D ~)d~ and if

D

with

w,

]

D

DT F = TDF.

M

D.

respectively,

M, N

(m > n); ¢ : M - N

N; w M

(resp. raN) an analytic m-form (resp. n-form) on

positive Betel measure on If

V, W

3

basis

onto with

unique

(resp. N).

Let

(resp. N)

defined by

~M

W, U = ker L,

V

with

%(M),

~x e A P ( % ( S x ) * ) = ~M/~0N : x ~ w x tion of

~

to

W

is

defined by

%(Y) :

~.

(resp. on N). ~M

(resp. ~N )

m,n

respectively,

T¢(x)(N )

written as spanning

and

WV/~W,

-l(y)

We assume

L,

a Am(v * )

such that for any

U~ ~0(Ul,...,Up) =

L = (d~)x.

M

@-l(y).

~d%

and for each

23

By.

We define

(~

6

V

is the

We then have

(Sx = @-l[@(x)}).

oJx = OJM,x/OJN,~(x)

on

onto

be the

an element of

We apply this remark to the case where

~y

M

(resp. WN).

(resp. WW)

is a nowhere zero p-form, say

tive Borel measure defined by

(J-)

~V

i s an analytic p-form on ~- (y)

is a distribution

then, it follows from a simple calculation

Ul,...,u p

~v(ul~...,Um)/~w(LUp+l,...,LUm). tangent space

and

0AV ~,, O, ~oW ~ 0,

of

M

M

~ 6 AP(u~), (p = m - n ) ,

Ul,...,u m

T

an analytic submersion of

are real vector spaces of dimensions V

(resp. An(w*)) that

M

D ~ D

are analytic manifolds of dimen-

m, n

(resp. w N # O) everywhere on

T = TF.

is also a distribution, and

Of course, all these are relative to

We now consider the following set up:

linear map of

If

DT : ~ ~ T(D*~)

sions

~M # 0

instead of

such that SM ~(DB)d~ =

is the adjoint of

differential operator,

F e C=(M),

T = F

a unique involutive antiautomorphism

differential operators on

(~,$ e Cc(M)) ; C~

a

C~

thus we often write

Cc(M))

Then

y e N, Let

by

the restricbe the posi-

22 LEMMA i.

For

c Cc(M), f

(~)

4

is the unique element of

=~

~(F o ¢)d~M

f Fd~]i~

In this case~ for any locally integrable function tegrable on

M,

and (2) is true for F.

Cc(N )

such that

~ / F ~ Cc(N)).

F

on

N,

F °~

is locally in-

Moreover, supp(f )_C~(supp(~)), and ~ f

is a nonnegativity preserving continuous linear surjection of c(r)(M) onto C r)(N) (0 0; QkA(D)

D.

If

For any

is a~ analytic differD

has polynomial

has polynomial coefficients.

be the map defined in Theorem ] t~{ing A-invari~nt distributions on

into distributions on

U".

Then

h(DiD2)¢ T = /i(D1)A(D2)~ T.

A"~ then

has polyno~ri~l

is a polynomial such that Q ( Z ) { 0

(Z c X + Uc).

be an open subset of

eoeffieients~ then for some integer T ~ aT

D

We thus have

analytic A-invariant differential operator ential operator on

Q

If

5r(-) o /h(D)

with polynomial coefficients.

Q(Z) ~ 0 ~ 5 r ( Z ) ~ 0

divides a power of

moreover;

which is a radial compon-

The relations (4) and (5) are ~lso valid now.

Z ~ U'c then

on

such that

such tha~

is an analytic differential operator on

can be carried out in

a"

By Lemma 6,

Sz(~ z) - i ~ A(D) z ~ ~+(v) ® s(u),

ential operator on

Let

Y.

~(V) @ S(U)

coefficients it follows from our discussion above that

r

~,

giving a Lebesgue measure

In view of our general discussion on radial components, if we define

as the unique element of

ent of

5

is nonzero on U'.

invariant, as we have assumed it to be unimodular.

(7) then

; 8r

be any analytic A-invariant differential operator on ~'= A • U'. For each

D' : Z ~ Bz(Dz) DZ

to be any n-form (n = dim(,~))

leaves

Y c a we write

A(D)z

Zd+e< r ~V)d®S(U)e

is then obviouS.

is similarly chosen as a p-form (p : dim(U))

A

Let

wQ

BZ

and

Let [~r~Z)~e the polynomial which is the

FIU"

COROLLARY 8. U = ~, V = [X~ a].

F

is locally integrable on Suppose

~

~DT : A(D)aT If

and

supp(aT) c_ supp(T) n U";

is A-invarian% and locally integrable U"~

X ~ a is such that

is its own normalizer in ~.

and

ad X

FIU" = a F.

is semisimple.

If A 0

Then we can take

is tile normalizer in A of ~

then A 0 contains the centralizer of X as an open subgroup; U and V are stable under 28

27

A0, ~ q

is a subalgebra, and

is a polynomial on

stable subset

'~

~c

of

~

as the distributions

[~,V] c V .

invariant under where

~

Let

q

q(Z) : det(adZlVc) AO~ ~ = X + U,

does not vanish.

(Z • ~c ).

and

U'

is the A 0-

The operators

are then A0-invariant ; moreover~ if

D

Then

A(D)

as well

has polynomial eo-

efficients~ ~ an integer k_>0 such that ~kA(D) has polynomial coefficients also. That ~ is a subalgebra, [ ~ , V ] c V and that ~ is its own normalizer in a are obvious.

We have

(DZ)a,

this gives

A0,

fa.z(ta)=Fz(t) a for t • ~ ( V ) ® S ( U ) , D'a.Z= (D{)a ~

(7) ~A(D)a.z=(A(D)z )a

for

a6A0,

Z• '~.

a•A0, Z• '~.

The case when

s : ~

A = G,

We now apply the results of X e @

is s.s.

real on u ~ [

in

~c )

S(~c)

to the case when

induced by

(resp. G).

(., .)

onto

t (1 +

I l x l l 2 ) r ~ ~H ( x ) = o

in He ~'

unifor~y

These relations lead to the first two conclusions of Theorem 9.

It is now

obvious that the °H (Hs 9') are tempered and that Cf is defined on 9' ~ f e $ ( ~ ) . The remaining assertions follow from continuity since C:(~) is dense in ~(~). COROLLARY i0.

The invariant measures on regular s.s. orbits of

tempered distributions on

in

~

are

~.

We shail presently deduce from Theorem 9 that for any fixed lim~, 9 H'H~ ~f(H;5(~))

G

will exist for all

provided t~e approach to

H0

~ £ S(~e) , f e ~(~)

H 0 e ~,

the limits

and will be unique,

is from within a single connected component of

~'.

However these limits will depend on the connected component used, and there are some subtle linear relations between the limits associated with the various components. Our main concern in the rest of this section is to elucidate this fine structure of the behaviour of

~f

in the neighborhood of an arbitrary point of

9;

this will be

carried out in two stages. In the first stage we fix an H O e ~ which is semire~ular, i.e., H 0 is such that there is exactly one root ~ e P with ~(H0) = 0. is the centralizer of H 0 in ~, then is singular or compact.

~=

In this case ~ is noneomplex.

If

[~,~] ~ ~ ( 2 ~ 9 ) or ~u(2,C) according as

The study of ~f around H 0 is then reduced, using a descent

argument~ to the study of the corresponding problem for 8; be done by direct calculation.

this latter problem can

Once this is done, the behaviour of ~f around the

"higher" singularities is determined in the second stage by elementary general arguments.

40

39

4.

Reduction to

~.

We shall fix a semisimple element X of ~. are as in ~ e o r e m 1.20 with of a vector space b c A ~ta L

W

over

+ (i - t)h e A

the centralizer of = Z/L;

let

and

dx~ dz

dz

U = U%~

V = G •U X

in

0 < t < i). G

and

Let

(as well in

dh

z ~ z)

~

Z).

Let

are Hear measures on

Let

~

a a A

be a CSA containing

G~ Z

G

and

dxdz~

dx = dx* dh, dz = dz dh.

i'19)

~i e Z (l_ 0

All CSA's of

du

~

considered

being a Haar

such that

(H ~ ~', g ~ Cc(~))

(~)

extends to an element of

Cc(~ ).

Since (u,Y)~g(u "Y) is an element of Cc(Zx ~), it is immediate that 0ZE g Cc(~). such that

Now ~ and L are both compact and so ] a constant b > 0 ~g(~.H)d~=b

~

Moreover, as all roots of

g(u .H)du, (~e,~e)

(g~Cc(~), H E ~ , T~(H)~0 ).

are imaginary,

42

s ,R=I.

This gives the lemma.

4i

5.

$8 ~ {I(2,~).

In which

We assume

in this

$8 ~ {I(2,1~.

n°

[X',Y'] = H'.

= ~ + ~ • (X'-Y'). on

ac

~(X'-Y')

= 2i

and that

~ ~

calculation since

~

take

(resp. where

Let

It is obvious

8

is imaginary,

Note that

~

of differential

coefficients. ~,

If

ZI

D~D v

of

operators

on

is the group

it is quite straightforward

morphism Since

of

8

such that

Z 0 c Z m ZI,

define the invariant

(io) here

~(~) L0

:

where

then a

~ in

i

or

in

Z0

LI, ~ n Z0,

Z0/L0, Q ~ ZI/LI, ~.

(12)

2.

(resp.

Z0

and

is the centralizer -(X'-Y'),

we have

of

in

b;

to be positive

Diff(~c) being the

polynomial 8

which are trivial where

T

8

on

is the autoT .y, = -y,

is reductive,

we can

(H c ~,~8, a(H) ¢ 0); u ~ u

i s the n a t u r a l map and hence

(g s C~(8), ~ = Z0 )

independent first

ZI).

of

~ = a.

Then

g. Let

On the other hand, L0, a

Z I = ZoLI, ~

(resp.

and hence,

Ll,a) as

if be the

L0, a =

Therefore

is a constant b

~

or

-i

Now, as

Z0 = Z

~zg,~ : bl ~g,~8 b I = bl(X,a ) > 0

~

that will

in the same way as for (2,~):

H)du*

we have

is a constant

of

An easy

8

T • H' = H'~ T .X, = -X',

8

Assume

of

and

ZI = Z 0 D T Z 0

g )- 9g,~ 8 = 98g

[Z : Z 0] : i,

centralizer

where

of

~8,~(~)~8(~)/'Z o/Lo g ( u * "

Z = Z I.

(Sc,bc)). to either

~

(resp.

are CSA's of

Diff(~c) ,

of automorphisms

is either

b = b(X,~) > 0

[Z: Z 0] = 2,

onto

with

map

Z

Z0

v

~c = bc~ac

[Z : Z 0]

(li)

~(H') = 2 b

(resp.

under

We choose

to show that

of

and and

be the linear

is no automorphism

Diff(bc)

integral

SO, when

~ ~

ill', ~ = ~ o

T I ~ = identity,

i s the c e n t r a l i z e r

Z 0 ~ Z0/L 0.

[H',X'] = 2X',

Then

v • be= Sc~ v • ( X ' - Y ' ) =

algebra

on

(Se,ac)

there

and that

a = [ + • .H' ,

(resp. ~)

that

is singular.

v : exp[-i ~ ( X ' + Y ' ) } .

We put

~

is conjugate

= 3)

such that

8.

Let

are the roots of

and we have an isomorphism

$~

such that it vanishes

(resp. _ ~)

b.

for

~ ~ b = ~.

i 2 = -i).

P

(dim(S8)

is the center of

shows that any CSA of

into

roots.

be)

is semiregular

[H',X'5' ]

~

Note that

is real and

a

X

We select a basis

[H',Y'] = -2Y',

function

that

Z0;

~-8, b o ~ = - 7 8 , b ,

(g ~ C~(~),~ : zI)

independent

as well as in and so

43

of Z I.

g.

If

Hence,

~ = b~ as

then

T . (X'-Y')

L0, b =

42

where

b 2=b2(X,b ) ~ 0

is a constant independent of

Let the Z-invariant element

~'

of

S(~c)

g

and

H.

be defined by

~ ' : H ' 2 + 4 X ' Y ' : H '2+(X'+Y')2- (X'-y')2 Then

co

and we get, from (7), (ll), (12), (13), for geCc(~), t,BE]RX: Z

(C+tH') = d 2

¢~(~')g,a

Z

(C+tH')

dt2 Cg,a

d2 z (c+0(x'-Y')) z ( c + ~ ( x z,))=---~g,b ~(w,)g,b de2 If K= { e ~ e ( x ' - Y') :e E IR], K is compact and (k,s,t) mk exp sX'exp tH' is an analytic diffeomorphism of Kx]Rx]R onto Z0, and the Haar measure on Z0 corresponds to

dkdsdt, dk being the Haar measure on K such that ~Kdk=l.

If

A = exp(]R • H') then L0, Q/A is finite, and hence we can normalize the invariant 96

dx

measure

on Z0 = Z0/L0, a such that

J'Z~(P(x*)dx*-~IK> 0 be s~eh that 9(a~b)=

0

defined

then, V C ~ a ,

(19)

~ Z (D,~,b)+(c)-(D~Z p-(C) = i~[~: z o ]-D (%,o)(c)

In partic~lar~ if (a) E~,a_

Dt~,b

z g ~ C~(~), 9g,be C~(b').

Then, for

Z s~ .D=-D, D~g,b

extends to an element of

has been essentially proved already.

is skew symmetric under For (b)

s

we note first that

all bounded on bounded subsets of

Note that if

and so must vanish b+ 5'

are convex.

Cc(b ). s

- E = -E,

then

on

The derivatives of

~g, b

are

Consequently these derivatives a~e uni-

formly continuous on bounded open subsets of

b'

The continuous extendability

is now immediate. It remains (19); the last of D~ gZ,b to Cl(b +) ~ s~to prove ~ s~ assertion would then follow from (a) because ( D ) =-D if J = -D.

46

44a

Moreover we need only prove that

(19~) Coo.

For, if this were done, we are already through when

[~. : Z 0 ] = 2.

Z=Z

0

°

Suppose

Then Z

and hence, from (!9~)

we get

S

On the other hand; it is easily seen that

(DT)V= (Dv) ~.

Hence using

G)

we

get

We shall now prove (19~). U ~ S(bc).

As (19~)

u = (X'-y')m

where

odd, (18) and

(a)

even,

It is clearly enough to do this when

is trivial for m _> 0

U c S(~e) ,

is an integer.

D = 5(u)

for

we may restrict ourselves to

Then

~(u) v = i m H 'm.

imply that both sides of (19~)

are zero;

if

m

is

m = 2r

If

is

then (17r) and (15) imply that both sides of (19~) are equal to

(-i)r i~ %~ ~(~,)rg (C+~X')du

6.

Behaviour of

Let

F

@f

around singular points.

be a vector space over

19

of dimension

be a finite set of affine subspaces of

~")

be the subset of all

any open set

W c F

sion

W(Y)

0

(resp. < 0); ('~)+ c CI(f~) r] Cl(fb),

the only connected components of

c ~ ,~ then

If

~(C) = 0

proving the first assertion. Then, if

101

vanish at

o
0

U P+,

~

of

the centralizer

A\A

x

4-

~ ; ~

~ ~ P+,

~

vanishes.

A ,

Let

is a convex cone.

H ~ ~+.

It is then

~ + ~ S ~ + ~ p +

is a positive system in

~ = ~ n F

is

these are all pure imaginary on ~.

stable under complex conjugation.

(~)

~

H~n$~

(resp. pure imaginary) for all roots

A

be one of the (finitely many) connected components of

Let

Since any CSA

to a e-stable one we may assume (and we do) that

then

(ii)

P+

U

P=P

is

Set

~

~cp +

It is then easy to see from (i)-(iii) above that mc = Z cp+ ~ ,

and that

In particular

[~,~] ~ n.

this shows that

n

p= ~ + n

n

is a subalgebra c @~,

is a subalgebra of

If we choose

P

~

containing

as in (iii) we find that

is a nilsabalgebra, i.e., all elements of

n

n

that

as an ide~l.

nc ~ Z

cP g~;

are nilpotent.

We have the following obvious relations:

(22)

= D n 6(~), Let

by

dk

K

~ = ~ + n + ~(n)

be the maximal compact subgroup of

the normalized Haar measure on

= fK fkdk

so that

K-invariants.

f ~ f

i.e.,

corresponding to fKdk =i.

is the projection map from

We now define, for any

(23)

K,

G

(direct sum).

v c ~(g), g v = gv,~

$(~)

For

K.

We denote

f c 8(~)

we write

onto its subspaee of

by

gv (H) =jr v(H+X)d~(X)

(H ~ ~)

n

where that

dR

is a Lebesgue measure on

v ~ gv

(resp. Cc(~) ).

(24)

maps

$(9)

n.

(resp. Ce(~) )

Furthermore, if

It is obvious from (23) that linearly and continuously onto

b~S(~c) ~

then

$(~)

differentiating (23) we get,

~(b)gv = g~(b)v

Our aim is now to establish the following reduction theorem:

58

gv c $(~),

51

THEOREM ~2. Denote by (resp.

Let

f ~ Cf

P~

be a positive system of roots of

(resp.

~, ~, P~).

u ~ ~)

Then there is a constant

(25)

~f(H) = ° ~

~

write

a-

Then

~I = ~ ~ ~ " LEMMA ~ .

n+nl,

and

Observe that

We fix

P

and

P~

Let

a0

and

(b)

If

Z = ~ + s0 + n0 "

~

prove (b) l e t So

£(n) c ~

Let

then

G

and

~N,...

Z.

of

N×N I

(a)

Vf

also.

We

C " nO c £ ~ c Q ~ '

Q~

in

A

~

E:hen [ = 8 + n + T .

nO

If

normalized by ~0 c £~

such that

proving that

8.

n0 =

~.

we have obviously

ad s0

and [s0,n0]

C • nlCZ

cQ ~ .

is a nil subalgebra ~. To

If

Xcrl, e(x)=(x+e(x))-x~+n.

be the analytic subgroups of

G

defined by

and

Z = K INIA 0

Using the fact that

exp

is a bijeetion from

NO = N N I

LEMMA 34. proper. that

To p r o r e

G= KN O A 0

with

Clearly it

~+n+e(n)=~.

is eo~paet while

easily that

~ ~ ~

is an lwasawa decomposition of

is an lwasawa decomposition of

[=~+cl0+rl 0 . +n,

as above.

is 8-stable.

Now we can select a positive system

Q = Q8 U P+,

.

be a maximal abelian subspace of

It is enough to prove that (a) nO is a nil subalgebra of

c nO .

U P+.

$, ~, P

such that

and is maximal abelian in

8 = II + ~0 + nl

~ = I + Q0+n0

P =P

We begin with a lemma that relates the

~.

~0 ~ ~ G ~

Suppose

then

~

and let

(f~S(~), H~')

f c Cc(g ).

lwasawa decompositions of that contains

c>0

(H)

The proof requires some preparation. is enough to prove (25)

S~

the invariant integral associated with

and that the map

N 0,

Let

The map

(n;nl) ~ nn I

of

nO

onto

NO

we see

is an analytic diffeomorphism

kl,n,n I .... etc. be points of ~: (k,n,z) ~ knz

ad(Tl);ad(n); . . . .

are the lwasawa decompositions of

K×N×Z

KI,N,N I ... into

G

Moreover there exist normalizations of Haar measures on

etc.

is surjective and G~ N and Z such

~Cc(G),

(26)

~ G f(x)dx =~K> i ~..j Since n

~

is the centralizer of

In partieular~ the

n.

ni

and

N

defined by

~°.

H0

The map

and there are suitable

H 0 ~ I+

Let in

~,

n.

Let

and let

0 < t I < ...

n i = IX : X ~ n,[Ho,X]= ~iX], we have

[n,~i] c ni+l

are ideals in

I

group of

n,

Select

I(~(Ho) . c ~ p + ] .

is the direct sum of the

nr+l = 0).

N. M = M + I .

onto

UCCc(n),

N- M = M + n .

be an enumeration of the set

Clearly

~'

N

We begin with the proof that < tr

V

Then N

~

for

[~,ni] c n.~ V i.

for all ~.

i

(nr+I =

be the analytic sub-

1

We shall now prove by downward induction on

l

55

i

that

53

~i " M = M + n i ; center(n)

for

i=l~

and so for

this will give

X c nr' (expX) . M = M + [X~M].

an invertible endomorphism of [X,M]=Y; ~.

1

-M c

then

M+Y[~r

M+n..

Let

1

such that

Jr= mr.

• M.

Yc ~.. i

i.e.~

So, if

Suppose As

Since

and

Then

~(M) ~ 0,

hi,

(expX) • ( M + Y )

• M.

i=r.

nr m

adM

~i+l " M = M + [ i + I .

is invertible on

Then

M+Ye~.

Suppose

defines

Y~ nr- , we can choose X e n r such that

i _> i

adM

[X~M] ~- Y(mod ~i+l).

( M + Y ) cNi+l " M~

N" M : M + n .

Clearly

we can choose

~ M (mod ~i+l).

X ~ n. I

So

(expX)

This carries the induction forward.

N

1

acts transitively on N

at

M

is

M+ n

and as

discrete and

simply connected;

so

n~n

n ~ n •M - M

~(n) = n • M- M.

dim(N) = dim(M+ ~); "M

is a covering map.

let

n ~ N.

The integral formula (29) is then immediate.

A simple calculation shows that

We can now prove Theorem 32. (resp. Z).

Clearly

L/A

and

Let

L

LI/A

such that

VHc~',

(resp. LI)

are compact.

co

el>0

Since

M+ n

is

will have to be an analytic diffeomorphism.

ally~

G

the stability subgroup of

(d~)n = - ad M

Fin-

for all

be the centralizer of

~

So there are constants

in

c >0,

,m

fCCc(9) , UCCc(8) ~

G

G

Z

Z

%

where

x

• H=~.

H=x

•H

etc.

But, by Corollary 35

~/' f(~.H)dx = / ~ f(knz "H)dkdndz : /~ ~(nz.H ) d n d ~ . G Further

KXIg, 0

m.-i

i, vj,..o~vj J implies that

such that

v.0

J

modulo

J.

is linearly dependent on

0

So

~

monic

qj c C[X]

such that

qj(vj) e J;

this

J ~ J(ql~...~qd).

Proposition 3. distribution on

U.

Let Then

U c F T

is

with an exponential polynomial.

be a nonempty connected open set and let S(Fc)-finite if and only if If

J

is an ideal of

sien, the vector space of all distributions

T

on

U

S(Fc)

T

T

be a

coincides on

U

of finite codimen-

such that

5(J)T = 0

is

finite dimensional. Let F.

v. (i ~ j ~ d)

be a basis of

F,

Then

58

t. J

the corresponding coordinates

on

56

- bj

(i _< j _< d, 0 < rj < mj)

fr,~

r r bltl+ m : t i ... t d "''+~dtd If we write qj(X) = ( X - ~ j ) J, then r,~ i d e ~(~(ql ..... qd))fr,~ = 0. So any exponential polynomial is S(Fc)-finite.

where

f

then

For the converse, let

T

be a distribution on

finite codimension such that

5(J)T = 0.

such that

If

J o J ( q l ' ' ' " q d )"

an analytic function on ~ m. Assume the

> deg(~).

(mod SI~)

is skew and

~(uj)z W = 0 ~ j ,

of degree

So we can find

In other words,

P(W -x)

is homogeneous,

that

to both sides,

% q ~ mu . i c S I +S •

w,

u

(7)

of degree

WW

by the induction hypothesis.

is divisible by

then

Since

~

are

is a subgroup of

elements

in it,

61

then

Ip(Wl)

is a free Ip-module rank : [W: W 1 ],

and one can choose a module

basis consisting of homogeneous elements. Select

x c Fc

regular.

to the representation of

Then the representation of in

corresponds to

Ip ® ~

invariant elements of relation

~

Let on

F

sI = i, s2,...,s r

Since

P - Ip ® ~

~

is equivalent

Ip(Wl)

it is immediate that

ander this isomorphism~ ~.

dim(~)

THEOREM ii. polynomials

in

P/P(W. x)

to the regular representation.

th e

W

which is equivalent to the regular + So the representations on W in P/P Ip and ~ are equivalent

representation.

W

~

being the space of W l-

Ip(Wl),

This leads easily to the assertions concerning = [W : W I ] following from Frobenius reciprocity.

~ c Fc

and let

such that

g(Z)

be the vector space of all exponential

~(u)~ = u ( ~ ) ~ V u

c IS .

Then

dim g(Z) = w.

is a complete system of representatives for

(i _< j _< w(~) = Iw(l) I = w/r)

is a basis for the space

W/W(~)

~(W(I))

and

If

pj

of polynomials

harmonic with respect to the finite reflexion group W(h), then the w functions sk Sk~ pj e (i < k < r, i < j < w(~)) form a basis for g(h). Suppose U c F is a connected open set that is nonempty.

If

T

is a distribution on

differential equations (8) are satisfied on ential polynomial on

U,

the space of all such T. sk Sk~ Let ~jk = Pj e u e IS

let

Obviously

uh c S

and the map

By Lemma i the

be such that

so that

enough to prove that

ba~s for

~s(W)

By Lemma 9, kernel of

S

~(u)(e~j)

dim(g(Z)) _< w.

and fo~

f ~ ~(~)

is the direct sum of

X l (u,~ u(h)).

u [ S, ~ f = 0.

~he map

So if

U,

T

As

~jk

are linearly independent. e Fc;

ux(0) = u(Z)

= u(h)e~pj. Fix

coincides with an expon-

is a linear bijection of

So

x 0 e F c.

and

Now; for

pj e ~ ( W ( Z ) ) ~

qj

SIs(~ )

and

Hs(W )

we have

and it is now

(i _< j < w)

~ = 0,

where

then

is thus injective, giving

onto

u~ e Is(W(Z)).

0

hyperbolic (resp. elliptic); elliptic) elements of connected,

sgn(w)

~.

(resp.

(resp.

~e )

+ ~+ = ~e ~ ~-' n

to either . Let

Further a

~

~hen

~ ~

that

~

or

s • H = -H

~

while

a

(resp. ± 8) and

8

for

and

v(X- Y) = iH.

Define

ponents of

8(~)

on

D

and

A = ~(~).

a'

and

~en,

for

by Lemma

Let

K

~ =a,b

3.7J

l) 0

G

Thus

is

9e

splits as

n = n+ U n" ~ {0]

of

"

be such that (~c,ac)

~

~(H) = 2

and

CSA's

where

is con

te

_8-1zd 2 / d82

X - Y. a

and

t

Let

and ~f,a

~.

Then

o 8 8

and we assume v - ~ c = ~e'

and so from Theorem are the radial com-

being the coordinates

and

~f,b

be the invar-

Write dm = --d8 m ~f,b(8(X-Y))

~m~(8(X-Y)) T'f,~

(resp. B(X- Y ) = 2 ~ .

(resp. (~c,bc))

= H 2, ~h = - ( X - y ) 2 ,

and

we also write

for

,(i) ,f,~ ,, ~f,~,

etc.

I(2) for ~f,~

V f ¢ C ~c(~),

be the compact group

T ~ c~(~) be defined by Lb)

(resp.

where

respectively,

H

(m) (tH) = d--T-m f,a dt m ~f,a(tH), m _> 0;

so, as

G .X', X' ~ ~e"

v = e~p~i~/4)(X + Y)).

Then

~'

with respect to

is then called

are not conjugate under any automorphism of

t -Io d 2 / dt 2 o t on

X'

is the set of hyperbolic

(0 -i

s =

b

Let

iant integrals associated with

for

9~

are the roots of

are positive.

We put

~± = ~X ' : x' ~ ~, w(X') o] .

(resp. ~) ~ a*c (resp. ~ bc)

2.14, we see that

a

~(X') < 0);

splits as the disjoint union of ~ orbits,

r~+ = ~ ~ ~ .

is regular ~(~)~0,

nilpotent ones.

X' ~ % , w ( X ' ) 2 ~ - ~(X') > 0;

is constant on each orbit

the disjoint union of invariant open sets

The set

of

X' e ~' is conjugate to a nonzero element of

~h

If

n

X'

[exp 8(X- Y) : 8 c m].

T(X') = /K f(k'X')dk

be the centralizer of

a

(resp.

b)

in

where G.

For

f e Cc(~) ,

/K dk = 1.

~t

let

~Q

(resp.

From Section 3, n° 5, especially

Theorem 3.18 and relations (3.15)-(3.19), we get the following result. LEMMA 21.

The invariant measures on

that the following results are valid. s-invariant element of an element of

G/L a

For any

Cc(a) (denoted also by

Cc(b ) ~/ r > O.

Moreover, let

Then :

68

and

G/L b

can be normalized so

f c C~(~), ~/~ extends to an ,~ ± ~ a (2r+l) ' ~s t ~f~ a) an d ~f b ex~ena o ¢ ( 2 ~ ) ( 0 + ) = limm_~j~(2~)(8(X - Y)).

66

(2r+l) (0) = 2i(-l)r+l(Arf)(0), ¢f,b

[--+~Ar--~ (uX)du 9f(2~)(0_+)= i(-l) r J0

9~m~(0+)-tf(m~(0-)=im+l

~i(m) (0) Tf~

From now on we shall suppose that wa,y that these results hold. LEMMA 22.

~

Let

a constant

dZ

(m>_0)

G

~f',a and 9f,b are ~ormalized in such a be a Lebesgue measure on ~.

e = e(g) > 0

such that V f c Cc(g),

co

t,f,a(tH)dt - i

8*f,b(8(X-Y))d8 oo

By L e = a 3.3

~

constants

e!,c 2 > 0

such that ~ f

co

~

Replace

f

co

f(Z)dZ = Cl f _~

= 2Ci
0,

[~,~,~]

and

_> 0 and

~v r = V r + z ,

2 . v o = o,

70

stand-

and let k s C\ ~ + . H, X, Y

the end-

defined by

2.v r = (~-2r)Vr,

It is

and the operator of multiplica-

be the set of all integers

be a vector space over

morphisms of

~% of

be a Lie algebra over 2Z +

E, h

E

~

again):

In other words~ the linear span

.~I(2,c),

Let

[ZI,Z2,Z 3]

are the corresponding linear coordinates,

easy to verify these commutation rules among tion by

Let

i[0]

be the vector space of all invariant distributions on

The measures

field on and

where

fn i i[0]

such that

elsewhere.

fn -> 0

-ic-l{~- ~TF} : ri®~(o) - ®{(o + )b++ ri®'(o) - ®{(o-)b'- 2r®b(ot-®b(o-)]6. Let

c n.

and

Indeed, let

2"vr+ l = (~-r)(r

+l)v r

68

(r _> 0). isfies

Then

c 3• e C

becomes

M

of

If

M NcM~

suppose

M

and no nonzero

is an m-module

X - v 0 = 0, H ' v 0 = h v 0

CV.

~hat

m-module,

e0v + C l ~ - v + --" + Cram . v = 0

Conversely~

such that

0, M : ~ r > 0

module

an irreducible

of the form

not all zero.

vector of ~.v

M~

a relation

Then the

is isomorphic

is a nonzero

where

v e ~

and

v0

Z e C\ Z~+ •

are linearly independent and

~

to

Mh

L

via the map that carries then

H.NcN~v

e N

a nonzero Let

r v

submodule~

sat-

with constants

Vr =

is a sub-

to

VrVr.

for some

r >0;

r

if

r > 0

N = Y~,

v r e N~ Vr_ I = [(k-r+l)r] -I ~ ' v r e N,

and proving

irreducibility.

If

and so

v = a0v 0 + .-. + amy m

c0v + C l ~ ' V

+ "'- + c ~ • v = 0, where c ~ 0, then r r r aivi' a contradiction. The converse assertions

Zi < m + call

[C~r}r > 0 We

v+ =

a standard

now

come

A~r~+,

0

m

basis for

back

= A~.

~+

Thus

a m ~ 0,

0 = c a v + + rmmr are well known

and

We shall

~.

to

Let

v 0 ~ N.

where

~.

Define,

(resp.

for

~-, ~ )

any

m > 0,

be the linear

span of the

v+

m

-

(resp.

Vm,

~m~A

m

0

Vm); ~+, gOc ~. 2>

•

0

~v-:

(~ +

,

-

2)v-:

~

,

.

+

Since

v-

f e Co(g)

and

In terms of linear E

5

vanishing

are measures

on

n.

that live on ^

So, as

coordinates

m = 0

Zl,Z2~Z 3

= - Z (8/~zi) oz i = - (E + 3), E* (m~)(f)

that

writing Hence

g(u)=

F(uX),

we

LEMMA 26.

find

3 + , 3-

and

30

~-, g0)

is isomorphie

(resp.

Vm,vO )

form a standard

are nonzero

constants

Cm,d m

~

If as

f) = 0~

M i/2

basis

of

and so

~

= E?

(EF)(uX)=

ana on

ng'(u).

Ev- = -2v-. submodules

M I/2,M 3/2)

(resp.

~ = v-, 5.

this shows already

~urther

independent

~+

+

for

E;

that

(resp.

such that

of

Simul~riy

of

3.

3+

and v+,m = 0,I ....

~-, ~0).

In particular,

there

~mv~ = c v -+, ~m 0 = d 5 V m. m

Given Lemmas

they vanish for any (~

= -~.

easily

Ev + = -2v +.

are linearly to

~

~, E = ~ z i ~/~z i

i.e.,

quite

Therefore

(resp.

on

~,

n~

being the adjoint

= - ~(~ + ~)f) = - ~ ( f ) ,

v+(Ef) = -~+(f).

on

24, 25, only the linear

m

m

independence

of

m

~,

~0

is not immediate.

3 + n 3- ~ 0~ then by irreducibility 3 + = $-, ~ C" v + = C "v- which is absurd + ~+ v and v live disjointly. If ~0 n (~+ + ~-) ~ 0, then ~0 c + ~- by ^

irreducibility~

and so, as ~ 5 = 0 ~

5 c C • v + + C • ~ ~ a contradiction.

THEOREM 27.

J = 3 + + ~- +30; no nonzero

Let

Let

T e ~.

G × U

~/

is

t eIR.

So

~/

of

U = {X + tY : t e JR} , N + = G • U

(x,X') ~ x .X' of

T' = TIN + .

element

onto

submer

Then supp(T')

C n +.

N +. Write sire,

X t = X + tY. and

N+=Z\(~e

So, by Theorem 2.7, 71

U

is

Is(~c)-finite.

and let Then U

~

be the map

[Xt, ~] + IR "Y = ~

n- U {0}).

supp(~T,)cUNn

Let + = IX].

69

Using

t

as a coordinate

support c [0], ^ ~(Xt) = t

while

induction Suppose

on

m

If

T'

we find that

where

~T' = 0 ~ & T '

m > i,

by the above,

~m-l(be~tl -

+~ Vm_l) = b~ +~ ,

~0

Thus

C- ~i

"

T+

~+

~

So

combination

= b~0

for some

~ b e C

~m-l(T~

~m-iT'

N +.

on

T - (T + + T') = $(u)~

u e S(~c) ; u e Is(~c ) that T e ~+ + Z- + ~ 0

for some

T - (T + + T-) e ~

This proves

~

at

= b~ +' ,

So

•

a e cx

t = 0.

On the

= (b/a)o +, ~ T '

N- ~ @ \ (9~ U n + U [0]).

with

We now prove by

+, : 0 ~

Similarly

T - (T + + T-)

~

I N+, i ~ m - 1.

= ~/a)v +'

while b y Lemma 26,

- bCm~ I ~m_l) +' = 0;

T = T+

such that

~

on

b e C, ~ & T '

such that

so that

~mT,=0.

of the

is the Dirae measure

on

On the other hand,

~^v +, = 0 ~ t o

~hen

+'

T' e Z i < m _ l

is a distribution

tm~T , = 0.

= ~r ~T'"

o~rT,

is a linear

~T'

m ~ 0,

v +, = v + I N +.

and let

~ +, = a$ 0

other hand

U,

obviously

that

m = 1

such that

on

so that for some integer

by induction,

T - e ~-

~

has support

~ [0]

such that and hence

by G-invariance, Lemmas

and so

24-26 finish the

rest of the proof. Define

LEMMA 28. Let jective,

+ 0 C " v m + C " V m + C. Vm"

Am=

Suppose

T = T O + ... + Tr, (h- Z)T

T = F ~,

Let

~

Is(~c)-finite on

~';

and

and for any

and

(A - ~)T e A 0

Ts e As

has a nonzero

THEOREM 29. variant

T e Z

s,

component

on

u e IS(@e),

~(u)F

As

At+l,

invariant

~;

the distribution

Tr ~ 0.

in

be a completely

distribution TF

V

for some

F

is locally

Then

A : Am ~ Am+l

T = 0.

is

bi-

a contradiction. open subset of

the analytic

defined

Z e C.

on

~

function

by

integrable

F.

on

9;

T

an in-

on

~'

with

Then

~

T = TF

and

on

~(u)T =

T~(u)F ~ ~(u)~F. If

X' e ~

with the case

is regular, X' = 0 e ~.

nomial

of degree

while

~:~

d~l

on

T = TF

in an open neighborhood

We may assume that

such that p ( A ) T = 0 .

~'

By(l~),

~]uen p(z) = (z - l)q(z) A T - ZT = ThF_} ~

by (14) again. since

THEOREM 90. distribution analytic

h e C

on

function

Let ~.

~ = g(2a) Define

on (-a,a),

~

If

and

T : TF

T

¢b as above,

on

~

(Z e C)

we use induction q

of degree

lhen: ¢(2r+i)

(i)

d - 1.

d. Then =

The last asser-

and coincides with

an invariant

As on

(h- I)(T-TF)

by Lemma 28.

Q

We are left

be a monic poly-

On the other hand

(ii) the derivatives 72

X'.

(A-~)(T-T;) ~ % .

d>l,

Is(@e)-finite

(a > 0) and

Hence

and some monie

As before;

~(~)T is invarian%,

A0.

T = T F.

b z the induction hypothesis.

-(AT F - T£F ) e ~ tion is obvious

for some

of Let p

If d = I, AT = ~T

(~-~)T F ~

s u p p ( T - T F) = n, T - T F e [; by Lemma 28

N = @(2a).

b~)%

on ~'.

Is(~c)-finite ~b

extends

to an

extend continuously

7o

across

t = 0 (r = 0,i .... ),

Conversely, let and let

TF

F

¢~2r+l)(o) = (-l)ri}(a2r+l)(o),

be the distribution on

the conditions

~

with

TF = F

Suppose first that

TLF = &TF.

~

T

on

~'~ invariant and

defined by

(i) - (iii) above, then

distribution on

ArT,

(iii)

be an analytic function on

TF

¢~2r)(0-).

is invariant and

Is(Zc)-finite.

and F

~b by

proving (i) - (iii).

satisfy (i) - (iii) above. £rF,

£rT F = T

p(A)T F = 0.

So

.

Hence

TF

finite.

F

p

on

Since the conditions

COROLLARY ~i.

Let

T

Then

( - a , O ) , Cb

Cb

we have

AT F = Tg F.

p(A)F = 0 Suppose

T-T F e Z

and

on T

Replacing ~',

then

is also such a

T-T F

is IS(Zc )-

IS(%)

Is(~e)-finite

distribution

~

is unique and satisfies

Is(%)-finite

such that

on

~

(aI - a2)

and

+ a2e

-~t

Case 2:

Then

~ = T

~;

such that

T

on

T e ~,

~

i0 FT(8(X-Y))

~ = 0.

b 2 = ia 2

and

(t > o ) ;

ngain,

6.

For

(t > 0);

~

~.

an invariant

such a distribution

such that

bI - b 2 =

= bleil0 + b2 ~il@ (8 e]R×), t FT(tH ) = tFT(-tH ) = dim(~)

In this case, for

dim(~0) :

Sk

AT = h2T.

bl,b2,al,a 2 e C

in particular,

i8 FT(0(X-Y))

T c S0,

= b I + b28

= 3~ bl,b2~al,a2

(@ e m × ) ,

c C

such that

tFT(tH ) = tF(-tH) = a I + a2t

~.

We resume the general case.

Let

~

the distribution

(arbitrary

~)

be a completely invariant open subset of

an invariant 1-finite distribution on ~'; TF,

on

~

h e C~ the vector space

such that

Behaviour around semire~ular points

on

distribution on

~(J)T = 0.

We can use these results to determine, for any

~ ~ 0.

and

5(J)~ = 0.

of all invariant distributions Case i:

t = 0

are defined everywhere, we get

be an invariant

the ideal of

= T

is

and

by

:

(O,a)

(i) - (iii) involve only behaviour around

J

~; T,

T

~(2r)(o+) ~b As

Is(gc)-finite.

and since exponential polynomials

Denote by

ale

replacing

8 = OVm = 0,t ....

is monic and

~'.

By ~heorem 29,

By Theorem 27, T : T F.

8 = 0~

~t

satisfy

Is(Sc)-finite

For the converse, assume that

Then by (14)

if

£r F is invariant and

distribution coinciding with

@b

= i @(2r+l)(O)'~

extends c o n t i n u o u s l y a c r o s s

8 = O~

and

¢b(0+) = ¢ b ( 0 - ) ;

c o i n c i d e s w i t h an e x p o n e n t i a l p o l y n o m i a l on each of analytic across

¢s

~'.

(-1)r@~ 2r+l) (0-) = ( - l ) r ~ 2 r + l ) ( o + )

]]%US ¢~m)

If

is the unique invariant

So, by (14) ¢~(0-) = ¢~(0+) = i ¢ ~ ( 0 ) ,

we get

F.

( r = O , l , 2 .... ). Is(%)-finite ~

a;

F,

f ~ f~, FfdX on

73

the analytic function on ~

(feCc(~)).

~'

71

Proposition open set

~i

)2.

with

Let

X e ~

X e aI c a

be

s.s.

and semiregular.

such that

T = TF

This is immediate from Proposition i~ LEMMA ~ .

Let

E

u e S(E)

let

nilpotent,

~u

operators on

be the derivation

i.e., for each

generating ~U"

~U"

Dill(E).

nomial p, we denote by

~f

u =.u I ..- u s

~ ~ ~ ,

the~

~(~s)~(Ul "'"

~u

M(p)

Then

where the

%(~(~))

~4.

locally integrable

for

and

so that

~, T

m, Let

DT

~;

~(~ re(p).

Now,

(r ~) ~(p)(D)(~(p)-×(p))m-r

0k(p)

3

k(p)>i

such that

--

~

+ re(p),

then

I = C[Pl,...,p~] ; for each

i,

generated by DT

($(p)-X(p)~tDT

then, writing

~ ki

such that

k1

k~

ql ' ' ' " q ~

is thus I-finite.

'

r ~r

~(D)=

for

r >k(p).

Hence if

+

Let

Pl ..... p% e I

qi = P i - X ( P i )'

$(qk)DT = 0

then

0

~ P ~

= 0.

if

we have

k _> k i.

Now assume only that

T

I = C[ql,...,q Z]

If

dim(I/J) < k I -.. k f < ~

J

and

is the ideal in

while

is 1-finite,

then from standard spectral theory we have homomorphisms

be such that

I

~(J)(DT) = 0.

write

~ = ~(I)T;

X i : I - £ and subspaces

g. c ~ such that ~ = 81 + "'" + ~ is a direct sun, and for each i, T' e ~. l r ! and p e I, 3 m : m(p,T',i) such that ($(p)-Xi(P)) k T ' = 0 ~ / k _> m. Then by the preceding Proposition

result, ~5-

system of roots of

Let

DT

is I-finite.

~j T

(~qc,~c) ;

~

The remaining

be as above; = I]~eP ~; Ps' 74

~ c ~,

a

assertions CSA;

P~

are obvious. a positive

the set of singular roots in

P

72

and

~'(Ps)

7[~(H)F(H),

the set of

H c ~

(H c ~' N ~),

then

such that ¢~

~(H) ~ 0 V H

e Ps"

If we put

extends to an analytic function on

that coincides~ on each connected component of

~'(Ps) q ~,

@~(H) = ~'(Ps) n

with an exponential

polynomial. Let

X 0 e ~'(Ps) n ~

X0 e Uc

~'(Ps) n ~.

and let

Let

let an

I~, ...,Mq

Br+l'''"~t

~i

in

either

X'

~

~

vanishing at

X';

has compact adjoint group.

If

] i, i < i < r

is

So (Proposition 13 and Corollary 20) in

X'

U.

~.

~ = ¢~

on

T

coin-

This ~ ¢~ extends dim(Mi) _< Z - 2 V i;

has only finitely many connected components on each

U n ~'

But then,

~

U. is

¢~

are

So Lemma 3.21 ~ ~ r9 s C~(U) ~(S(~c))-finite and so is analytic

This proves the proposition.

Let W(~c)

in

is an exponential polynomial, so that the derivatives of

bounded in some neighborhood of every point of such that

such that

or to then

X'

Now,

on

N. J

M i,

~i

U.

U ~ ~'

and

in the latter case, the centralizer of

X'

@~

B1 ..... ~r L i (resp. N.~

X' e "U = U \ ~ _ < i _ < q

cides with an analytic function in a neighborhood of

moreover, by Lemma 16,

Let

r + 1 _< j _< t)

to an analytic function in a neighborhood of

of which

where

P.

~

such that

which are either equal to an

L i• .

is regular or it is semiregular and P

in

~ (I < i < r) (resp. 8j

be all the subspaces of

~

= {~i ..... ~t }

the complex roots in

intersection of two or more of the

the only root in in

be a convex open subset of

Ps = {~i ..... ~p}' P \ P s

are the compact roots and be the null space of

U

Pk

be the set of positive compact roots and let

generated by the Weyl reflexions LEMMA ~6.

Let

lwasawa type. ponents of

Ps

be empty.

Wk c W(~)

~'(Pk)

and

Wk

under

Then

~

is conjugate via

which are all convex open subsets of are complex,

is connected for every convex open subset W(~);

be the subgroup of

G

to a

CSA

of the

acts simply transitively on the connected com-

empty, i.e., if all roots of (~c,~c) F N ~'

Wk

s , ~ ~ Pk"

this is in particular the case if

~' F

~

~.

If

Pk

is also

is connected and in fact of

~;

and

7~

is invariant

is the underlying real Lie

algebra of a complex reductive Lie algebra. We may assume Let

9 = T +

~

centralizer of is a root of

~

semisimple and

in

(%,~c),

~.

Then

to be stable under a Cartan involution

then

~

be such a root.

Put

a = ~ n p.

~ : (~ N ~) + (~ N ~). is real on

(~c,~c) ~ I ~ = 0

that

If

S = 0

shown for 5(~)T F - T$(w) F

S

So, if

~

~',

such that

being the

is as in Section 2,

on

9

having singular supports

is any distribution

Sectinn

For this,wefirst derive

(4.14) and shows that it belongs

4, one has

and constructed

in this class, it will

around each semiregular point;

in

F'

on

is easily verified to be a subal~ebra of

n°3, it is enough to prove that ~(w) c ~ in view of Lemma 2.12.

in a very specific manner.

: 0

D [ Diff(te)

emerge

as the latter has been

5(~) e 91.

For the second step we use the descent method and reduce the proof to showirgthat no nonzero invariant

I-finite distributions

pretty much as in Section 4,

2.

An expression for

We fix a G-invariant that is real on

9 × ~

with respect to which,

n ° 5,

with supports

using the theory of

coincides with the Cartan-Killing

gives rise to an isomorphism

this is done

8(~)T F - T~(~) F-

symmetric nonsingular bilinear form

~c

c ~;

~[(2,C)-modules.

and v ~ ~

center (gc) of

S(tc) 78

(.,.)

form on

on

~gc × ~te'

are mutually orthogonal. onto

P(tc)

9c × 9c and

~his form

that commutes with the

76 actions of

G

~ ~([c); Let ~,

on these two a!gebras; so that, in particular,

we d e f i n e 2 c ~

m ~ I

by

~(X) = (X,X)

(X { gc ) .

be completely invariant, open~ and

having the following properties:

(i)

T

v e I = Is(~c)

T

an invariant distribution on

is I-finite on

~'. (ii)

is semisimple and semiregular, ~ an invariant open neighbourhood such that point

TI~ ,

X 0 e ~,

is i-finite.

and select

an invariant

9 e C=(~)

~

and

TF

V 0 = VT, T

containing

such that (i)

9 = i

invariant distribution

and

p c I, $(p)~

= 0

4.17 that

is a bounded open connected % s P t ~' Pt

V 0 c ~.

7

is a

We choose

on a completely invariant open set co

¥:~-T(~)

(f ~ Cc(~)) is an

on

9'\supp(q0).

is locally integrable on

there is nothing to prove.

proof of T h e o r e m

and

in

We fix a s.s.

as in ~heorem 1.20 where

We need only prove this around each s .s. point. X / supp @,

X' ~ X'

on all of ~ and ~ = T on ~i" We write ~ for t~e function ~F

on g'; ~ is invariant, s C~(~'), For any

If of

be as usual.

X0, X 0 e U0~

(ii) supp(~)=v 0. ~en

with x0 { ~ c ~

LEMMA i.

F

U0 = % , T ,

bounded convex open subset of

~

Let

~,

Suppose

CSA ~t

G .F t and

being a positive system of roots of

~t(Fl~' n ~t), ~t = ~t (~l~t)'

Pt : P~t"

X e ~

X s V 0.

V 0 Q ~' = ~ l < t < m

subset of a

Let

~hen, by Theorem

be

s.s.

If

It is clear from the where, for each

Ftc~.

(~c,~tc);

~.

t;

Fix t and let write

Ft

mt =

Ct =

2.14, ~ H

c Ft

(~(p)~)(H) = Irt(H)-l(~(pt)~t)(H ) = 7[t(H)-l(~0@t)(H;~(pt)). But,

as

constant some

Pt

is bounded, all derivatives of

Ct > 0

C > O,

such that

~(p)~

b(p)~

~}t

are bounded on it~ and so

is majorized by

is majorized by

Cl{l -I/2

CtI~t I-I

on

rt.

on

V 0 n ~',

{

~

a

~hus, for being as in

(1.i). mis prove~ the 1emma, as I{1-1/2 is locally integrahle. LEMMA 2. of

Let ~ c ~

be a CSA;

(~c'~c)); Y~ = 7r~(~I~');

Ps'

7[~ = ~ c P

~

(P'

a positive system of roots

the set of singular roots in

P,

~'(Ps)

the

co

subset of

~

where none of them vanish.

~hen

~

extends to a

C

function on

~'(Ps)' and ~ u ~ S(~o), ~(~)¥~ is hounded on ~'(Ps)" For the first assertion we srgue as in Lemma I, using Proposition 4-35. second, it suffices to prove that for each u c S(~c) , ~ ( u ) ~ F t is the finite subset of all x e G

For the

is bounded on ~ ' ~ V 0. If

such that x - ~t = 9, then (in the notation of

Lemma i),

~'nv0:

U

~t.rt

l

~. < 0.

:L components

connected T+ (~i~);

M0

of

of

E'

whose

(x C M0).

(~(u)h) +

on

M~, (~(u)h)+(x)=

It is easy to verify that

~d

t~at

h-+ ~ ~(M~)

LEMMA ~.

Let

of them

~ ~(E'),

~,

dE dM i

then,

M.

h+

V

If

h e B(E')

M 0l"

We

and

(~(u)h)(x')

'-x,x' sr+(M~) that

(~(u)h) + = ~(u)h +

Finally,

E.

for

let

~hen ~ uniquely determined

such that if

h2 ~(u)hld~'

~ l~H s ~e"

Zf ~

If we define

is real or imaginary, then

~ (Ha) > 0

+ ~;

i+ dMi, ~

there is a linear isomorphism

E I < j < _ ~ ~(H.~,o.)H.~,3. : H .

are those that contain

[?f,i~(Hi, j) (SiYi)

I,Cf

if

H~

is a root or to he

H~

and for given

g c B(~i(Pi,s)),

~ ~ HZ

of

~c

onto

(~c,%c), we have

or

iH~

i,~,

according as

the haZf-spaces

M+

g[,~(H) : lim~.~0 + g(H + ~H~)

(H ~ M~,p. LEKMA ~.

Let

Lebesgue measures

~(f) ~

r @cS.

dMi, ff on

7M

lT = 0

is regular in

is never zero anywhere on

supp(I~l

singular in

be

O~l + ~2' a neighborhood of XO.

if

sufficiently small that U.

Tfl

is invariant and

by the induction hypothesis.

By the work of Section 4,

Z-invariant distribution on

and let

Tfl

(YI s c, Y2 ~ $~)

We shall prove first that if

around

g, dim(d) < dim(~).

Then

supp(g) c ($~) ' ~ supp(f I ® g) c ~'

supp(Tfl) ~ ~2 \ ~'2 ~ TfI = 0

~

fl ~ Cc(~l)

on

T(f I ® g) = 0 ~v~fI ~ CC(~I), g [ Cc(~2) , ~ T = O

on

T = 0.

Replacing ~2 by G • ~2 we may assume that ~2 is

g ~ T(f I ® g) (g ~ Cc(~2) )

Is($~e)-finite.

point of

an

dim(~).

even completely invariant.~

the distribution

in

T

Let X c ~ be s.s. Write X 0 = X I * X 2 where X l S c, 0 (resp. ~2) be an open neighborhood of X I (resp. X2) in c

~i

such that

invariant;

X # 0,

Then

and

c = center(B) ~ 0.

and let

(resp. $~)

~,

G

g •X ~ ~ ~,

and is

for

there is a 0

on

By Theorem 26,

91

THEOREM 28. invariant

Let

g~ be a completely invariant open subset of

I-finite distribution on

coincides with

T

on

~' = ~ N ~'.

~.

Let

~hen

T(f) =

E

on

~,

THEOREM 29 .

of

T

Let

invariant distribution

~

be a completely ~,

each s.s. semiregular point of ~', ~hen on

and ~ ~';

TF

the distribution

exactly one invariant and

TF

1-finite

TF

was l-finite~ used only

~.

Let

So we get

invariant open subset of

on F

~'

and

T

an of

the analytic function defined by

f ~ f~, F f d~

95

~

and in some n e i g h b o r h o o d

(f c C~(~))

1-finite distribution

is that distribution.

an

i.e.,

around regular and semiregular points.

on

T

(f c Cc(a)).

We also observe that lheorem 17, which proved that the 1-finiteness

and

be the analytic function that

T = F

fFd~

~

on

~

defined on

~

T

by

that coincides with

on F. T

6.

local structure of invariant ei~endistributions the behaviour of the function

F

on

~:

around sir~ular points

In this section we work with an invariant 1-finite distribution a completely invariant open subset coincides with Let

T

9 c ~

~P = % e P

on

be a CSA

and

H ;

~.

P

P

F

denotes the analytic function that

a positive system of roots of

more generally,

if

~

of roots of (~c,~c)inPis

and

we write

~p,~ = 1~ p

If

~eP

"Zgp,~/~ = ~p,~/ .

, s

of

defined on

~'= ~n ~,

~' ~ P = % e P

9 c ~ ~ ~, the set

~

T

~p,~/~

under the Weyl group of

(~c,9c).

We write

is a reductive algebra with

a positive system for the latter,

I~ep\p "~p,~ %ep

ThUs,

~,~/~,

and similarly

~p,~/~, are invariant

(~c,9C) .

Now we define

(i)

¢9,p(H) = Fp(H)F(H), Y~(H) = (~(~p)®9,p)(H)

Then

¢9~ P

and

~

behaviour of

#~,p

P.

Expressions for

Let

Xe~

be s.s. U

for amy

#~,p

lq@l

and

We select

_ = U~ V

by Lemma'4.18

= V

~T

LEMMA i.

Let

g4 = I

7,~

be

c ~

r = ~ dim(~/~),

~,p

Let

3

=

for different

~

9.

be as in Section 2~

Is(~c)-finite

Ihl ~/2

~ e Is(~c )

X.

on

~

such that

Then

7

~

is

n°3.

distribution on

is also Z-invariant and

be a CSA containing ~p,~/~

and

does not

as in Theorem 1.20 and such that

ca.

is a Z-invariant

and

Y~

Y~.

~e Is(~c), 5(~)(I~II/2~T)

such that

and it is obvious that

Our aim in this section is the determination of the

and the relation between

i.

convex; let

~ N 9'

are analytic on

depend on the choice of

(H~ ~ n 9').

U.

Then So,

IS( @c)-finite • a constant s = g(9,P) (-l) r

n ~;

2

s ~

= Zfp,Q/~

where

=

for all

9, P

as

above. Let

r = ½dim(~/~)

iant polynomial

q

on

(r de

is an integer). By Latona 2.]0 we can find a Z-invar2 such that q : ~ . As U is connected and as ~

vanishes nowhere on it, 3 a constant £0 = s0(~) = + i such that s0q= I~l I/2 on 2 U. Further q9 = (_l)r (~p,g/~)2 on U N ~ and neither of these vanishes anywhere on

U n ~

constant

while

U n ~,

g I = ~l(~,P )

£ = a0g I.

Then

s4 = i

is independent of Is(de ) Then

9

being starlike at X, is connected. Hence we can find a 2 gl = (-l)r and q9 = sl~P,g/~ on U n ~. Tske

such that

and

and P.

(cf. Section 2, n°3).

S~p,~/~ = i ~ i i / 2

on

U N ~.

Now we have a natural isomorphism of Let

~Cls(~c )

g~9 = % , g / ~ " 96

correspond to

_v 21= 2 ( - ir) g 2 =sOs

Further

~(~c)

(-l)re0q

with

under it.

9~

Let

(2)

~ be as above. l%ll/%T

=

Then by Lemma 4.5 we can write, on

2 l i so that (D'j~i )( E~ H ) J -from either side. Thus in either case,

extends continuously across

then

0.

So

D~,p

extends continuously to

un~. Let

H 0 c ~ N ~.

neighborhood P

of

H0

First assume in

H 0 c ~'(R).

~ N ~'(R).

can vanish somewhere in

F;

Then

let

Let

F

81,...,8p

be the imaginary and

the complex roots with that property.

Let

~,...,Mq

which are e q u a l

of some

be all the subspaces of 7i

~

be a bounded convex open

only imaginary or complex roots from

Lj

71 ,.--,Tq,

be the null space of Bj in [j

and

either to the null space

or to the intersection of two or more of the

Lj.

If

H e F\~0,

e(~) : c(b).

V

and the function

Y.

We shall now reformulate Theorem 5 in a more elegant manner. LEMMA 6. that if where

here

~ a unique analytic invariant differential operator

X ~ ~'

and

A f~ = f l A n ~. X~A

and

~

is the CSA of

is open in

~,

~

containing

X,

then for any

(T f)(X') = (~(~p)(~pf~))(X')

99

V

on

~' s~ch

f c C~(A)

( X ' ~ A N ~);

96

For any CSA

9c

of

~c

let

E9

= ~(~'p) ~ Vp

where

P

is a positive system

c of roots of the Weyl

(9c,9c);

group of

(Etc)He S(~c)

9c

The invarianee of (H{ 9c , x c G c

Etc

~X :

Etc F

striction of

~

(XCGc;

H,

and is invariant under

(Et) x X{~c)"

9c

~'

S(tc)

i),

Yn c V ~

@, ~'= ~ n ~,~

Suppose for each CSA

be the s.s. component of

plate system of mutually nonconjugate

that

~';

Y c ~, Yn c ~'

We may therefore assume that

Yn = Yn" Hn

such that

are CSA's of

~ = f

X

~'.

f(Yn) =

i(n):i

f(Hn) = fti(X)

exists.

As

ftl(X) . . . . .

f~r(X); limn,

f(Yn)

itself

i(n):i exists.

To prove that

limb, ~

Hn = X

observe that if

p

is any invariant

i(n):i polmomial on i(n) : i

%,

P(~n): P(Yn)"P(Y)

as n -

~.

are all contained in some bounded subset of

point of the set

H n.

Let

~'~T'

be such that

100

So for

~y

i,

9i.

Let

H

0 < T' < Tj

and

the

Hn

be a limit CI(9,') c 7"

with

97

Then

CI(Vq/, T,)~

and Y e ;IT/ for all sufficiently large ~T n ',T' for all sufficiently large n and hence H e C I ( L ), ,.T ,_

H n e L , ,_T T'

c V

were arbitrary this implies that

Since

H

is semisimple,

THEOREM 8. invariant

N' = 0; i.e.~

~

o D)F

such that

T = F

on

~.

Let

~'.

~

Replacing

DT

by

T

VgF = Y~

on

~ n ~,,

then

5.

that equals

X :l ~ C

is the (Chevalley)

on

9

and

T

an

be the analytic function on

Y

on

~.

~,

~ n ~

and

D c Diff(gc) ~ In particular,

Y

for each CSA D = i.

T9 F

is the unique con~ ~ ~.

If

~

is any CSA of

be a homomorphism.

h

is regular.

2

be a completely

If

~c c ~c

such that

is a CSA, and X(p) = p~ (h)

P ~ P~c (p e I);

We shall now prove the foll~wing

X

converse to

3 and 8.

THEOREM 9"

Let

invariant open subset of

2' = £ O ~'

where

X :I ~ C

is a regular homomorphism.

on

given by

TF(f ) = / 2 , F f d g ( f £ C c ( 2 ) ) .

c 9, ~ p F ~ (ii)

~.

with regular ei6envalue.

invariant analytic function on

v F

and

~',

and the result follows from Lemma 7 and Theorem 5.

isomorphism, then ~ h e be

is called regular if

£

F

we come down to the case

Ei~endistributions Let

Y~

Since

is a nilpoten% of

Then for any invariant

extends to an invariant continuous function on

tinous function on

Theorems

N'

But then

H = X.

extends to an invariant continuous function

~,

where

be a completely invariant open subset of

l-finite distribution on

~' = £ Q 9' ~

Let

H = X + N'

n

such that Let

function on

(i)

~ n ~,(R); 2.

and

F

an

be the invariant distribution

Suppose that

extends to an analytic function on extends to a continuous

TF

9

8(p)F = X(p)F V p e I

Then

for any

here

TF

CSA

F~ = FI~' n

is 1-finite on

~(p)T F = X(p)TF~/p e I. In view of Theorem 5.29 it is enough to prove that

each s .s. semiregular point of

~.

TF

is I-finite around

~efore starting the proof we obtain the follow-

ing lemma. LEMMA i0.

stants, c

s

+c

s s The

s, s' e m

Let

* ~c"

element of

then =0

If

be a CSA;

~,

i s t h e Weyl g r o u p o f

Esc m cseS~ V

vanishes

a root of

(gc,~c)

(£c,~c);

and

on the null space of

cs ~

s'?~ = sh

~,

a regular

are con-

if and only if

is obvious.

s ~ s',

For the

sk = s'Z

'only if'

part,

on the null space of

we n o t e f i r s t

on the null space of

~,

then

101

that

~ s' = s s.

s' = s s ~ s '~ = s~ - 2(s%,~)/(~,~) -I = sV~ on the null space of if

and (s e t0)

sen.

'if' p a r t with

~c c gc

m

s'h = sh- c~

~;

for some

for In fact,

conversely, c e C.

So

98

(~,~) = (S'k,S'~)

= (S~,S~) - 2 e ( s k , ~ )

e = 0, s'~ = s~, ~ s s'Z = s~s2b for

or

[l,s }/~,

= s'

as

s' = s~s.

~

+ c2(~,~)

is regular.

If

Sl~...,s r

then the restrictions

distinct and the restriction of

= ()~,k) - 2 c ( s ~ , ~ )

So

bi

of

E cseSX

sih

is

to the n~i]: space of

Ej(Cs. + es s .)e ~. O

We now come to the proof of Theorem 9.

9', T

V = V

~

and suppose further that

F U = F IU N g'

tribution on

U

J~jl/2t u

Zs(~c)-finite

is

defined by

compact ad,joint ~roup. (~c,~c), and

FU,

then

on

Let

u,

~

~

be a CSA of

~,

and

U = U ,T'

V c G.

and if

tU

By Proposiis the dis-

In view of Lemma 4.18, if

would be Z - f i n i t e P

on

~.

Case l "

has

~

a positive system of roots of

the Weyl group of (~c,~c). Then P = [B), 8 is imaginary + = IR • iH~. Write F- = %, + { i t H ~ : 0 < + t < T~(H~)-I]. Then

U n ~' = F + U F-, U r] ~ = F + U F- U %,. As analytic function on U n ~', ~pF~

U N ~.

E s(S)Cs es~

U N ~ c ~T(H), ~pF~

On the other hand, as

3

constants

bs = s ~ l o "

~l = ~l(~ n ~)

and

Sinee

c

(s e m)

e s~

on each

such that

7pF =

s sB

comes from

is skew symmetric with respect to

Let

extends to an

~(p~)(TpF~) = p~()~)~pF~ (p e I)

is a linear combination of the exponentials

connected component of U n ~' So s~ E s ~ s(S)Cse on F + U F-. Since

(se~).

U,

let

Xe S

stud m,

~ n ~

on

as therein;

t U = ~TF.

are

o~ 0

is convex and

is locally integrable on

~

The lemma

We fix a s.s. semiregular

use the notation of Theorem 1.20, choosing %,,T tion 2.16,

If

showing

is a complete system of representatives

follows at once.

= G • U,

+ c2(~,~).

c = 2(s%,~)/(~,~),

G, F(sBH ) = F(H)

s~.

( s ~ , ~ ) ~ 0,

and so

This implies that

s~[(~ n ~)

= as81

c

= c

S

S~S

where

as

is a constant

{ 0.

Hence we o b t a i n f r o m Le~ma 1, the

following formula, valid

~/ H e F + U F-

with

H = C + H', Ce%,, H' e ~' N ~ ( T ) :

=

S ~

b s (C) ~

e-as~(H')

e asS(H')

E(S)ese

8(H')

s em

On the other hand, function

fs

on

- e

as ~8c

with

(e s

have

(I~II/2~)(H)

(e z - e-Z)/2z

invariant under the adjoint group of

s )/aas~

=

= ~e~ c'e

Zs

® f

on

be 0 ~ e .

on

tends to an ana~ff~ic function on

U n ~,. ~ U.

~

~ ~I(2,1R).

be the CSA's

z2~

~8c

3

an entire

that coincides

e s = ~(

)%%,

we

In other words, As both sides of this equation are in-

this must be true on But then

F

The extension is therefore I-finite on

Case 2: e, b

Therefore, with

e~e~S(C)fs(~')"

variant under the adjoint group of

V = G • U.

is an entire function of

102

y

So

FU

ex-

V.

We select a standard basis

defined as usual, and let

U n 8'.

extends anmlytically to

[H';X',Y'] for

~.

be an element of the adjoint

Let

99

group of

~c

that maps

elementwise. We write

~

= ~ oy

Let

Ps

bc

(~c,ac) on

--

into

i H';

y

fixes

be a positive system of roots of (~c,~c) and let P b = P Pa

vanishing at

=~Pa

(resp.

and

z~ =%b.

(~c,bc)).

X.

Pb

Let

Then

~

vanishing at t0a

Choose regular

(resp.

he ac

~v/ P e I. + we w r i t e r~ = ~/ + [ t H ' : 0 < + t < 2 } ,

oy.

is real, X.

~D)

We write

be the Weyl

such that

8(p)F =

V ~ B'

NOW, i f

the Weyl reflexion 8,

X ' ~I '

and takes

is imaginary and is the unique root in

group of

of

%

for the unique root in

F~ = ~Pa' 7[b = 7[Pb' ~

p~(Z)F

onto

s

sends

r+

to

r-.

As

+

then

s

(~)(H)

= -(~)(%H)

=

D

~d

comes from the adjoint group

we can conclude that for suitable constants

(4)

r]

U ~ a' = r~ u

Cs(a ) ( s e m

),

and ~ H e

F+

a(S)Cs(~)e s~(H). G

The situation is different for eigenhomomorphism of

I.

b.

The linear function

On the other hand,

U ~ b c b'(R).

y o~

now labels the

So, if we write

-r

r~ : ~ + [it(x'-Y') :0 < Itl dY

(X~ ~)

For any tempered distribution T, its Fourier

^

transform T is defined as usual:

In particular T is inv2miant if and only if T is so.

Moreover

/\

(1)

~(u)~ : (~ ~ i)T,

where

]o i

and

(~ o (-i))~ = ~(u)~

u o (-i)

are the polynomials

(u ~ S(~c)) X ~ ~(iX)

and

X ~ ~(-iX)

on Zc"

We also need the notion of a distribution being tempered even when it is not defined on all of

~.

If

~ ~ £

is said to be tempered on

G

is an open set and if

~

T

is a distribution on

a tempered distribution

T'

on

£

~

T

such that

T = T'I~ , i.e., (Hahn-Banaeh Theorem) if and only if ~ E.I e Diff(qc) such that

I~(f)l --i < •

(2)

~1(~)~2(~2)~(~ : ~)

=

~

Cs(~: ~)e

s e~ 2 107

(~i ~ ~" i=l,~)

10h

where, for each

H 2.

~ c ~i' C s ( ~ : ")

Cs( ~ :~2)

fact

We shall therefore consider THEOREM 4. Let

it in

property: onto

The functions

(~H~)

t ~]' X ~2"

if

G • U2

~Cc(U2)

e ~i x ~ We choose

L2

C s ( ~ : H2) cs

U2

U1 × U 2

is the centralizer of

U1 X U2.

~(~) ~ e~

~

and in

that contains H2e ~(R).

~I' x ~(R).

be a connected open neighborhood of

sufficiently small so that it has the following ~2

is an analytic diffeomorphism.

3 f ~ c (a.u2)

as defined for

are locally constant on

and let

such that

~hat follows ~thout co=ant. %vi ( ~ H 2 ) e

is a locs/ly constant function on

depenas o ~ y on the co~ected co~onent of

in

G~

the natural map of

G/L2x U2

It is then obvious that for any

~ = *f,2 on U2; we shall use this fact in 3 Cs(~) ~ C such that Cs(~ : H2) = Cs(H9

Clearly

We shall prove that the

Cs(")

are constant on

U1.

We prove first that Os(") e C~(UI ) . Since *f,l ¢ C~(~]') %J f e Co(G- U2) ,

2 c~ is in C~(UI ) for ve S(~2c)~ ~ oCt(U2). ~(~) : ~

Let

i

jU 2 B(H2)e

d~92

(He [92c)

Then it is clear that for any polynomial q on ~2c' the function

HI - r

se~ 2

is C~ on U I. use Lemma 4.9. on ~2c" Lemma.

Cs(H1)q(isy • H1)~(sy • ~ )

Note that ~ is a holomorphic function on ~2c for all B e Co(U2).

We now

Let gut : t e ~23 be a basis for the space of m2-harmonic polynomials

Then the matrix (ut(isy • ~))s,te~ 2 is nonsingular for each ~ ¢ ~i by that S~bstit~ting g = u t (t era2) it follows that for each s e~02, Hl~Cs(Hl) •

5(v){)(sY'Hl) is in C~(UI ).

If % e U 1 and ~#0, the holomorphicity of ~ implies

that for some v, (~(v)~)(sy'Hl)#0.

So the Cs(. ) are in

~,et

C~(U1 ).

i<sy-~,~> u(H1 : H2) =

~, Cs(Hl)e se~ 2

(HleU1, H eeU2)

Then w e C~(U1 XU2) and ~ a constant c ~0 such that

t,~,:~(%) = o "[u ~ ( ~ : ~)~,f,~(:~)d~

(% e ~_, f e C~(~" ~))

2 Bat, by Lemma 3.7 and (i), (u e Is)

So

108

i05

/~(~;~(U~l):H2)~f,2(H2)d~ 2 = / U2 As

f

~(~ :H2)%(iH2)~f,2(H2)d~2 • U2

was arbitrary and

~(iH2) =~(iy -I .H2) , we obtain

V(~;~(U~I) :H2)=~(iy -I " H 2 ) w ( ~ :H2) vj,vj' 6 S(~lc

So we can find homogeneous elements deg(u)

) with 0 <deg(vj),deg(v; )


As the c s are polynomials, Lemma 4.1 ~

( % cU1)

is the constant

c' (se~2). y-ls-ly

2.

Behaviour of tempered invariant ei~endistributions on CSA's.

Let

T

be an invariant eigendistribution defined on a completely invariant

open subset of q.

Our aim is to make a closer study of the function defined by

in the special case when Let

~

T

is

integrable function on

£.

Let

(~c,~e)

tempered distribution on

~,

tempered distribution on

~.

We may assume that

F

I~

3

be a CSA,

c ~' ~

Note that

~conj = cv

2 f d~ ] ~

and

F

an invariant locally

= ~ n ~', F~ = F ! ~ .

belonging to a positive system.

then for some integer

a finite set

F~

~

~ = G •~.

by Proposition 2.16 and that

T

tempered.

be an invariant open set

the product of roots of

assumption on

and so must

on

imply that for suitable constants

i<sy'~,H2>

ss~ 2

H2

the constant term of the corresponding

(~(U~l)cs)(~).

the differential equations for

LEMMA 5.

= 0 •

j

all vanish by Lemma 4.1.

(3)

(Hi6Ui~ U S I s )

F~

vmF~

such that V f

Here d~ is a Lebesgue measure on ~ and ~ f s C ~ ( ~ )

~

be

defines a

defines a

c = + i. c C~(~),

sup'Ef,

109

Let F

is locally integrable on

for some constant

£ c Diff(gc)

~

m ~ 0,

If

is defined by

By the

~

zo6

If

W

is the normalizer of

e Cc(G )~ If

~eCc(~)C!~)~

a unique f or

in

G~

then

= 'w,-l~seW ~s ~

f~ e

some

such that constant

We now estimate image in

~

invariant under the action of

G•

is W-invariant.

on

G*;

c ~ 0,

supp 7 *

We now choose

such that

= i.

then it is clear from Len~la 3.1 that

f~(x • H) = 7*(x*)~(H) ~f~ = c~ V

sup lEf~l for

contains

~f W

E e £.

and let

x e G, He 2~;

and

~ so

8 eCc(2~).

Let 7

V

C a G

be a compact set whose

be the function

x ~ T*( x ~ )

on

O.

Then -i

(~)

supl~%l ~

~up

I%(~'~;~)I =

sup

xeC,He2~ F~rther~

E

I~

-i

(~;f)I

•

xcC,He2%

is contained in a finite dimensional a finite set

So (4) leads to the following:

G-stable

subspace of

£' c Diff(~c)

Diff(~c).

such that V

e Cc(a~),

(5)

-i

r supl~%l ~ Ee£

If~ (~;~')1

s

sup

E'e£'

xeC,He~ -i

We shall use the work of Section 2 to estimate the derivatives FH

(He ~')

be the linear map of

Then ] an integer

q~0

and

@ ® S(~c)

into

S(~e)

(~;E'). Let

fB

defined in Theorem 2.5.

a i eP(~c) , u ie S(~c) , vie S ( % )

such that

(He ~,)

rH{~(H)-q~ai(H)h®ui] = ~ EH' being the local expression of -i

E'

at

H.

So

f~ (H;E')=F(H)-q i~ai(H)7(x;vi)~(H;ui). Using this in (5) we find integer q ~ O and Pie P(~c) , w i e S(~c) (lfii!t) such that V 8 e C:(~),

D

supl~%l ~

Ee£

~

l > 0 V

~ ~ fl' H2 e F 2

In particular, if ~2 is fundamental, f2 = ~2 ~ and if we write Cs(fl)=Cs(fl:~2),

Cs(r l) W o ~ sy • ~ ~ ~2"

111

108

Lemmas 5 and 6 imply the first assertion.

If

92

is f~ndament~, 92= 9~(~),

and ~m<sy -H1,H2> ~ 0VH2 ~ 92~sY .H~ ~2. COROLLARY 8. on

bC ~ be a fundamental CSA and

Let

G • b'

unless G • X meets b. c is not elliptic~ then ~X = 0

X e ~'

Proposition 9.

Let

~ e Ip

In particular, if

Xe ~'. b

Then

~

vanishes

is of compact type and

on the regular elliptic set.

be as in (i.I).

Then 3 a constant

C>0

such

that i

1

(x,Y~ ~,)

I~(x: Y)l ~ el ~(x)l-~l ~(Y)l -~ where

~(X: ")

is the a n e m i c function defined on ~'

invariant open set and

T

locally integrable function any

CSA ~

r> 0

of

~

F.

For

and any norm

nc~

T

be

an

to be tempered it is sufficient that for

II'II on it, there should exist constants

C>0,

such that

I~(~)IIF(H)I If

by ~X" ~et

an invariant distribution on it defined by an invariant

~= ~,

and

T

~ c(1

+

ilHII) r

( ~ nn

~,)

is Is-finite, these conditions are also necessary.

By Theorem 7, we have the estimate

ICs(H1 : H2)e

valid for

sc~2, ~ c F 1 ,

H2eF 2.

first estimate.

SUppose that

tempered on

If

an

~.

F

Zpje J

and 6 imply that Re hi(u) ~ 0 V u F

described. Write

l[(Hi)12= Iffi(Hi)l ½, we easily get the

As

~= ~

~ Ics(F1 : F2)I 1

and that

T

is invariant, Is-finite , and

is a connected component of

exponential polynomial

~, let

i<sy'~,~2> I

on

c F.

r.

~',

As tF=F

and

F~=FI~', FgF

for all

is

t ~l, Lemmas 5

This gives the last assertion.

For arbitrary

be an invariant locally integrable function satisfying the estimates Let

~i (i ~ i ~r)

be a complete system of mutually nonconjugate CSA's.

Fi = ~9 i' ~f,i = ~f'~i' Fi = Flgl A ~.

The temperedness of IK~V~ARK. Let

T ~= ~

Then 3 constants e i ~ 0

such that

is now immediate from Theorem 3.9. and let

F

satisfy the above condition on each

CSA.

The argument above shows that the function IF1 also defines a tempered distribution.

So, by Lemma 3.8, ~ an integer

m> 0

such that

( I + H .11)-m IFId~ < ~. In particUlar P

T(f) = J Ffd~

112

(f ~ ~(~))

z09

3.

The class

~

of invariant open sets

From now on for the remainder of the section we shall assume that CSA

b

of compact type.

that

~(u)T = ~(iX)T

T

be tempered,

has a

We shall study tempered invariant distributions

(ue IS) ,

X

our results we look at the case let

~

being elliptic and ~= 61(2,]R).

c g'

such

With notation as in Section 4, n°5,

invariant and 5 ( ~ ) T = k 2 T where A / 0

tion on g' defined by T.

T

In order to motivate

is real and F, the f~nc-

Then

iSF( A(X - Y)) = bleik£ - b2e-i}~ ( ~ e l~X), tF(tH) = alert - a2e-~t

(t > 0)

b I + b 2 = aI + a 2 Since T is tempered, Lemmz 5 implies that for ~xne m > 0 tm(aleht-a2e-~t ) is tempered on (0,~). So, by Lemma 6, al=0 (resp. a2=0 ) if h>0 (resp.~ 0

Z,

C

and reductive in

(~c,bc)

in

Pb"

Fix

he~'.

5(u~)T = ub(h)T

~.

If

WZ(b )

If

V

=

T

is

u c Is,

is the subgroup of the

the space of distributions o s (s ~ W(bc) )

~,

be an open neighbor-

~ = ~[C,a]n ~ (cf. (6)), ~ , b

~8 such that

If

b

and let

Write

has a unique such extension to all of

Weyl group of

such that

T

defined

are constants such that

Cts = Cs kJs e W(5c) , t c WZ(b), ~ s unique distribution T on ~ such that (i)

T is tempered and Z-invariant

(ii) ~(u~)~=ub(~) (u~I s) (iii)

if F is the analytic function defined by T on the regular set ~

-i F=~,b

~ ~(s)e e s~ seW(be ) s

on

of ~,

bA ~ .

Since ~8 e ~(~) by Lemma ii, the first two assertions follow from Lemma 17 and Theorem 15.

For the last assertion, let sI = I ,

sentatives for

Wz(b)\W(bc).

(16)

~ =

has the required property. COROLLARY 19 . ~ ,-Ib Z s S ~ b c )

Let

£(S)cseS~

T

s2,...,s m be a complete set of repre-

Then 7 TM l~j~m

E(sj)Csj

T~ sjk

It is the only such distribution by Theorem 13 . be an automorphism of

be invariant under

120

~.

~ Then

leaving T

b

invariant ~nd let

is invariant under

~.

117

Let the function ~(. : .) be as in Section i. ~(-:.) on b'×b'

We take ~i = b and determine

Then by Theorem 4 we can find locally constant functions bs(. )

on 3' such that

(17)

~%(:~)~%(~)~(-i~ : H) =

~ ~(S)bs(~)eS:~(H) s cw( ~c )

(H ~ b', ;~ ~ ;~').

Let S l = l , s2,...,s r be a complete system of representatives for W(b)\W(be). then clear from (17) that ~.(9~)~ .~. = ~ ~ .< e(s .)bs (~)Ts. A. -in h

±

~ j

in (17) and noting that ~_iSkH~ ( l < k < r [ a ~ d

r

j

j

TsjT~ ( l < j < r )

It is

Replacing 9~ by Skh

span the same space,

we obtain the following: THEOFJ~4 20. functions on

3'

~ an invertible matrix with inverse =

TKEOK~M

21.

(ajk( " ) ) l < j , k < r of locally constant

(aJk(.))l 0

(20)

such that

l;~(x)l _< el ~(x)l

-½

(~e~,, x~ ~,)

The function 7[b(Fkl5') is a linear combination of the e th (t e W(bc) ) with + i coefficients.

This shows that the bs(h: .) are integers in view of Lemma 14.

the bs(h : H) are locally constant in h follows from Theorems 20 and 4.

That

For the

last assertion we argue as in Proposition 9. Fix a connected component Define, for (21)

3+

of

~'

and write

h e CI(~+), the invariant function ~+ =

F7% 3+

bs(~+)=bs(h ') on

B'

~+).

by

~(S)bs(~+)eS(X°Y) on ~'

That this is valid and leads to a G-invariant locally integrable function is clear from (19) , (20), and the limit relation +

(22)

F~ (x) =

li~

~,(x)

(xe~')

3 + 9l' ~X

Proposition 9 together with the remark following it imply that the distribution ~+

5+ B

is well defined and tempered.

Moreover, by (22), we see that

121

l17a

3+

3+

~(u)T~ : ub(~)T ~ (24)

3+

F~

=

~b

-i

(u c Is)

~~

~(s]e s~ "

on

b'

"

sew(b) a constant

The estimate (20) and the limit formula (22) then give the following: C> 0

such that +

(2~) We must remember that let

i

IF~ (X)I ! Ct{(X)I -~

~ = ~I(2,]R).

element of

3

Then

S'

3+

T~

3+

in addition to

has two connected components

~

i0 ~ ( e ( X - Y)) = 1

F+ 0

~. and

that can belong to the closure of both of them.

distributions defined above and

(26)

depends on

(X~ ~', ~ Cl(~+)) For instance~ 0

is the only

T -+ be the 0 the corresponding analytic functions. Then

(0~]R X)

t~(tH)

= +1

Let

(t>0)

The formt~lae (26) also reveal that the uniqueness theorem no longer holds 3+ h, i.e., if h is singular, T h is no longer determined by the 3+ restriction of F h to b'. for singUlar

The results of this section are due to Harish-Chandra. transform theory in n°s i and 2 see his papers in n°s 5-5, see [ii].

122

[2], [3];

For the Fourier

for the theory developed

Ii8

8.

Let

G

and

G

The limit formula f(O)=~($(~b)gf,b)(O)

be as usual.

The main result of this section is a formula that

enables one to calculate, for any element inVariant integrals to the group. serve

Cf~..

f e $(G),

the value

f(0)

in terms of the

Using the exponential map one can carry over this result

As we shall see later~ the formula thus obtained

on the group will

as the starting point for the proof of the Planeherel formula for i.

The distribution

J~

Let

~ c ~

P,

be a CSA;

a positive system of roots of

~]~eP ~; ~Y~=If~,P=1]~ep H ; ~=W(~c) , tion of Sections ~ and 7.

For

the Weyl group of

f e ~(~),

Cf,~

a connected component of

(1)

~'(S),

~D,r,i0:f ~

then for any

tim

(~c,~e); ~ = ~ , p =

(~c,~c).

We use the nota-

is as in (3.2).

3.26 and their corollaries we then have the following:

G.

From Theorems 5.23,

D c Diff(~c)

if

and

F

is a

H 0 ~ CZ(F),

(f ~

(D,f,~)(~)

~(~))

F ~ H~H0 B;

is a well-defined tempered invariant distribution on all singular imaginary roots ~;

B,

denoting this extension by

(f ~ s(B))

D~f,~

then D~f,~

if further

Ds~=-D

for

extends continuously to the whole of

again, the distributions

are well-defined~ tempered and invariant for each

f ~ (DCf,~)(H)

H c ~.

In particular3

if we put

(2) then

J~(f) = (~(~)¢f,~)(0) J~

(f ~ ~(~))

is a well-defined tempered invariant distribution on

L~MMA i.

The distributions

TD,F, Ho

are l-finite.

~.

In particular,

J~

is

I-finite. If

E ~ Diff(~c)

and

E0

is its local expression at

H0,

we have

TE,F,H 0 = TE0~F,H 0 Let

ue I s

and

E = D o (~o i).

If

Eu, 0 6 S(~c)

Eu

is the local expression of

H0 , ^

(3)

^

5(u)TD' r'Ho = Tsu,0' P'HO

On the other hand,

EU, 0

D0c S(~c)iS the l o c a l

is also the local expression expression

of

is the algebra of all ~0-invariant polynomials on if

L b = {q : q ~ I~, (8(b) o q)0 = 0}

D

of at

D O o (To i) H 0.

where

So ~

~c' and, for any

if

b e S(~c),

where the suffix indicates the local expression

123

at

119

at

H 0,

it is sufficient to prove that

and let

d = deg(b).

dim(l~/Lb) < ~

for

every

b.

A simple calculation shows at once that for any

Fix

b

q c I~,

5(b) o(q-q(H0))d+l c (q- q(H0)) .Diff(gc). Thus,

Lb

contains

(q-q(H0))d+l

it is clear now that LEMMA 2. If

9

If

for any q e I~.

Since

19

is finitely generated~

dim(Ig/~) < =.

9

is not fundamental,

J~ = 0.

is not fundamental~sthere is at least one real root, sa~v ~.

g = 5('6~9)~f,9. We claim that

g ~ = -g;

prove the claim~ we observe that

s

this implies at once that comes from the group

G

Write

g(0) = 0.

To

and so~ in view of

S

S

Lemma 3.4 and the relation Z E ~ = - ~ ,

we need to show that

(eRF~) ~ = SR~9,

i.e.,

S

that

£R~ = -ER"

If

system of roots of

PR

is the set of real roots of

P~

S

7, n°3.

So

that

Let

where

9

J9 = ~5 ~ 5

(~(~'9)~f,~)(O)

PR

is a positive being as in Section

S

FR~=-FR

LEMMA ~.

then

(8e,9c) where 8 is the centralizer of 91~ 91 FR=~BcpR B.

be fundamental.

But then

Then there exists a real constant

being the Dirac measure of

= ~(0)

S

gR~ = (~R/IFRI) ~ = - ¢ R .

[

¢

located at the origin~

such i.e.~

for all f~g(~). ^

It is enough^ to prove that By Lemma i

J~

grable invariant function F

is constant on

~

that

(resp. F) y ' ~ c =9c ,

stant

a # O,

is a multiple of the Lebesgue measure on

~.

F

on

B

that is analytic on

~'.

We must prove that

g'

First we show that let

g~

is invariant and 1-finite and so it coincides with a locally inte-

F

is locsdly constant on

be a connected component of and write

~

= ~9 °Y'

there is a constant

~'

~ = p o y.

c / 0

B'

Let

(resp. As

~ c @ 9').

Fix

~ conj = a ~

such that for all

be a CSA

and

y e Gc

such

for some con-

g e C c ( G .~),

~G.P g(X)dX = c~_ F[(~)¢g,[(~)d~ . F If

~(H: .)

B'

and coincides with

a s (s e~)

(H c F)

is the locally integrable function on ~H

~

that is analytic on

there, we can choose, by virtue of Theorem 7.4, constants

such that, for all

(H,~) e F × 7,

e

s e~D

s

So ~ if

124

i <sH,y.~>

120

s E~

where we write

where

s

for the constant value of

a' = a i d ~ s ~ ( S ) a s

is locally constant on

and

d=deg(~i).

SR

on

r.

This shows at once that

In other words,

F=a'

The next step is to show that the constant values of ed components of continuously to

(~c'

e )'

we

~'

are the same.

~.

If

I ~ ~

is any CSA and

F=a'

on ~,

~,

then

~=F

F

~

~.

J~ =

is real. ,First we note that

If we write

P

~

on

i s r e a l on

~',

~8

g

~.

~

Proof that

such that

Thus

~

~

for

g

of roots of

also.

Then

m.

and

We need to check

= ~ ~,p-X-, F~* = 7T~,p-X-,

~f~

is r e a l ,

is real.

Thus

I~

and if

(Be,be)

and we define

~b,~b

Let

and let

6

b c !

are imaginary. as usual.

Let

P

c ~c.

Let

Pk

c > 0

it is

be the constant

t = IPI,

then the corresponding root subspaee

(resp.

T,

is a positive system

lo-b, dX)dX = (-z)tc lb %(~)¢g, b(H)dE "

or

This

be the corresponding

is a CSA of

(4)

(go,be),

sign.

i s o f compact t y p e .

be a C~rtan involution

~= i + ~

gcCe(~) ,

is a root of

~{~

rk(z) : rk(~).

such that for all

~

6'

extends continuously

t h e case when

when

rk(~)=rk(1)

with

on

are as before and

b y a d e s c e n t argument based on S e c t i o n 3, n°7 •

c = center(g)

All roots of

(~e,be)

J~

has CSA's of compact type.

that is the identity on

Cartan decomposition. one

c fc

~ = 7-1V F

~,~ F

> 0.

being invariant

i s nonzero and t o d e t e r m i n e i t s

(-l)½dim(z/i) ~ > 0

We assume that

If

J~

is a real differential operator on

pC6nj

We c o n s i d e r f i r s t

The g e n e r a l case i s reduced t o t h i s

extends

is a constant Now~

for some constant

~(~) °7 h

proving that

Our aim now i s t o p r o v e t h a t

F

~'

for the positive system

be done i n two s t a g e s .

2.

F

on the various connect-

Therefore,

will

on

on

Q.

In fact, if

t en

(~({~)°~)¢f,~

So

is a positive system of roots of

nor on

~'.

= a'

is constant on

At this stage we know that that

on

~=~/-18(~)(a'TT~)~

proving that

Q

~

and 1-finite, there is a continuous function We claim that

F

~(T~I,Q)(T~ I ,Q)

7=

Obviously this constant depends neither on

to

G.F.

Clearly it is enough to prove that

know from Corollary 4.7 that

(cf. Section 6).

on

~'.

Pn)

be the set of all 125

~eP

g~

is either

for which the first

121

(rasp. second) alternative holds. (5)

Put

q =~dim(g/~),

Then

q = IPnl

dim(~)

amd

amd

m : IPkl

dim(s )

obvious that involution

~

of

are integers

are both

q = ½dim(%) ~

m=~dim(I/b), ~ O.

Moreover

~ dim(b) mod 2.

where

equals

{@

~ = dim(b). dim(~)

is even while

From the definition of

is the subspame of

~

q

it is

on which a Cartan

identity.

We shall prove in Lemma 8 that

(-l)q~

is

> 0.

Before doing that we wish to

show (Lemma 4) that the results of Section 7 already imply that

~ ~ 0.

The argu-

ment used for this is however not delicate enough to determine the sign of

~.

The

proof of Lemma 8 does not depend on Lemma 4, and is self-contained. L~A

4.

g

is nonzero.

Suppose K = 0. By Theorem 7.20, ~ locally constant functions ak on b' such that, if ~H is the linear f~/nction H ' ~ i ( H , H ' } on b,

TXH: %(H)

(He b')

1_ 0 on ( ~ Mc ~', M+n

Yen

and h ( M + Y ) ~ 0 .

and hence M e G

• ~'

Then h ( M + Y ) ~ O

~')Xn.

In fact, suppose

and hence M + Y c G "

~'

But N . M =

Consequently the centralizer of M in ~ is a CSA of

that is conjugate to ~ under G, hence under Z by Lemma i0. then (_l)~im(n)Q(M)l~ > 0 by Lemma 9, proving our claim.

Thus M e Z • ~';

but

So

(-i)q ~=(-l)q~f(o)= (_l)q~ ~ • (-i)~d~m(") (~(~)gf)(o) > o. i . ThUs we have extended Lemma 8 to the general case.

We also observe that if C c c

there is a corresponding formula for

it is obtained at once from

the preceding on replacing

f

by

fc

(~(~)¢f)(C); where

fc(X) = f(X+C).

We have thus proved

the following theorem. TH]~OB]94 ii.

Let

~c~

be a

CSA

and for any

Ce c

(the center of ~) let

J~,c(f) = (~('~'~)¢f,~)(c). Then

J~,C

J~,C = 0.

is a tempered invariant distribution on Suppose

~

is fundamental.

Then

J~,C

~.

and

m~,

9f,~

J%,c : ~5~,C $~,C

is the Dirac measure on

~

located at

C.

q

(-i) ~ > 0 where

q~

is the integer defined by q@ = ½[dim(G/K) - rk(G) +rk(K)]

K

being any maximal compact subgroup of For Theorem Ii see Harish-Chandra

is not fundamental,

and not on the

and there is a real nonzero constant

such that

where

~

depends only on the normaliza-

tion of the invariant integral used in the definition of positive systems involved in ~

If

G.

[8 ].

131

Then

9.

Invariant differential

i.

Formulation

operators

that annihilate

all invariant distributions

of the problem

The results proved so far are essentially what we need for studying invariant analysis on a real reductive group, with one important exception:

we have still to

develop the technique for carrying over the results from

~

to

going from

G

is noncommutative

~,

~

regarded

of examples

to

G

is a highly nontrivial one because

as a vector group, is commutative.

However

that the theory of invariant distributions

Fourier analytical

questions on the Cartan subgroups

one

The problem of while

s e e s from the study

on G can often be reduced to

of

which sme abelian and

images of the Cartan subalgebras

It is therefore

quite reasonable to expect that there is a close relation between on

G

and that on

~,

~

G

are homomorphic

invariant analysis

of

G.

under the exponential

map.

and that this relation can be studied

through the exponential map. To illustrate what is involved, is an eigendistribution erential operators

on

iant distribution

T

let

G.

on

~

that are suitably normalized of ~

8

in the algebra

and D

f of

f/~

do not have constant coefficients,

8~ an element

canonically

for all invariant and ~,

D T

D

of

determined by

~(Is) D

~ ~ ~'

and

such that

D ~ D

theory.

0.

3.

T

is

operators

S i nee

However,

T

coin-

and this function is of ~

is a CSA of

Consequently we can attach to each element

would become an eigen distribution

the help of the preceding

of

where

DT=DT

on

~ n B'

~(Is)

is an isomorphism.

have the same action on all invariant distributions

which asserts that this is so.

that diff-

the theory that we have de-

as the unique element of

~ e C~(~);

G

to an invar-

f~ of

immediately.

~ ' = ~ n B',

component of

is an exponential polynomial.

T

®

of differential

puilbacks of the elements

cannot be applied to the study of

on each connected

~

on

of biinvariant

map to pull back

of the algebra

cides with an invariant analytic function on the form

~

defined on some invariant open neighborhood

eigen distribution for all elements

the elements

be an invariant distribution

We use the exponential

an

veloped on

O

for all the operators

for

$(Is) ,

D

is in fact

such that

D@=Dq0

If we know that on

~,

then

D

D T = DT

on

and so can be studied with

The main theorem of this section is Theorem 2 3

We shall now proceed to outline the main steps of

the proof of this theorem. Let

~

be a completely invariant open subset of

iant differentiaA Let on

T

operator on

~

such that

be an invariant distribution

dim(~).

~.

~

and

D

an analytic invar-

for all invariant

To prove that

DT= 0

Y' = O. i e [i,...,£]\ F.

130

Write

h t=exp(-tH)

(t > 0).

while

E%(h t • Y')=E%(Y')

for

the orthogonal projection and so

E%(h n • Xn)

EFY'=EFX

enoch,

and

llht

3.

Then

E%(h t " Y') " 0

~eP(F) O-P(F).

so,

for

• Y' [[ < a,

So

B " ~0 +~'%c~P(F) ~h"

tends to the same limit as

t -+~

for

h t • Y' ~ EFY'

B~t if

%cP\P(F) where

EF

is

l ~ iP(F) , %(log hn) - 0

E%(Xn) , i.e., E%Y'= E%X.

llht .m'll-lIErxll S [Ixll< a.

t~+~,

i .e.,

as

Hence

So, for t large

Y' ~ ~a"

A key estimate

Proposition 2. pendent of

a,

and

Let

a>0.

b, c

(4)

Then there exist

dependent on

xll _< h,

a

m > l , b>_a, c > l

such that for

II H _
0; Let il,...;i v be

a l l such indices

i.

For amy i = i v ,

e h(log ~) ilxhlt = e~(hiz(til_

J~ = c2(1 + IIh "XII ) •

So

LZ* ) + ... +~. ~ (t. ~ -T.*~ )+?\z(tz-T£))IIXht I .(~ +.

(%(i + llh"xIl)) --~ "'+%¢e~(~t l+...h~tpIlx~il _< (l + llh" XlI)-l

e h(l°~ h) IIxhll

_< (l + Ilh. xll)-l Ilh"xll _< So combining both cases,

e h(l°g [) qlXhlI _I

Let

b

U

of

i

with

be a Haar measure on

U = U -I G

and

and

M=dim(n)+l.

F

Then

and

~ a con-

such that b(G(t)) ~C t M

Let A+(t)= G(t)N A +.

Then

that, with ~+(t)=log A+(t)

G(t) =KA+(t)K

and

#(G(t)) = C1

(t~l) and so 3 a constant

CI > 0

such

t El, ~

n

(eZ(H) - e-~(H)) m(~) dH

a+(t) ~P where

dH

is a Lebesg~e ~eas~re on

a and

m(~) e2~(H) ~

m(~) = dim(~)°

lle~ HI/2 ~

But

t2

(He ~+(t))

~cP so t h a t

e ~(H) ~ t

and

0!Gi(H)~Zogt.

Thus

b(G(t)) 4 CI tdim(n)( l°g t) 4 L ~ M A 7"

~ a constant

one can find a finite set (i)

(ii) (iii)

IFtl

Ft = G(t)UN £. So

Then

~V

L~MMA 8. b ~ a , m~_l

V

of

i

Select

For

t>l_

let

such that

t El,

£tc£

such that

UCG(tl).

VcU

and

(w~Ft).

IFtl.

VV-I ~ £ = {i]. Then So

m b(G(tt~)) ~ c t l 2M t M. ~a(t)

be the set of mull X e ~ a

be as in Proposition 2 and a

tI > 0

It remains only to estimate

are disjoint and~G(ttl)UCG(tt~)

depending on

a finite subset

G(t)cFtU.

I I T I I ~ t l t V W { F t.

#(v)lFtl

C~_I

for any

as in Lemma 6,

for mULl T ~ F t .

Take a compact neighborhood

Let

M

S CI tM

I}~II~ CI t

FtCG(ttl)N£.

the sets

with the following property: such that~ with

G(t)CFtU

Write Then

CI E 1 FtC£

(t ~ i)

M

as in Lemma 6.

with the following property: such that

139

for each

with

IIXIl < t.

Then

3 a constant

t~_l,

there exists

134

(i) ~a(t)~rt. (u.%) (ii) Irtl _< ct (iii) IIII _< c tm For

t>_l

write

t ' = c ( l + t ) m,

c

being as in Proposition 2.

Then, by

that result and Lemma 7, ~a(t) c G(t') .~ b c Ft, " U ' ~ b Tske

Ft = Ft,

and use Lemma 7.

5.

Estimates for an invariant distribution on the sets

Fix

a > 0

tribution on

and let

~.

~a(t)

Since

the estimates for

T

~a(t)

be as in Lemma 8.

Let

has compact closure

T

thus obtained will vary with

depend polyuomially on

t.

T

~a(t).

be an invariant dis-

is tempered on it.

t.

B~t

We now show that they

In what follows we shall use the symbols

independent of

We shall also need the following easily

an integer

t ~ 1 if

v

and

r(~) ~ 1

choose a constant

f s Cc(~a).

is a finite dimensional representation of such that for any matrix coefficient

c(~,f)

f

T~a

with

> 1

proved fact:

which may depend on

B,s

or without indices to denote constants

G,

then there is

of

~

If(x)l S c(~,f) Itxllr(~) Let

T

(xsG).

be an invariant distribution on

the sets

~a(t)

be as in Lemma 8.

elements

ql,...,q r e S(~c)

(8)

we can

for which

(7) L~MMA 9"

but are

~.

Fix

Then we can find constants

such that for all

IT 0

and let

B ~ i, s ~ i

and

f e C~(~a(t)) ,

supl (%)fl. ll

such that ~f6Cc(~a(t)) , (t_>l, y6 Pt'

l<j i such t h a t ~ f e C c ( ~ a ) ,

JT(fmn) l _< B 4 ( l + n Taking

d:s3+2

)

-d+s 3

and noting, as

for suitable constants

IT(f) I _< %

( iiXii2)Sl+dI(~%)(f))~)l

~ ~p l + l_< j_d7)"

V

given

146

Obviously DSkr

a~d ~

k r-s

=

(0


i, k --

orthogonal group of k r ~c(2r-d-l)(v).

is a tempered distribution on

invariant under the

V

r

V.

It is an analytic function on

Let

r > ~ , {c S, and %=deg([).

3 a constant

Cr, ~ > 0

such that for all

i f ~ moreover

~ 2r-d-l]

If

V \ {0].

r >~

is homogeneous.

Suppose

Then

x c V\ [0],

sup(1 + 11~112)s I k r ( ~ ; ~ ( ~ ) ) l

0 v~e~

S

f~(f)

is a where

(f~e~(u).

#q,

such that q e S 0.

as a topological vector space under the topology induced by

any

x

such that

x~U.

~q(f) =7~ I~(q)flPwdx

V ~ ] e SO •

and

is the open ball with center

and Cl~(~,~(x))EU

"~(x) =

finite module over

Let

weC~(U)

let

(8) Then

and

is a nonempty open set;

We assume that we are given a function

o 0

(12)

] qox(y~(O) I _< c~s(~:) -r

If we now replace

(~3)

such that

~

by

f~x

f(x;~(~)) =

(x c u, y ~ v).

in (i0) we then obtain, ~/f e C~(U)

l~

/-'

z_<j_<m~ s ( ~ , s ( ~ ) ) 154

and

x e U,

~js(X-y;~(O)(f~x) (y; ~(~j))dy.

r,

149

Let

t=maxl<j
l

C C ~0 k~s(X-Y; ~(O)~x(Y~ ~(~J'q))f(Y;~C q))dY 1e(l + i/raini _< j~..J _~ ~ ~j(x) I)-r V x e V". So we may as sume that

Then

f e ~(~',S0,w), ents of

This is in

w = m.

I]xjH=I ( l J j J q ) .

B(~)f

Let

It is enough to prove that if

is bounded on each of the (finitely many) connected compon-

f e ~P(v',S0,w )

and

V+

a connected component.

By changing

some of the

k to -k we may assume that each k is >0 on V +. Since the J J J set where all the %. are >0 is a convex hence connected subset of V', we have + J V = Ix :kj(x)>0, l < j < q } . We apply Lemma 3 with U = V + and

S(x)=~mn(1,kl(x ) ..... %q(X)) (xeU); ClB(x,s(=)) c u ~¢x~u, and i n f a c t l

gives

.

--

--

~(=)_>e(i+l/~(=))

integers b > 0

-r

for

x~U

since

and so

such that SUPxeu g(x)blf(x;~)l < ~

satisfy the considitons imposed on

W

kj(y) _> kj(x) - IIx-yll,

~j(y)_>s(=)

and

157

for

ycClB(x,~(=)).

~(x)_>c.~-r(=)~ for every

~e S.

This

~x~U. Th~s, Now

~/ in Lemma 6 if we take

U x0

and

s

152

arbitrarily in

U.

This proves that

~(g)f

is bounded on

U

for each

~ e S.

The argument just outlined also proves Proposition 8. V+

Let

Bl,...,b N

be nonzero real linear functions on

the set of points where all of them take positive values.

such that for some constant

c >0

and integer

Let

V

~P(v+,So,W)~ ~(V+),

be

r~O,

W(x) ~ c(1 + 1/min bj(x)) -r 1Sj_l

with the following property:

given

q £ P,

lq(x)l i l + I % ( 0 1 + ' ' ' + lqm(X)l kl~...~km~P

that

we

such that

such that

qkj = El_l, then we have

q(~) = -%(~) - q2(~)q(~)-l ..... qm(~)q(x)-(m-1) which implies (18). Proposition i0. feg(V') of

Let notation be as in Proposition 7.

if and only if

g(V')

~([)(qf)eLP(v)v[eS0,

is already induced by the seminormS

Suppose

f~C=(V')and

Proposition 7 we find that deg(~)

~/~'eS

with

qfeM=(V')VqcP deg(~')_.

i.e.,

ind(~)~

= dim(~R) -

~I

Let

8

sponding C~rtan decomposition,

It is inde-

@

is a Cartan

is the direct sum

8.

is positive definit%

Since

ind(Q I~i )

is

dim(~i)-

We say t h a t

~

~ is fundamental

(resp.

Finally, we say that

~

(~c,~e)

a @-stable CSA.

equivalent : ~

where

QI~I O ~

be a Caftan involution of and

n £i"

by

(resp. maximum). a~l roots of

is fundamental

163

is

Yhis said; we come to

we have (26).

then - ~ _!< i n d ( ~ ) _< ~.

4:dim(~),

(i)

Then

being the Caftan decomposition associated with

We now define the index of

Iwasawa)

8(~) = ~

is negative definite and

dim(~ I O ~)- dim(~ I O ~),

f

(q = +1)

We may assume that

that fixes

while

and

can be represented as

is then defined as the integer

the proof ,of Proposition 14.

]R

such that in terms of the corre-

pendent of the choice of the basis used in its definition.

If

Then

is of

are pure imaginary. 9,

~ = ~+ ~

the corre-

Then the following are

(ii)

~ N ~

(iii)

is a CSA of

(~c,~c)

T

has no r e a l r o o t s . form a single conjugaey class.

All fundamental

CSA's of

subgroup of

corresponding

G

~

to

single conjugacy class under

~,

If

K

is the analytic

then the e-stable fundamental CSA's form a

K.

For proving this we may without losing any generality assume that simple. rk(~)

If

~

is a @-stable CSA,

since

(~c,~c)

~ N ~

has no real roots.

Hence we can find

ind(~)=dim(~)-2

is an abelian subalgebra of Then no root of

He ~ N ~

is the centralizer of

H

such that

in

~,

dim(~ n I)

~.

So

(~c,~c)

Hc ~'

while

(ii) ~ (i).

is semidim(~N~)
0 the product being over the p o s i t i v e roots of the symmetric space with the multiplicity of the root function

f

~.

to be rapidly decreasing is to require that for every

sup

m(~)

as

Consequently, the natural definition for a s_>0, h e A +,

If(klhk2) l = O(e -p°(l°gh) (1 +lJlog~l]) -s)

k I ,k2eK Using the spherical functions

~

and

~

this can be rewritten as

If(x)l = O(~(x)(l+~(x))-s) ~rther,

C~

for a

function

f

to be in

C(G),

(xeG)

we require both its left and

right derivatives to possess this decay rate at infinity on

G.

C(G)

becomes

a Frechet space in a natural manner. Section 9 contains the proofs of the basic properties of the Schwartz space and its dual, the space of tempered distributions:

the natural imbedding

Cc(G) c~ C(G)

(Theorem 9.2);

and the density of

like result that which the L2-norms of

C(G)

in

C(G)

is precisely the space of all functions

ll(l+~)r(afb)ll2

~) (Theorem 9°9);

Cc(G )

are all finite

f

the Sobolevon

G

for

(r>0,a,b arbitrary elements

the criterion for a positive Borel measure to be tempered~

i .e., the theorem that a positive Borel measure only if it is slowly growing in the sense that

183

~

on

G

is tempered if and

f

E(l+o)-r d~ < ~

G

for some

r>0

function

f

(Theorem 9.11);

the important result that a 8-finite K-finite

defines a tempered distribution on

G

if and only if it satisfies

what we call, following Harish-Chandra, the weak inequality:

namely,

If(x)l ~ const. ~(x)(l+q(x)) r for some

r~ 0

(Theorem 9.13)~ and finally~ that

C(G)

(x eG) is closed under convolu-

tion under which it is a topological algebra (Theorem 9.18).

We note that both

in the case of the weak inequality as well the rapid decrease described earlier~ the function

~

appears in the first power, and that the difference lies only in

the exponents of

(i +~).

At this stage we know that the K-finite matrix coeffi-

cients of an irreducible unitary representation whose character is tempered~ satisfy the weak inequality.

Since the characters of the discrete series representa-

tions are easily proved to be tempered, this proves that the corresponding matrix coefficients satisfy the weak inequality (Theorem 9.15). Given a psgrp M I =MA.

Q

of

G~

let

Q=MAN

be its Langlands decomposition;

let

Then there is a rather close connexion between harmonic analysis on

and that on

M I.

This connexion is a

analysis and dominates it completely. f(Q) [ C~(MI)_

G

leitmotif that runs through all of harmonic For example, for any

f e Cc(G)

let

be defined by

#Q(lOg a ) < f(Q)(ma) = e Then

f

f(Q)

,~

f(man)dn

is a continuous map of

for the dual map taking distributions then

T(Q)

finite)

mI

•

on

into HI

is invariant (resp. S-finite) whenever being the Lie algebra of

turns out that there is an integer r>0~

C (G)

one can find a constant

--

Hl

Cc(Ml).

If we write T i~ T(Q)

to distributions m

aeA)

T(Q)

on

G,

is invariant (resp. 8(ml>

(Proposition 6.31 ) . 9]Irthermore, it

q~ 0

C >0

(meH,

with the following property:

sueh that ~ / m { H ,

for any

a£A

r

ePQ(log e) fN ~(man)(1 + q(man))-(q+r)dn ~ C r ~l(m)(l +~(m))-r(1 + a(a)) -r (Theorem 9.23).

f I~ f(Q)

So the map

above can be defined at the level of the

respective Schwartz spaces and is continuous; tempered distributions on

MI

the dual map

to tempered ones on

G

Sections i0 through 12 study the invariant integral on this is done on

Cc(G ).

T ~ T(Q)

then maps

(Theorems 9~24, 9.25). G.

To begin with

In analogy with the corresponding theory on the Lie

algebra we proceed to define, for each

CSG L

f ~ 'F

f,L

184

of

G,

a map

( f ~ C~(G)) U

- -

(ef. (i0.4)).

Its elementary properties (Propositions 10.2 through 10.4) such as

continuity; symmetry~ behaviour under differential operators from those of the invariant integral psgrp where

A = LR, then,

g I~ 9g

on

~ a constant

B.

c~ 0

Furthermore, if

8

Q=MAN

(h) ?(a),L

(Proposition 10.6).

As

LI

(h~L')

is a compact

CSG

of

the study of the invariant integral to the case of a compact However, it is not sufficient to work with tegr~l has to be extended to case when

L= BCK

C(G).

is a compact

is a

such that

'F~,L(h) = o'F~, V f eC~(G)

resemble

C~(G)

M

this reduces

CSG.

only; the invariant in-

It is certainly sufficient to consider the

CSG

of

G

and prove that for some integer

q>0,

sup 1'I~(b)1 [ "(xbx-1)(l+~(xbx-l))-q dx
_0,

with

A=L R

Thus, let onto

G .

L

then enables Us to extend these results to an arbitbe a e-stable

we can find a constant

~ I '/~:/h)/Z~+(h) I .

(Theorem 12.3).

q>0

C >0

G = G/LR;

x ~ x

the natural

with the following property:

such that for all

for

haL'

r

.

.

~-(h x ) ( l + ~ ( h

-(q+r)

x ))

.

dx

_< C r ( l + ~ ( h ) )

-r

This allows us to define

'F whenever f e C(G); 'F will f,L f~L 'F will be a continuous map f,L Of course; in the previous ease, when L = B ~ C(B')=8~(B'). In

be in (the Schwartz space) (Theorem 12.61 .

CSG;

Then 3 an integer

C(L')~

and

f ~

partieular, the invariant meast~res on the regular eonjugacy classes will be tempered (Corollary 12.7)~ and the properties of the invariant integral extend to the Schwartz space by continuity (Theorem 12.8). It is thus seen that the estimate (*) is basic to this chain of conclusions. Now, using the differential properties of 'F (f s Cc(G)) and the estimates f,B from classical analysis obtained in the Appendix to Part I~ it can be shown that o

f ~ 'Ff,B is a continuous map of C (G) into with the topology coming from the seminorms

(**)

f ~

[

~ (B')~ C

l~fldx [B']

185

G)

being equipped

(z ~8)

i0

(Proposition i0 .i0).

So it is a question of proving that the seminorms (*~) are

continuous with respect to the topology that

Cc(G )

inherits from

C(G).

This

however is a deep result and proving it is the object of Section ii, where it is deduced from the following estimate:

there exists a constant

c> 0

such that

,j'K 1B0m)dk _< e ~(x) for almost all 11.2).

xcG,

IB

being the characteristic function of

The continuity of the seminorms (**) in the topology of

immediate consequence (Theorem 11.5).

G[B'] (Theorem C(G)

is now an

The proof of Theorem ii .2 is technically

quite subtle and makes use of the distributions

%*

constructed in Section 5-

We refer to Section ii, n°2, where the idea of the proof is explained in detail and illustrated with the example

G = SL(2,]R).

To complete the theory of the invariant integral it remains to study the behaviour of Sl(h0) L)

with

if

Si(h0)

all

'Ff~L(h )

in the neighborhood of an arbitrary point

be the set of singula¢" imaginary roots

~

of

(~c,lc)

h 0 e L.

Let

(! is the CSA of

If Sl(h0) is empty, 'F is of class C ~ around f,L s~ is nonempty, this is no longer true; but if u c ~ and u = - u

~(h0)=l.

~ c Sl(h0) ,

h0; for

then -61 (e

61) o uo e

extends to a continuous function around

'Ff, L h0

(Theorem 12.9).

In particular,

-81 (e extends continuously to all of C (G)

in

e(G)

oqNo e 8!) 'Ff,L

L

(Theorem 12.10).

we need only consider

the consideration to

f

iant neighborhood of

h0;

with

f s Ce(G ).

supp(f)

In view of the density of The results of Section 2 reduce

contained in an arbitrarily small invar-

we then go over to the centralizer of

h0

in

(formula (12.30)) and appeal to the theory of Section 3 of Part I. If the differential operator flexions

s

(~ s Sl(h0)),

u

above is not skew with respect to the re-

then -8 (e

eSl Io UO

),Ff~L

will no longer extend to a continuous function around across the kernels of the

~

(~e Sl(h0) ).

When

h0

h0;

there will be jumps

is semiregular, one can

take a very close look at these jumps because the reduction method brings this down to the case of the invariant integral on

~I (2,1~).

More precisely~ let

h 0 = b e B be semiregular, and let the positive root ~ for which ~ ( b ) = i singular. We can then find a noncompact 8-stable CSGL containing b; L B

are the only e-stable

CSC's

of

G

containing

I86

b.

Then we can find a

be and

ll

sufficiently small open neighborhood property:

F

of

b

in

LDB

with the following

'F extends to a Coo function in an open neighf,L borhood of F in L, and the jumps of 'F and its derivatives across the f,B points of F are, up to a nonzero multiplicative factor, equal to the values of (the

for any

f c C(G),

extension of) 'F at the same points (Theorem 12.11) In particular, f,L if G has a compact C S G B C K , and 'F =0 for all noncompaet CSG'sL f,L (f e C(G)) then 'F lies in C°°(B) (Theorem 12.12). ' f,B C ~°

Suppose

BCK

is a

CSG

respect to the corresponding

of

G

CSG;

and

P

a positive system of roots with

let ~ p = % ~ p H

be as usual.

Then we have

the following limit formula~ proved by reduction to the Lie algebra: stant

c(G) >0,

independent of

P,

~ a con-

such that V f c C(G)

f(1) = (-1)%(G)('~p 'Ff,B)(1) -Sp where

~=e

o~o

e6P and

q

i s the i n t e g e r

½dim(G/K)

(Theorem 12.13).

This i s one of the most fundamental r e s u l t s of harmonic a n a l y s i s on f(x) = (r(x)f)(1), r(x) method of recovering

being right translation by f

x,

from the invariant integrals

G.

Since

it gives a very explicit 'Fr(x)f, B.

If we compare the theory of the invariant integral for

G

with the corre-

sponding theory for a compact group, the possibility of that jumps across the singular points of the two. L,

We saw above that if the integrals

then this does not happen. psgrp

any

A function

being assumed to have a compact

°C(G)

of

C(G),

a cusp form, over, if

'F f,L f c C(G)

are all = 0

for noncompact

is called a cusp form if for

Q=MAN~G

~N G

B

'F may develop f,B is the most striking difference between

f

f(xn)dn : CSG.

0

(x ~ a)

The cusp forms form a closed subspace

stable under all translations (Proposition 12.14).

If

f

is

'F = 0 for all noncompact CSG's; 'F lies in C~(B); moref~L f,B is in C(G) and is 8-finite, it is a cusp form, while groups not

having a compact

CSG

do not possess any nonzero 8-finite functions in

C(G)

(Theorem 12.15). The theory of the invariant integral of elements of

C(G)

is then applied

in Section 13 to study when an invariant distribution is tempered. integer

r> 0

There is an

such that

ID(~)I -~ ~-(xl(l+~(x)) -r dx < G (Theorem 13.2), and the main result in this Section 13 is Theorem 13.1 which asserts that if

G

is an invariant locally integrable function on

fines a tempered distribution if and only if for some constants

187

G,

C>0,

it des>_0,

12

1

le(x)l _< CID(~)t -~ (l+~(x)) -s If we know in addition that

®

homomorphism is regular, then

(~')

is an invariant eigendistribution whose eigen®

is tempered if and only if it is majorized by

i

eonst. IDI -~- on

G'

(Theorem 13.11).

At this stage we know that the distribu-

tions

%.

constructed in Section 5 are tempered; and their Fourier coefficients

%*,8

satisfy the weak inequa&iby (Theorem 13.4). With the conclusion of Section 13 we have a rather complete picture of the

behaviour at infinity of the most important invariant distributions associated with the group

G.

Section 14 complements these results with a detailed study of

the asymptotics of a very different type of object, namely, the matrix coefficients of the simple tempered representations of

G.

More generally, the goal in Section

14 is to obtain a detailed picture of the possibly oscillatory behaviour at infinity of a ~-finite K-finite function on

G

satisfying the weak inequality.

It is technically convenient to work with functions values in a finite dimensional complex vector space module for

K,

U

f

on

G

which take

that underlies a bi-

and which are spherical in the sense that f ( k l X k 2 ) = k I .f(x) .k 2

If we write, for

keK,

(xsG,kl,k 2oK),

ucU

"~l(k)u:k -~,

~'~2(k) =~ .k,

"~= (T1,'~ 2)

O3

and if

G(G :U :T)

is the vector space of all

C

maps

f

of

O

into

U

that

are spherical, ~-finite, and satisfy the weak inequality, the results of Section 14 tell how the members of

G(G : U : T)

behave at infinity of

G.

The entire theory in Section 14 is dominated by Harish-0handra's notion of the constant term of an element of

G(G : U : T)

psgrp;

feG(G:U:T),

TMA=~IKNMA.

Then, given

called the constant term of

lim for all

mcMA;

here,

and for some constant fq

a~ ~

f

along

Q~

along a psgrp. ~

fQeG(MA:U:TMA),

g>0, ~ ( l o g a ) _ > ~ ( a ) f;

(Theorem 14.1).

~

sense:

if

the group

The process that takes

f

Q CQ is another psgrp so that 1 MA (Proposition 6.20), then

t88

of

(Q~A)~

The spectrum of

indeed,

(zf)Q = ~Q(z)fQ (Theorem 14.3).

be a

= 0

c*(loga)~+ °° for each root

is very closely related to that of

Q=MAN

such that

Idq(ma)f(ma)means

a unique

Let

to

(z e 8) fQ

~=*QN

is transitive in the following where

*QCF~&

is a psgrp of

13

f% = (fq).q (Theorems 14.4, 14.6). difference

dQf - fQ.

Finally~ one can obtain rather precise estimates for the To formulate these, we fix a minimsl psgrp

consider only psgrps containing

Q0"

one-one inclusion preserving correspondence with the subsets simple roots of

(Q0,A0)

(Theorem 6.9).

corresponding set of simple roots.

Q0 =MoAoN0 and

These are then the standard ones and are in

Let

QDQ 0

F

of the set

be a psgrp and

E of

FeE

the

Write

~Q(lOg h) :

rain ~(logh)

(h e A0)

~e~F Then there are constants

(t)

C>0, r>0, ~>0

such that

IdQ(h)f(h) - fQ(h) l _< C ~(h)(1

for all

h e C4(A +)

portent, i.e.,

(Theorem 14.2).

IEl =dim(A0) ,

following manner:

for any

C~(Ao)

in

of all

h

If we assume now that

then the estimate (t)

~ e E~ t > 0

CZ(Ao)

standard psgrp attached to

+ ~(h))re -~Bq(z°g h)

let

stlch that

Ft,~

G

has no split com-

can be rewritten in the

be the

"sectorial" subset of

[Q ( l o g h ) > t pQ0(logh), Q~

E\ [~], then there are constants

being the

C>0~ ~>0

such

that (tt)

If(h)-dq (h)-ifq (h)l _< C~(h) l+~t

for all

h e Ft, ~

(~ eE)

(Theorem 14.8).

for some

behaves when

t>0

he C~(A~)

Since

((14.19)),

(tt)

C%(Ao)

can be covered by the

Ft,~ gives decisive information on how f(h)

goes to i ~ i n i t y .

Since

G=KC~ (A~)K,

this is enough

for obtr purposes. In the remainder of Section 14 these results are applied to the study of square integrable eigenfunctions. G

has no @ l i t component.

From (it) it follows that

and only if the constant terms fQ= f

for

Q = G);

moreover, such f,

then

each

G

f

fQ

are

0

f

lies in

L2(G)®U

Q/G

(of course

for all psgrps

and this is entirely equivalent to saying that are cusp forms (Theorem 14.9).

must have a compact

a,b e@~

Let us assume; for brevity of exposition~ that

we can find

C>0

CSC~

If

G

if

f e C(G)®U;

admits a nonzero such

and there exists a

~ >0

such that for

with

l(afb)(x)l < C ~-(x)l+~

(xeO)

(Theorem 14.9). If

f eG(G : U : T)

must be nonzero. fQ~0 sion).

and let

Let Q=MAN

Then, for each

is not square integrable, then some constant term of ~(f)

be the (nonempty)

set of psgrps

be a minimal element of a e A,

the function

189

f ~,W

~(f)

Q

of

G

f

with

(with respect to inclu-

given by

14

f__

(m) = f_ (ma)

Q,a is in

G(M :U : ~ )

f~,~ ~ 0;

that

G

and is indeed even a cusp form; there are

and so, in particular,

a e-stable

(mc~)

q

CSG~

such that

has a compact

~

LR= A

CSG,

aeA

for which

is necessarily euspidal~ i.e., there exists (Theorem 14.10).

and that

f

If we assume in the above

is an eigenfunction whose eigenhomoi

morphism is defined by a regular element of sponding to the that

f

cients

CSGBCK)~

is in ®b*,#

L2(G) ® U

(b

is the

CSA

corre-

then the above theorem leads easily to the conclusion (Theorem 14.12).

of the distributions

forms (Theorem 14.15).

(-l)2b *

®b*

In particular, the Fourier coeffiare in

L2(G)

and are in fact cusp

This~ as we had mentioned earlier, is a major stage in

Harish-Chandra's construction of the discrete series. At this stage, everything needed for the determination of the discrete series is available. procedure.

Section 15 does this by essentially following H. Weyl's classical

The asymptotic theory of Section 14 already shows that

discrete series if and only if it admits a compact G

has a compact

for any

b

eB

CSG B C K , ~

and let

~2(G)

CSG

G

has a

(Theorem 15.7).

be the discrete series of

Suppose G.

Then,

the distribution

(-1) q ~(b )%~ is the character of a class

(~(b*) = sgn~. (logb* +6)) w(b*) e ~2(G);

the map

b* ~ ~(b*) from

B

to

82(G )

is surjeetive;

and

~(bl) = ~(b2) ~ b2 = Sbl ~s~-~ f o r some s e W(G~B) (Theorem 15.8). once f o r a l l ,

and

is a constant

d(w) (~e62(G))

c(G)>0

Finally~ i f

dx

i s a Haar measure chosen

are the associated formal degrees~ then there

such that for all

b

eB

d(~(h*)) = e(a)l w(o,~)l d(b*)l ~ (log b* +~))1 (d(b*)

is the degree of the irreducible character

The constant of the classes of

G.

e(G)

~(b*)

b*).

This is Theorem 15.9.

appearing in the above expression for the formal degrees depends of course on the normalization of the Haar measure

Shppose we choose the Hair measure to he the so-called standard Haar

measure of

G.

Then we have~ for all

d ( ~ ( b * ) ) = (2~) -q2 - ( q - ~ )

where X(h*)=logb*+6,

b

eB

(w/~°)d(b*)([~(X(b*)) I/~

w=IW(G,B)I, w°=lw(s°,B°)t

(Sk))

(Theorem 17.7).

In Section 16 we study the space of cusp forms in greater detail. be such that ~ 2 ( G ) ~ and let °L2(G) be the discrete part of L2(G) to the regular representation. We then have the orthogonal projection

190

Let

G

relative

15

oF:S2(G) ~

%2(a )

Then the main theorem (Theorem 26.11) of this section asserts that °L2(G) N C(G)

and that

°E

is a projection of

C(G)

tinuous with respect to the Schwartz space topology.

onto

°C(G)

Moreover,

°C(G) = that is con-

°C(G)

is a

nuclear Frechet algebra (Theorem 16.21), and its topology is the one given by the Hilbertian norms

f ~ I1~m~ ~il2 where

LI'II2

(from

@)

denotes ~2-norm and

~ is a suitable K - i n w i ~ t

of second degree (Theorem 16.20).

tion algebra of

C~

el2iptic

operator

The analogy here with the convolu-

functions on a compact Lie group is quite striking

This

theorem follows from the following' remarkable property of the discrete series representations:

given an irreducible representation of

finitely many classes of g2(G)

g2(G)

containing it;

contains the trivial representation of

K,

there are at most

we also note that no class of K

(Theorem 16.15).

There are two appendices (Sections 18 and 19).

Section 18 contains certain

estimates of solutions of some ordinary differential equations.

Section 19 con-

tains a study of certain representations of polynomial algebras associated with finite reflexion groups.

191

i.

Our

first

aim

now

Groups of class

is to carry over to a connected

semisimple Lie group with

finite center a substantial part of the results on invariant analysis on reductive Lie algebras.

One of the basic tools will be induction on dimension,

use this smoothly it is necessary nor connected

in general.

of connected

s.s.

some of its properties.

Definition of

Let

G

This

and as usual we write B

G°

first

G

~;

G

done

by

is not necessarily connected,

for the component of identity of

is reductive and write

subgroup of

was

space (el. [Ii]).

Z

be a real Lie group with Lie algebra

assume that

We shall introduce this class in

in his Princeton Lectures on Schwartz

i.

analytic

These groups form a class somewhat larger than the class

Lie groups with finite center.

this section and discuss Harish-Chandra

and in order to

to work with groups that are neither semisimple

defined by

~i"

~i = [ ~ , ~ ] = $ B

G.

We shall always

, c=center(~).

If we write, for

a,b e G,

then from the general theory of Lie groups it follows that

GI

GICG °

is the

[a,b ] = aba-lb -I,

is the group gener-

ated by all elements of the form [a,b],a,be@ °, i.e., G I = [G°,G°], the commutator group of

G° .

G

we often write closed in by

c.

operates on x "X

G,

and

Finally

the class

~

~

(i)

(ii)

C°

C

Ad;

is the kernel of

is the analytic

A group

G

for

Ad.

[G :G °]

finite center.

subgroup of

G

is a connected

1:

(i)

real

s.s. Lie group,

is said to belong to

~

is reductive

Gc~

if and only if it has

with

Ad(H)Canalytic

G°CHCG

[A :A °] finite~ (iii)

if

and if

H

...XO(n) cz;

GxAc

~

if

(ii) G c ~ H ~

Gc]J and A ~

for any

G s ~, H a closed normal subgroup of

subgroup of G c defined by

(iv) G ~ A d ( G ) ~ ;

(x ~ G)

G(i)[N (liiin)~G(1)X

any abelian Lie group with

and

C ° are

defined

From (iii) we see that

Proposition

G/Hc~

G

~c"

Ad(x) l c = id

H

and

is finite

(2)

subgroup

X c ~xeG,

C

if

is a real Lie group and its Lie algebra

We note that if

H£ M

Ad(x)(X).

Ad(a) ~ Gc center(G1) is finite

(iv)

and

by the adjoint representation

is now defined as follows.

G

(iii)

~ for

denotes the connected complex ad0oint group of

(Ge ~)

(i)

Xx

c O = exp c; moreover

Gc

The class

or

sub-

be ( ~ = L i e

algebra of H),

is the normaliser of

H ° = A d ( G ° ) = A d ( G ) °. 192

~

in

Gc,

G then

then

is

17

Elementary;

the finiteness of

is an algebraic set in Proposition 2:

GL(z )

Let

Suppose

G

is finite and group).

If

~

fined over Using

~

be a complex algebraic group,

be its Lie algebra, and let ~

is the Lie algebra of

and if

(det) -I

~

G

G

(when

G

is of class

G ~ = ~ n GL(nj ~),

Let

Gc

be a real form of

then

is considered as a real Lie ~.

G~

In particular, if

is of class

as an additional coordinate and changing

~ c SL(n,C).

connected

closed in the usual topology such that [G : G ° ]

is reductive, then

~,

assume that

Let

is a subgroup of ~

H

(el. Whitney [i]).

~ c GL(n,C)

in the Zariski topology. ~.

[H : H ° ] in (iv) follows from the fact that

~

is de-

~.

n

to

n +i,

we may

be the connected complex adjoint group of

~.

The

Zariski-connectednes~ of ~ implies its oormectedness in the usual topology, so that A d ( ~ ) c G c. by

Let

[~,~]

center. G ~ ~.

~i

(resp. GI)

(resp. [%~]). So, since

Mn(C),

the set of points of algebra c ~

be the analytic subgroup of is a complex

s.s.

center(Gl) Ccenter(~l) ,

Suppose now that

matrix algebra

~i

~

is defined over

then ~

~

and so is a real form of

we see that ~.

~.

~

As

~

~;

[G~ :G~]

~

~

(resp. G)

defined

group and so has finite center(Gl)

is finite.

So

is complex conjugation in the and as

operates on

it is clear that

~.

If

is stable under

fixed by

defined by G ~

matrix

~

G~=

~Mn(~),

and if ~

G~

is the

is

~-Lie

is the set of fixed points for

is finite by Whitney's theorem, we are

through. 2.

The exponential map

In this

n°

we make a few elementary remarks on the exponential map.

proofs follow from the results of Chapter 2,

Varadarajan [i ];

The

as the details in-

volved in these arguments are elementary, we omit them. Let

V

be a finite dimensional vector space over

m = ~l (V), M = GL(V),

and write

exp(m~M)

II or

C;

for brevity we put

for the usual matrix exponential.

An

elementary argument using the characteristic polynomials of matrices shows that if is any closed (resp. open) subset of values of

X

lie in

(3)

is closed

the set of all

(resp. open) in

m.

X ~m

E

such that all eigen-

For any

a>0,

we write

m[a] = {x =xcm, lm~l < a Veigenvalues ~ of X].

By the above remark, Proposition 5: in

E

C,

M

for

O 0.

ad Y.

(18) morphism of

~,T

So,

will be open in

For small onto

G ,T

and

T (with or

Z ,T;

Z y,~_=xexp ~7. r,

7,T,

ZT, T c "Z

and

the natural map of

will be open in

G,

212

aT, T= a[zT,.r].

Y ~ x exp Y G×Z

and for any open

G.

we

Y e ! such that IX1 < T for all

Let

~7 .r=7+ (~)(-r),

ZT, T is Gx-stable.

of 0 in ~

For any s.s. Lie algebra !

write (ef. I, Lemma 1.18) I(T) for the set of all eigenvalues of

of 0 in ~ that are

In what follows 7 (with or

without suffixes) will denote such an open neighborhood

mersive.

into

G c.

We continue with the notation of n°3. [Gx :Z]

Now, by (i) of Proposition

~i[~]

Gc[~] n ~,

into disjoint open subsets each of which is a closed.

and so, by

Thus Gc[ ~] n B c

C

manifolds gives the conclusion that the natural map of

Gel ~]

-ig[~]=a

Hence g [ ~ ] = z [ ~ ] C ~ l [ y ] .

,T

is a diffeo-

into

G

ZI c Z/,T,

is subG[Z I]

It

37

THEOREM 18. (i)

For all sufficiently small

Z

is an open subset of

morphism of (ii)

If

yeG

"Z,

~,T

and

we have the following:

Y~x

expY

is an analytic diffeo-

~7,T

onto Z T,T y[Z7, ~] n Z7, r ~ ¢,

and

then

YeGx;

in this case, y[Z , ] =

Z (iii)

If

ZI c Z

is G -invariant and closed in

~T

Z,

X

G[Z 1 ]

We need to show that (ii) and (iii) are true once we take ciently small.

and

T

G.

suffi-

We need a lemma.

LE~MA 19 . Let that for some

7

is closed in

y

x

be as above.

Let

A

be the set of all elements t cC ° -i t), yxy =xt. Then A is a finite

(depending possibly on

such

group. A Z)

is obviously a subgroup of

words, we shall assume that that

t~l

G°

center(Gl) = [i]

is some element of

G°~GI X C°

G

C ° . Passing to

it suffices to prove that the corresponding

(as Lie groups).

such that

~

A.

As

and prove that

is trivial on

GI

G ° = G I C°

and

k(t) N

G,

where

and

~(t) ~i.

det k'(x)det Z'(t) chosen

or

}~ such that

[G:G°].

But as

det(h'(t)):l. Z(t N) ~i.

As

group and a compact torus, its elements of order

N

First we use Proposition ii (with

Gc

centralizer

(Gc) x

(x exp ~ ) ~ ,

of

then

x

is a bounded sequence in now take

7

and

T

7 n (c n $~) + 7 ~ c

in

Go

ze (Gc) ~

of

0

such that

(b)

Gc,

instead of a

if

then

in

For L,

of

commutes with each So

det(Z'(t)) =

det Z'(yxy -I) =

tN~l,

we could have

finite subgroup.

z e G,

write

z=Ad(z).

the Lie algebra of

~c n $~c'

(a) if

ZnCGc,

{Zn[X] ]

be the representation

is the product of a vector

form a

~qis done, we come to the proof of ~heorem 18.

~. ~ c ) to find an open neighborhood

of

t

If

C°

we have ~

we have

Z(t)N:I.

tN= i.

Suppose

representation

Z'(t) = Z(t) • id.

yxy-l=xt

Thus

Hence

~'

Since

In other

is finite.

G I n C ° = [i],

Let

Z.

a simple calculation shows that N=deg(k')=

(which is of class

A

Let us consider a one-dimensional

obtained by inducing from the representation

element of

G/ceater(Gl)

subgroup there is finite.

z c Gc

an~Xexp

a

Gc

being

invariant under the and

z[x exp a] n

are such that

is a bounded sequence in

[Zn[an]]

Gc

also.

We

so small that in addition to (i) we have the following:

(a') 7 =

(note that

Gx leaves

the two spaces stable, a c t i ~

c = (~ n ~ ) +

trivially on

c, c)

the sum being direct, and

(b') exp(~ ~ G) is one-one on 7 N c;

if C l : e ~ ( ~ n O, Cl~ 1 n A: {l] (c') ~,~ n ~% c a. Suppose now y~a, Y1,Y2~ ~T,T_ are such that

y[xexPYl]=xexpY

to

~[x] = x .

G,

we see that

in particular t=l

t c CICI I c C° . As

by (b').

Hence

y[x]=x.

It remains to prove (iii) • ceO.

Then

gn[bn]-~.

Hence

So

2.

Let

y[x]=xt

teA But then

where

Y.=Y'+Y'~j 3 J for some A

(Yje$~,Yje t cG

Passing

t exPYl=eXPY2;

is as in Lemma 19, we must have

y[Z ,T] = Z ,T

obviously.

Suppose

b n c Z I, gn c G

{gn[~]]

is bounded in ~; 213

and

¢).

are such that as

~[x]

gn[bn] is closed,

38

there is a compact set

G ~

G/(G)~

B c G[x]

such that

gn[x]eBVn.

As the natural map

is a submersion we can find (I, Lemma 1.12) a compact set

such that the image of

BI

is

B.

~ gnT C Bl

Hence

Passing to a subsequence we may assume that

gn' ~ g ' e G .

BI c G

~'[~]~ -- On~[ ~ ] V n .

such that Now

~C' ~ ~n~n

7~x V n

and so, arguing as in the proof of Theorem 17, we may pass to a suitable subsequence ~nd ensure that Hence

gn = g ~ f h n ,

hn[b n] c Z 1 b' c ~ .

with

also.

D/t

We then get

COROLLARY 20. ~l~Tl

~,T

where

f

is fixed and

hneGx

for all

n.

SO g [b ]=g'f[b'] where b': in n n n n ~ f- g'-l[c]=b' say~ and hence

g~ ~ g ' , h n 6 % . b~=f-±g~-±[gn[bn]]

c=g'f[b']£G[Zl]. 3

are such that Choose

g~-ig n : f h n

sufficiently small

%m

with the following property:

C~(71) c ~ and 0 < 71 < 7,

then

C2(%I~TI ) C G

if

T.

sufficiently small so that in addition to the properties of

Theorem 18,

exp E

llary 1.6).

Then

is closed in

Z

whenever E is closed in ~ and c 6 % T .

G[x e x p C Z ( 8]% ,~T l )

As this is contained in COROLLARY 21.

~

is closed in

and contains

G7, T ~0,T0

G

G

the corollary is clear.

such that the collection

basis for the invariant open neighborhoods of

(Coro-

by (iii) of Theorem 18.

[G

}

is a

7 , T T C y O ,T < T 0

x.

Obvious. COROLLARY 22. isfied. function

Let

7~T

be such that the properties of Theorem 18 are sat-

Then, given any function F

on

GT~T,

(resp. analytic),

f

on

invariant under

so is

F;

Let

7,T

if

f

Zy,~, G,

invariant under

such that

Gx,

F IZ%, T = f.

is locally integrable, so is

~

If

a unique f

is

C~

F.

Clear. COROLLARY 2~. (19)

be as in Theorem 18. 7 : g[z] ~ g [ x ]

is well defined, analytic from commutes with the actions of G

such that the map

neighborhood of

x

G G.

n ~ nix] in

X,

(20)

Z

~T

onto If

(geG, X= G[xb

N c G

and is submersive.

to

)

It

is a regular analytic submanifold of

is an analytic diffeomorphism of

(n,z) ~ n[z]

~-l(n[x]) (noN),

and structure group

zeZ%T

N

onto an open

the map

is an analytic diffeomorphism of [n} X Z % T

Then the map

NX Z

( z c Z y, T, n e N ) onto

i.e., (G%T,X,7)

G . x

214

m-l(N[x])

t h at

s end s

is s, G-bundle with fibres

39

COROLLARY 24.

If

x

is regular,

are none other than sufficiently If

McZOG'

compact

is compact,

LcZ

such that

G[M]

Z is the CSG containing x, and the Z

small G -stable open neighborhoods of x x is closed in G. If E C G is compact,

7, T in Z.

G [ Z \ L] ~ E = ~.

The first assertion is obvious. same as in I, Corollary 1.24.

For the second one, the argument is the

For the third we may go over to

Ad(G)

and use

Lemma 12. Exactly as in the case of the Lie algebra, we can use these results to characterize

completely invariant open sets. An open invariant

called completely invariant

if for any compact set

We now have the following results.

set

~ c G

B c ~, C£(G[B]) c ~

is also.

The proofs are omitted since they depend on

Theorem 18 in the same way as the corresponding

results on

9

depended on

Theorem 1.1.20. THEOREM 2~. containing

Let

x.

Then

x eG 3

iant open neighborhood THEOREM 26.

Let

ing are equivalent: (iii)

if

be

of

y

as above and

an invariant of

x

Q

and

and

f ~ C~(G)

O

an invariant open set

such that

f =i

in an invar-

supp(f) ~ ~.

be an invariant open subset of

(i) O

is completely invariant

y ~ O, 3 an invariant

neighborhood

s.s.

f c C~(G)

supp(f) c O.

G.

(ii) if

such that

In particular,

f=i if

~hen the followy ~ O,

then Ys c

in an invariant x ~G

open

is s.s. as above,

the sets

G are completely invariant for sufficiently small T,T. If O is 7',T a completely invariant open subset of G and ~ c ~ is an open invariant subset containing

all the semisimple points of

~,

then

~ = ~.

Finally we have THEOREM 27.

Let

y c G.

~hen

G[y]

submanifold

of

G,

morphism,

G

being the centralizer

and the natural map

is a locally closed regul~r analytic G/Gy ~ G[y]

of

y

in

G.

is an analytic diffeoMoreover

C£(G[y])

splits

Y into finitely many G-orbits. It is enough to prove the second statement. Baire category argument 0i

are orbits,

set, for remaining

l0,

Let

k

B

be the subgroup of Fix

beB

and let

'b be the set of points of

b

B

~

~.

Then:

(iii)

(Bnz)UOBnz~¢, then u ~ \ snn= Uuc~(~n z)U, B, nn= Uuc~(s'nz)U 8

and

Ye~Y

¢

centralizes perty for lation.

are stable under

are sufficiently small~

is regular in b~w.

b~adY Z.

As

B~Z

~t,

~. as

centralizes

BO ~

for some

b C ~j = b.

ueB

and

is regular in

G

for

B

and

¢

b ~ Y e b.

are sufficiently small,

expadY

Property (ii) is just the slice pro-

obviously contains the union on the right we should prove only Let

Xe b

and

b I = (bexpX) x ~2

b I = (bexp y)t e B

to

u

for some

Ye0~, t eG.

are pure im~inary.

in the analytic subgroup of

with

~i ~ ~j

ad Y

x=t~.

Write

b 2=bexpX.

being the centralizer of

and so 3 x'

Clearly

b2eBDZ=bex

y=bexp Y

if

So (i) will be proved if we verify that

is elliptic, and so all eigenvalues of

bl, it takes

~;

To prove (iii), in view of (i) it is enough to prove the first re-

the reversed inclusion.

then obtain

b.

be as above, sufficiently small.

regular in

B'nz

and

~l

~= (b exp-) O,

~ = ~[~,¢],

centralizing 8,¢

y~.

RI = ~n (CG°).

be an open neighborhood of

Define

(ii)

bexpYcB~Z~Ye

have bX 'X

Let

B' n Z=bexp ('bn~), BnZ:bexp ( b ~ )

As

Y=X u

G.

~(~),

proving that

(i) (24)

some

in

G (b) [~b,~b ]

y s ~ n (CG°) and y is So Ye = z [ b ] f o r some

Hence, as

R D (CG°),

both sufficiently small.

Proposition ~5" Let

also

b ~ l , Ye ~ ~l elements of

Since each

bj

G

defined by As

in

x ~.

in the analytic subgroup defined by ~i X 'X bl=b 2 also. Then u = x ' x e B and b l = b 2

p (bOw).

220

Then expY

This shows that ~.

We

ta/ces b 2 to As

bj.eB, we

such that where

3.

Descent from

G

~o

Z

and

In this section we develop the method of descent from latter two being the group of class element

x

as in Section 2.

M

G

to

Z

and

8, the

and its Lie algebra associated with a s.s.

The theory is however much deeper than in the Lie

algebra, mainly because the radial components are much more difficult to calculate and one is able to determine only the corresponding endomorphisms of the space of invariant distributions induced by them; even this (which fortunately is quite adequate for invariant analysis) depends in a fundamental manner on I, Theorem 9.23. 1.

The differential of the conjugacy map of a real Lie group

In this n connected; ~,

a

we work with an arbitrary real Lie group is its Lie algebra,

ac

ac

(a c c ~).

as differential operators on the smooth functions right~ the actions being denoted by

for

af

and

a = X 1 .--X r

fa

at

x ¢A

not necessarily

the complexification of

the universal enveloping algebra of

values of

A,

f ~ af

and

a

(a c ac), and

The elements of

~

act

(on A)~ both from left and f ~ fa

(f eC~(A), a ~ ] ) ;

are denoted respectively by

f(x;a)

the

and f(a;x);

(Xje a)

(1)

f(x;a) = (srf(x-tl

where

f(x-t I ..... t r ) = f ( x e x p t l X I ... eXptrXr) , f(tl, .... tr-X ) = f ( e x p t l X I ...

exp trXrX)~

and the suffix

tI . . . . . t r = 0.

If

operator of order at

x

~CA

< r

0

...

~,

E

then, for any Ex ¢ ~

is an analytic map of

~

is an analytic differential

x ¢~

such that

the local expression of E

f(x;Ex) = (Ef)(x) V f ~ C~(A);

into the subspace of

and the correspondence that sends

E

N

to this map

of elements of order x ~ E

--

is a bijeetion.

X

We often write

f(x;E)

refers to the action if and only if

for

AxA

(Ef)(x).

If

a~b ~ a[b ] = aba -1

y[Ex]=%[x]

for the automorphism of

~

~ of

(xcn,ycA); induced by

is invariant (as usual, invarianee A

on itself), then

here, we write

is a Lie group whose Lie algebra is ~ QX a~

acX ac,

and Universal enveloping algebra ~ Suppose now

~(A×A

~ A)

E

is invariant

u ~y[u]=uY=Ad(y)(u)

Ad(y) (y ¢ A).

onical.

® ~,

with complexification

all i s o m o ~ h i s m s being can-

is any analytic map.

Then for any

(x,y) e

co

A×A, into

8tr) 0

indicates that the derivatives are taken at

is an open set and on

is the unique element

x ~E x ( r,

tr)/St I ... 8r)0,f(a;x ) = (srf(t I ..... tr:X)/St I

. . . . .

we have the differential ~

(d~)(x,y)

such that V f e CT(A),

221

defined as the linear map of

'~®~

~6

(fo ~)((x,y);~)= f(~(~,y);(d~)(x,y)(~) ) Let us now consider the map

~((a,b) ~ a[b])

(u ~ ~ ® ~)

of

A×A

into

A.

For any

x~A,

the diagram A×A

(2)

@

>A

xxL

;i

Xx((a,b))= (xa,h) ix(h ) =~bx-1

A × A -------~.~ A i s o b v i o u s l y c o m m u t a t i v e , so t h a t

(3)

ry = (d~)(l,y)

(d~)(x,y) : Ad(~)ory,

We wish to determine g(a;u,b;v)

for

F . We often also write, for g ~ C~(AXA), u,v ~ ~ Y g((a,b);u ® v). Note that since @ O ( i x , i x ) = i x O @ where (ix,i ~

is the antomorphism

(a,b) ,- (xax-l,xbx-l),

(4)

we have

Fx[y] o (Ad(x) ® Ad(x)) = Ad(x) oFy. Proposition i.

endomorphism

v~uv

the endomorphism tion of

Fix

(resp.

For any

v~vu)

L(y-I.x-x) + ad X

~ in ~. Let c N in ~. Then

tion of

y sA.

(5)

q

u e~

of of

let

L(u)

~I. For any ~.

Then

(resp.

X ¢ as'

let

ay(X~Cy(X))

X ,~ ay(X)

~ = f • ~.

y'=y,

so that

and prove (5).

If

(X ~ ac)

u = i,

we are differentiating

It is enough to do it for

deg(u).

f(x,y') = f(xy'x -I)

So we may assume

u=Xw

For f e C (A),

where

at

x = i,

deg(u)=r~l

X e a, d e g ( w ) ~ r - l .

Let

Then, by (3) and the induction hypothesis,

~(exptX;w,y;v) Differentiating with respect to

t

= f(~(t)[y]; a(t)[~]). at

t=0

we get, on noting that

y e ~ (ty -I • X)exp(-tX), ~(l;u,y;v) = f(y;(y-i .X-X)~) + f(y;(adX) (~))

which gives (5), bec~se

he

is a representation is elementary verification.

(dm)(l,y)(l ® v ) = V = a y ( 1 ) ( v ) .

e(t) = exptX, ~ = Fy(w ® v).

~y(X)

is a representa-

(u,v ~ ~).

We need only prove (5), and we shaAl do it by induction on write

be the

denote also the extension of this to a representa-

y

ry(u ® v) = ~y(U)(v) That

R(U))

~y(U)(V) = ~y(X)(~) = (y-i. X-X)~ + (ad X)(~) .

222

~(t) [y ] =

47

2.

Radial components on "Z.

turn to our group

Ge~.

We take

dmslruction and elementary ~roiPer~ie~. G=A

in

is the universal enveloping algebra of

~c"

n°l. x

~, F be as before; @ Y is a s.s. point of G. Let nota-

tion be as in Section 2, n°s 3 and 4; in particular any subspace

m

algebra over

mc

map

of

S(~c) ~ @,

ents

Sr(mc).

sum.

If

m

~,

and Let

~r (g)"

ic~lly on

Let

of

r ,

If

m

~(m)

in

~

~y

S(mc)

(the symmetric

under the canonical symmetrizer

~hen

~(m)=~+(m) + C ' I

> 0.

~(q) ®~(~)

K+Z--~I K

-'Z

Then, for any

(l,m).

y c "Z, the re-

is a linear bijection onto ,

F<m)~ then Y

"B <m) denotes th; inverse of Y

"B (m) Y

depends analyt-

y.

Proof is by induction on a basis for

~.

Fix

ye "Z

m.

Let

~,-..,X_~

and write

be a basis for

q since det((Ad(y -I) - 1) ry(X®v)~ ~,y -1 - X - X ) v ( m o d t e r m s of d e g r e e ~ r ) , and h e n c e , i f

.

.

Xik®YJl . . . .

YJ~) mXjll - - .

This implies easily (since

~= q+ ~)

q, YI,...,Yt

X [ = y i . X . - X . . Then the X ! also form J 8 J J lq) ~0. Now, if X e ~, v e @, deg(v) Er,

a basis of

Fy(Xil

is a direct

is the subalgebra generated by

~

For

for the images of the homogeneous compon-

be any integer

to ~

q= range(Ad(x -I) - i).

for the image of

~+(m) = ~ r > 0 % ( m ) .

y

~<m

(r ~ 0)

is a subalgebra,

r (m)

~(m)

c S(~c) )

~r(m)

Proposition 2. striction

we write

regarded as

We re-

Let

XJlkYJl " ' "

k+~=m,

then

Y'3~ (mod terms of degree < m).

that the range of

F (m)

is all of

~(~)

(rood ~ r ~ m C~r(~))" So, by the induction hypothesis, F (m) Yis surjective. By Y F (m) is a bijection. Since F (m) depends analytically on y, Y Y

dimensionality, so does "B (m). Y

Proposition ~.

For each

y 6 "Z,

the restriction of r to (~(q)®(~(~) is Y "B maps any u e @ of degree < m into y y ,~ "BY - ( u )" i s a n a l y t i c on "Z. "B depends on y

a bijectiononto@ ; its inverse ~ _ + l < ~ C~k(q) ® ~-~(~),

and

_

Y

only through

Ad(y).

~(q)®~(~n~l).

If

gl=[g,g]~

Finally, if

then

qCgl,

and

"By

maps

onto

zeG x, u ~ ,

(Ad(z)®Ad(~))('By(u)) = "B[y ](~[~ ]).

(6)

Only the last two statements are not immediately obvious. leaves

~ and

q

invariant;

iant, and (6) follows from (4), that

~(gl)

qc~l.

As

~l

Ad(z)®Ad(z)

Since

G

is the direct sum of

position 2 shows equally that

~+~%(~)®%(~0%) ~(q)%~(~ n %).

hence

FY(m)

leaves

If z e Gx, Ad(z)

~(q)®~(~)

centralizes eenter(~)~ d0 ~l

and

q~

the argument of Pro-

is a linear bijection of

onto r ~ % ( % ) ,

so "By maps ~(%)

223

invar-

it is clear

onto

~8

We are now in a position to introduce the radial components. served that map

~(q)

is the direct sum of

By(@ ~ ~(~))

for each

(7) It

ye'Z~

£ •i

"By.(u) ~ 1 ® ~ ( u ) is clear

analytically on pends on

y

y.

~d

(s)

Ad(y)

Ad(Gx) leaves

2

C" i

and

~+(q)

it by

~ n "Z

5x(E)

~.

Then

With

~i

E

E

is an open subset of

as above~ we write

whose local expression at

"E (x,y)

G × ~i

that Z

G[~ 1 ],

is

"By(Ey).

Proposition 4.

By(Ey).

We denote

is Gx-invariant. ~ N "Z.

Then the

the latter being open in G.

Since

Fy

maps

that

"E

and

= (ET) o ¢.

u®v

Let notation be as above. E

differential operator

on

f

"%(Ey) E

onto Ey,

are ~-related~

i s invariant,

f o~

is a

~,

5x(E )

where

u c~+(~).

We thus obtain

~hen for any analytic invariant

is an analytic Gx-invariant differential

Moreover~

(ll) for

an invariant

depending only on the second coordinate; it is therefore killed

by all differential operators of the form

~ N "Z.

E

for the analytic differential operator on G X21

If now

operator on

is

contained in

onto

i.e., Vf

eO~(G[~l]), ('E(f ~))

(6) gives

and

8x(E )

E~

GX21

de-

E:

we find, on using (3) and the invamiance of

function on

~.(U)

(y ~ ~ n "z).

It follows from (9) and the invarianee of

induces a submersion of

G

y e 2 N "Z

(~x(E))y = By(B)

9

depends

~ a unique analytic differential op-

and call it the radial component of

Suppose now

~(u)

(z ~ ox~ u~o).

whose local expression at

(10)

stable,

is a G-invariant open subset of

analytic differential operator on erator on

u,

fixed

(u { ~(~l))"

z[By(u)] = B[z](z[u])

Suppose now that

for

and that

By(u) ~ ~(~ n ~l)

Furthermore~ since

map

that

Moreover~ it is obvious from Proposition 5 that

through

(9)

So ~ a unique linear

~+(q). £~,

(~od ~+(~) ® S(~)).

deg(~(u)) ~ deg(u),

that

and

such t h a t ~ u

We have ob-

( ~ ) I % = ~x(~)(f 1%) a~

open s ~ b s e t

~l

of

~ n "Z

This justifies oar c~lling

and any O - i n v ~ i a n t

5x(E )

f ~ C~(Q[~Z]).

the radial component of

E.

We shall now extend this result to include the actions of the differential operators on distributions also.

Let

wG

(resp.

224

~0Z) be the biinvariant

49

exterior differential form on on

G

on

G × Z.

(resp. Z)

and let

G

(resp. Z) that induces a given Haar measure

WG× Z

be the form that induces the product measure

Using these forms we can define the

sponding algebras of differential operators.

formal adjoints

on the corre-

This allows us to let the differ-

ential operators act on the distributions, and further, to identify locally summable functions with the distributions defined by them.

In particular~

(resp.

G

IZ, I G × Z )

Suppose now

~i

can be regarded as the Haar measure on is an open subset of

I, Section 2 to the map

@ : G×~I

~ n "Z.

~ G[~I]"

sponding to

~i T

such that

8T = IG ® ~T

(resp. Z, G x Z ) .

We can then apply the results of

We thus have an injective map T ~ a T

from the space of G-invariant distributions on tributions on

iG

G[~ I ]

into the space of dis-

is the distribution on

via the correspondence set up in I, Theorem 2.2.

G×~I

corre-

From the re-

sults of i, Section 2 we get THEOREM 5. T ~ aT

For any Gx-invariant open subset

~i

of

~ O "Z,

tributions on E~E1,E 2

G[~ 1]

into the space of Gx-invariant distributions on

are G-invariant analytic differential operators on

(12)

~x(E)°T = %T Proposition 6.

the corresponding

x

is regular so that

~

is a

Then the relations (ii) determine

a differential operator on

~ n "Z.

Moreover,

E ~ 5x(E)

~Igebra of analytic differential operators on If

~i c ~ n "Z

ential operator on show that

F = 0.

is an open subset of DI

CSA 5x(E)

such that

tralizer of Fh = 0

on

lar in the the of

CSA ~ 0

in

t

in

G),

CSGZ

so that

via the map ~

"Z

and

F

5x(E)

DI

F,

containing

fix t

into the

is any analytic differg c Ca(G),

5x(E)

we shall

is uniquely

gives at once the pro-

t e~I"

If

N

and stable under

h c C~(N); Gt N Z

Y ~ t exp Y,

we must remember here that

is open in both

is a suffiGt

(the cen-

h'c~C~(n);

Z

and

G t.

t

that

is regu-

Going over to

we obtain a Gt-stable open neighborhood n

and an analytic differential operator

F'h'=0VGt-invariant

~

it follows from Corollary 2.22 and Theorem 2.25

N V Gt-invariant

is

for any invariant analytic differential operators

To prove the assertion concerning

ciently small open s~bset of

Z

is a homomorphism

F(g I ~i)= 0 V invariant

This will be sufficient to prove that

5x(EIE2 ) = 5x(El)Sx(E2)

and

uniquely as

~ n "Z.

determined by (ii); the unique determination of

EI,E 2.

If

~x(E1E2)% = ~x(E1)(~x(~'2)%)"

Suppose CSG.

~l"

~,

from the algebra of G-invariant analytic differential operators on

perty

the map

is an injective linear transformation of the space of G-invariant dis-

F'

we must prove that

maps onto some subgroup of the Weyl group of

225

on F'=0

n

such that on

n.

But Ad(Gt)

(~c,~c) and so, certainly F ' p = 0 V

5o polynomials on point of

n

~c

invariant under the Weyl group.

we can find

~

such polynomials

Since around each regular

(Z= d i m ( ~ pl ~...,p~

forming a

local system of coordinates on ~, the fact that F' annihilates ail monomials aI a~ Pl ...pz implies F ' = 0 on n O ~' Hence F ' = 0 . This proves that F = 0 . 5.

Action of radial components on distributions

For any invariant open subset

~

of

G

we define

set of all analytic invariant differential operators invariant

f ~ C~(~).

/A(~)

G

to

THEOREM 7" ~hen

Let

~

~ \ supp(ET)

x e~

to be the

such that

~.

Ef = 0

We shall now prove the following

be a completely invariant open subset of

be s.s.

T

on

supp(ET)

G

and

Eel,(2).

~.

is an invariant open subset of

of ~heorem 2.26~ to prove that Let

~

I, Theorem 9.2 3 .

E T = 0 V invariant distributions Since

on

is clearly a two-sided ideal in the algebra of all

invarian% analytic differential operators on analogue in

/A(~) = AG(~ )

E

~

it is enough~ in view

does not contain any s.s. element of ~.

We shall use the results and notation of Section 2~ n°s 3~4.

In partieular~ ~/~T have the same meaning as therein and are sufficiently small so that the results following Theorem 2.18 are valid. ET = 0

on

L~MMA 8. ~,~

Suppose

such that

butions

k

on

5

is a Z°-invariant analytic differential operator on

5h = 0 V Gx-invariant ~,~

If

CSA, p

is any polynomial on ~e

invariant.

Hence

~(5)h=

(~c,~e).

Then

5k = 0

for all distri-

Z °.

It is clearly sufficient to prove that if

the radial component

group of

h ~ C~(~a/,~).

that are invariant under

We use I, Theorem 9.2~. any

It is enough to prove that

G

~(~) = 0

on

invariant with respect to

0 Vpolynomials

h

~c~

is

~' Q ~1,,~ (of. I, Proposition 9.1@.

on

~e

Gc,

P I ~c

is G x-

invariant under the Weyl

As in the proof of Proposition 6 this gives

~(5)=

0

on

~' N ~7,. We can now prove Theorem 7. that

~ET' = 0

ator on (Ye ~ T

on

~T ).

Gx-invariant

Z/,T.

Let

$

that corresponds to

Let

T'= T I G

.

It is clearly enough to prove

be the analytic Gx-invariant differential oper5x(E )

on

Z

via the map

Using (ll), Theorem 2.25, and Corollary 2.22, we have ~ ~ C~(Z

invariant under

Z°.

T).

By Lemma 8

5k =0

dz

of the form q0dz

on

Z

where

Z

. However~ it is clear that ~

5x(E)~=0 V

for all distributions

We now transfer this result to

is a real analytic function

k

on 87~T

via the map Y ~ x exp Y.

%,~ Now~ this map will not~ in general~ take the Lebesgue measure Hear measure

Y ~ x exp Y

dY >0~

dY

on

~/,~

to the

will go to a measure invariant under

G . X

A simple calculation shows that for any analytic differential operator

226

L

and any

51

distribution

b

on

Zy, T,

Ldz " b = ( - i L ~ ) ~ d z "b'

where the suffixes indicate

the respective differential forms with respect to which the actions of the differential operators on distributions are ca&culated. invariant and so, for L~V~dA 9. and

y c91

Let

L = $x(E),

open subset

92

#= -I~T,,

tx(E)~T, = 0.

~

is Z°-

Thus, by (12), ~ET' = 0.

~i c 'Z be a co~0etely invariant (under Z) open subset of Z

a s.s. element.

g c C~(~I)

we get

If

of

%

Then we can choose a completely invariant (under Z)

with the following properties:

and is Z-invariant,

~

h e C~(G)

(i) y c ~ 2 c 9 ! (ii) if

and G-invariant, such that

gl92 = hl~2 y Then

is clearly ~ic~.

placing

s.s. in

G

also.

Let

~i

be the centralizer of

We now perform the constructions of Section 2, n°4, with

x,

to get a slice

But then, for

71,TI

action around y.

W l = y e x p (~l,~l,Tl)C~ I

for the

G

in y

~.

re-

action around y.

sufficiently small, this will also be a slice for the Z-

Shrinking

71 and ~i we get s slice W2= yexp (~1,72,T2) and an

e C~(G) such that ~ is G-invariant, ~ = i on W 2, and supp(~)~G[Wl]. Z[W2].

y

Take ~2 =

If g ~C~(~I ) and Z-invariant, and g l = g I WI, the slice property gives an

hl~ C~(G[WI ]) which is G-invariant and eqQals gl on W1; iant, ~ C~(G), and

h=g

on

W 2.

So

h=g

on

if h : ~ h l, h is G-invar-

92 .

Using Theorem 7 we can now obtain a significant sharpening of Theorem 5. Notation remains unchanged. THEOP~N i0. ~

Let

~

be a completely invaz-iant open subset of

'Z a completely invariant open subset of (i)

Suppose

E

such that S(hl%)=0

(ii)

Suppose

and ~ i c

is an analytic Z-invariant differential operator on

for all a-invariant h~C~(a).

all Z-invariant distributions

G[%].

G,

Z.

EI,E 2

~

on

Then ~ = 0

~i

on %

for

£i"

are G-invariant analytic differential operators on

Then, for any Z-invariant distribution

~

on

El,

For proving (i) it is elea~ly sufficient to consider o~ly Z-invariant Re C~(~I ). place of

For, if this were done, then Theorem 7 applied to

G, ~

and

E)

Z-invariant distributions &tppose

~s C~(~l)

to prove that E X= 0 y

~

on

and Z-invariant.

There exists

and

E (in

for a&l

~i" To prove E ~ = 0

at all regUlar points of

is s.s. and Lemma 9 applies.

in the lemma.

Z, ~i

gives the desired conclusion, namely E X = 0 ,

So we can find

h c C~(G)

~i" ~2

Let

on

Y ~ ~i

~i

it is enough

be regular.

with the properties described

and G-invariant such that ?J ~2 = hl~2"

227

Then

52

Hence E ~1~2=

o.

The proof of (ii) is i~mlediate now.

~=

Take

~(~z~) - ~(~l)~(~)

In view of (i) it is enough to prove that

h ~ C~(O).

E(hI~l) = 0 ~ G - i n v a r i a n t

But this is clear from Proposition 4. 4.

The homomorphisms

~/~,Z B

Our aim now is to calculate operator on

G.

8x(E ) when

E

is a biinvariant differential

~he results are analogous to those of I, Section 2, and are ex-

pressed in terms of certain natural homomorphisms from the center of center of

~).

@

into the

Furthermore, as we did in the proof of ~heorem 7, it is often

necessary to go over from ZT, ~ to ~T,. The o ~ i 0 g t c a n ~ e r

of differential op-

erators can be expressed in terms of a certain natural isomosphism of the center of

~(8)

of

G

with

Let

8

in

Is(Sc ).

We shall now define these maps

be the center of ~.

Suppose

~.

~cCBe

Since

is a

Ad(G) CGc,

CSA;

and ~, the left ideal of i 6p = ~ ~ ep~. Then, for each

P,

(Gc,~c);

@

let

z e 8,

such that

z m 8(z)

ment that

B(z)(h-Sp)=y(z)(h ) ~ h c

(mod ~).

8

is also the centralizer

a positive system of roots of

generated by the positive root spaces; 3 a unique element

Define the element

7(z) e ~(bc)

B(z) e ~(bC) by the require-

here we are identifying ®(be) can, onically with the polynomial algebra over ~c" ~hen T(z) does not depend on

the choice of

P,

is invariant under the Weyl group of

is an isomorphism of

8

under the Weyl group. a

CSA ~cB,

9:;

We write

we write

(Be,be),

with the subalgebra of elements of

~S/~

7= ~B /~ " If

bc

and 7(z~T(z))

~(bc)

invariant

is the complexification of

instead ~f c ~Bc/bc"

It is clear that if

x c Ge,

~e/~c(Z~: ~e/~e x (~) (of. Varadarajan [1D. Suppose in

~)

B"

Let

mC B 4

is a subalgebra such that

be the center of

be the Weyl group of

~(m).

rk(m) = rk(~)

Select a

(Bo'~e) (resp (~e,~c)).

that there is a unique algebra injection, denoted by such that ef bc.

bBc/tc =bmc/bc o bB/m.

We now define S(~c)

unique that

b~/m

m

is reductive

and let

Since ~ m ~ , ~g/m'

of

8

m (resp.

it is eZear into

~m'

does not depend on the choice

From I, Corollary 4.10 we get,

Proposition ii.

with

Obviously

and

CSA b c C m c

ZB"

Let

~c

be a

in an obvious fashion.

{E IS(Be ) Z

8(m) is a free b~/m(8)-module of rank

such that

CSA

of

We may identify

By Chevalley's theorem, given

bBc/~: (z) = {~c"

is an isomorphism o~" 8

Be.

[~ :mm ].

with

We write

Is(gc )

228

{= Zg(z).

~(bc)

z E 8, ~ a It is clear

that does not depend on the

53

choice of

be.

If

m~

is a subalgebra satisfying the conditions described

above, it is obvious that ~or ~ y

z ~ 8, ~ ( z ) = ~m(#~/m(~)), ~--~ (v s ZS(~c))

being the natural restriction map of 5.

Determination of

Fix a

s.s.

point

THEOREM 12.

6x(Z )

x

Let

2

iS(£c )

for

into

ze

and use the notation of Section 2, n°s 3,4. be a completely invariant open subset of

2N 'Z a completely invariant open subset of invariant distribution

(13)

~

on

x

Let

z e ~.

Z

and

21c

(on ~1)

~i = Zq/,T (for

21=Z[B,c ] (B,~ sufficiently small when

is regular, then

G

Then, for any Z-

#~/~(z)o I~y1/2)~

In particular~ these formula~ are valid for

If

Z.

21'

6 (z)~=(t~t-t/2o

small) and for

Is(mc).

~,m

sufficiently

G = ° G and x is elliptic).

is the CSA containing x, 'Z is the subset of regular

points of Z, ~ is the CSA corresponding to Z, Vx=DIZ ~ and one has the more precise formula

(14)

6 (z)= Iv I-1/2o %/8(z)o Iv I1/2 We shall show first that (13) follows from (14).

Assume therefore that we

have the formula (14) for radial components on the regular subset of any CSG.

In

view of Theorem 9 it is enough to prove (13) V Z-invariant ~ e C~(~I ) which are of the form q0=~lflI where ~ e C~(G[~I ]) and is G-invariant.

Further we need only

prove (13) (for such q0) at all regular points t e ill" Fix such a t also, let H be the CSG of G containing t, and let ~ be the corresponding CSA.

As ~ ,

DG(Y) = Dz(y)v/y ) Finally, we put

~H = ~IHA ~i

Now, by Proposition 4, calculate

6t(z),

as

t

~(t;6x(z)) S e t us now d e n o t e b y

~'

vt

G

or

~(t;6x(Z))=~(t;z)=q0H(t;St(z)).

is regular.

We now use (14) to

1/2)

the differential

®(t;6 ,) = 1Vx(t)I-1/21Vt,z(t)l-1/2

the group

We get

IVt,G(t) I-i/2

~/~(~/~(=)) = ~/~(=),

Z

is calculated.

operator

"Z. We apply (14) to (Z, #£/~(z)) instead of (G,z).

inwri~nt ~ d

(y s H n Z)

and indicate by the suffix

with respect to which the function

we have

~

1~1-1/2o ~o~(~). I~l 1/2 on

. bt~IVxl1/~%@ i~oting that

is Z-

get

~l(t;p£/D(z) o I Vx,HI1/21Vt,ZI1/2)

= lvt,S(t)1-1/2 ~}~(t;#~/~(s) o ivt.olm/2). We now take up the proof of (14);

in this case

229

Z

is a

CSG~ ~_ its

CSA,

54

and

•Z

is the set of regular points of

reduce the proof to the case when @ (BI) ,

~=~(c),

c

~

that

z~.

As

•Z,

z.

To this end, let

center (~). 8

Then

~c~(~)~ 6x(z)=z;

v =D. We x ~ = c e n t e r of

8 = ~ i ~,

and so, as both

into the algebra ef analytic differen-

it is enough to prove (14) V z

translations by elements of duces to

It is obvious that

is s.s.

being as usual

sides of (14) define homomorphisms of tial operators on

Z.

in

81U ~.

on the other hand, as

C

and

~/~(z) = z,

Suppose now that

z~.

Then

D

Suppose first

is invariant under

the right side of (14) also re-

~/~(z) c ~ n ~ l ) ,

and it follows

from Proposition 3 that both sides of (14) ame differential operators of the form ~ j (~j oAd)~j

where

v'c~(~l)0

and

~j

is analytic on

Ad('Z).

Consequently,

it is enough to show that both sides of (14) act alike on all functions where

g c C~(Ad(Z)).

But it is clear from the con@ruction of

unchanged when we replace loss of generality that

G

by

Ad(G) (for

G ~ G e.

Now,

Gc

be a simply connected covering group of of

G

(resp. Z).

x

and define

D

(~,~)

/A=~_ I] c p ( ~ - l )

isms of the

CSG ~

P

c

G,

(~

and

~

(resp.

is obviously in

of roots of

where the

and

~

corresponding to

~ ; c

-

(~c,8c) ~P

~

Z

/.

is real for

a locally constant function a = eI~l ~/~

there.

an algebra

B

lar point

Z)

y e Z,

c

of ~, ~ 2(=rk(~))

C~

Z.)

and let

~

Select

x~

D.

i ~=~

p~ .

are complex analytic homomorph~ ,

as ~(y):

c c 4 ~124 we see that /A = ([D I / )

be proved for

functions on

~


0 on w; ljlI/2 is analytic on ~. LEMMA i~. ~ = [I ep~.

Let ~ c 9

Then,

~

be a

CSA;

a constant

(17)

P, a positive system of roots of (9c,~c);

e=c(~,P)#0

such that,

IJ(X) I 1/2 Tr(X) = e I D(x exp X) I 1/2 For

Xc~, ~(X)4=(det(adX),.)2.

Hence, as

(X c ~ R w). j

is real on

9,

which is just D(~expX)2= (I j(x)lZ/ 2 ~(x))4= j(x)a(det(~x)~/~)2 ~/~ (ID(~e~x)ll/2f. ~is gi~es (Zn, as ~n~ is connected. TKEOPd~4 14. Assume

7,T

Let

~

be the isomorphism

to be sufficiently small.

distribution

~

on

Let

~ ~ Is(gc ) z e ~.

introduced in n°4.

Then, for any G-invariant

~,

It is clearly obvious that we need only prove (18) N/G-invariant Fix such a

~.

x e x p Y ~ ~(Y)

Let on

~ ~.

be a If

CSA

xexp Y

and

Xe ~ ' ~ .

Let

is regular in the

~

~ e C~(9).

be the function

CSGH

corresponding to

we get from (14) the relations

(i~)

(T~) (x)=~(~exp x;z): b(~ e~ X) I-~/~(I~I~/~) (~ e~ X;~9/~(~)).

The funotion X' -- (l~ll/~) (~e~X')(X'~ ~') is, by ~e~ma l~, the funotio~ e-llj~l I/2 ~ ' ~ ,

with

J~=Jl~, ~ = ~ I ~ .

So the right side of (19) becomes

I~(x) I-~/~(x)-~ (I ~1 z/2~ ) ( x ; % / 6 ~ ) o ~)

231

~,

%

la(x)1-1/2 ~(x~x(=)o IJl 1/2) by z, Theorem 2~4, since ~(z)

which is precisely is the element of and the set of all

IS(gc) such that k (z)~= ~/~(z). Since ~ X as above is dense in ~ , we get (18).

COROLLARY i~. and

~

and

O O,

VxcO' (iii)

for all

bcB'~

(3)

'~(b)%.(b)

Moreover~ the distribution

%.

~ ~(s)(s[b* ])(b). soW(®,B)

=

satisfies the differential equations

(z ~8). Here

D

~/b

and

are as in Section 3, n°5 •

and (i) is equivalent to saying homomorphism of

z ~ ~/b(Z)(V)

supp(Gb{ ) c C G °.

of

S

into

C,

cg°

is open and closed in G

We recall that

v

beim6

an

Xv

is

arbitrary

the element

bc . There are two parts to this theorem~ existence and uniqueness.

is somewhat simpler and is moreover needed in the existence proof.

The uniqueness We therefore

take up the uniqueness first. 2.

Uniqueness

It is actually necessary to formulate and prove the uniqueness in a sharper form than needed for the proof of Theorem i. Theorem 13 of I~ Section 7, n°4.

If

We proceed in complete analogy with

v e b~× c we call the homomorphism

Xw

of

8

i

into

C

lies in

elliptic

if

(-l)~b*

the homomorphism.

and is regular; in this case

s ~ W(bc),

An ideal

elliptic homomorphisms

80;

v ~ (-l)Nb *

for all

80

of

~ ..... %

sv

is regular and

so that ellipticity is really a property of 8

of

in this ease we can find a subset

is said to be of the elliptic t ~ e 8

into F

C

{Y~,..__.,~]__ such that

of

We shall now prove the following theorem.

Here

if

such that ~ ] l < j < r ker(Xj) c

E(G)

8 0 = % c ~ e)r (.k _

is the class of invariant

o

open subsets of THEOREM 2.

G Let

introduced in Section 2, n 5~ e g(G)

and let

®.

be invariant distributions on

l

(i=1,2). _-Si®i=0-

Suppose

8i ( i = 1 , 2 )

Suppose further that

is an ideal of ®.m=0

out side

244

8

of elliptic type such that

~NCG °

for

i=1,2

and that

69 i

Then

(5)

®1=®2 Let

e l ( b ) =Q2(b)

on

V b ~ B' n ~.

®=e l- ®2' 8o =~n82"

Write

80®= O, ~ b ) = 0

for

~Ol(b)

all

Then

= ®2(b)

Vb~'

i t is enough to pro~e that 80

is an ideal of

8

0

in

c

(= eenter(~))

b{BO2

and choose

valid with in 8

b [ B O D,

and

®

9.

there is an

such that

8=8b

~i= ~[~,¢]

and

and

a >0

~!

on

u n G,

it is sufficient

to

and an open neighborhood

G (b)[~b~b]b ~ ~

~= Cb

2ClDI -~

is majorized by

and

®= 0

on

8b

G (b)[~b,~b].

of

Fix

so small that Propositions 4.~ and 4.4 are

~i =Z[~,~],

~ of course being the centralizer of

Using the descent theory of Section 4, n°l

on

e l = ®2 on ~ C ~ ° .

of elliptic type, 1

b ~ B' n 9,

To prove that ®= 0 on ~D CG ° in view of Proposition 2.34 show that for any

n ~.

b

we now obtain a distribution

which is i~variant under the adjoint group of

~.

It satisfies the

differential equations

(6)

~(~ (.B/~(z)))e = 0

by (4.k).

If

Vl,...,vr

is the intersection of the kernels of Xvl' "" "'Xvr where A, are regular elements lying in (-l)2b , and J0 is the ideal of Is(Sc)

generated by ideal of

80

~ (~ /~(80)) ,

IS(~c)

u~u(svj)

of

!S(~e )

into

Now,

C

~i'

from I, Le~ma 7.17 that

(scW(5c), l0

such that

I%~(x)l ~ Cdim(b~)lD(x)l -¢

for all

xeG'

and

The new fact

b

cB

here is that the majoration (20) is uniform with respect to b *.

But I, Theorem 7.21 gives a uniform majorization for the distributions we now fix

b cB

T~,.

If

and substitute this estimate in (13) we obtain the following:

there is a constant

Cb > 0

such that

I~)(Y)l ~ cb dim(b )I~(Y)t for all

Y e 8'~ b

eB

using (14), (4.6)~

From this we get the required uniform estimate (20)

since

CG °

is covered by finitely many

2 (b) (Proposition

2.34). Let

G

be the group of all automorphisms

put ~ = ~ p

%,

and for

naturally on

B*~

the action being denoted by

define an action

G,b

~e

~ ~[b*]

(21)

demne

~

~(~)

of

G

such that

by J = ~(~)~. ~,b ~ ~ b

~.

a B = B.

We

~ operates

In addition we shall

analogous to (i) by setting

~[b*] = ~b*~8_8.

It is easily seen that

(22)

X(a[b-X]) = dX (b*).

In partic~lar~ morphism of duces

s,

~[B*'] = B*'.

W(G,B) then

a(x)

Each

element

in a natural fashion: ind~.ees

s~.

Also,

250

if

acG,

of

G

s cW(G,B),

induces an autoand

xcB

in-

75

(23)

('{)~ = c(~){~_~6m

To prove (23) we may replace

G

by

Ad(G).

connected and is a maximsi ictus of Ad(G°).

.

Moreover, by Theorem 1.17,

Ad(K) °,

so that we may replace

Ad(B)

Ad(G)

In other words, there is no loss of generality in assuming that

connected, semisimple, and alent to the relation

B

is a torus.

(~)~ = E(c)~

on

But then

5, ~

B = exp5

G

is

by is

and (23) is equiv-

being defined by (18).

If we

write

(24)

~ = ( n g(~))~

eZ/2

g(~) -

and observe that

%eP

_

e-Z/2

z

~eP

is invariant under

&,

the required relation is

an immediate consequence of the defining relation

~=

~(~)~.

Proposition ii.

If

~

g(~)

Let

~e ~

and

is induced by an element of

(2~)

b* e B

G~

Then

we have in particular

eb. = 0

I%.(x)l

(54)

if

contain-

being as

(b~B')

such that i

~ C d i m ( b * ) l D ( x ) l -~

xeG'~ b * e B *

~eG,

(%*# ~(°)%[h*] =

261

86

in particular, if

c

is induced by an element of

(%)

%* = ~(°)%[b*]

G,

"

e

with aproposition which extends Proposition 12

We shall conclude this n to singular

b .

Proposition 2~. write

~

for

CG °.

Let

b* e B*, v = k(b*).

Denote by

®

As usual let

v fined by Theorem 24 corresponding to the character

(b) for

Let

c eC

and

Yi ( l < - i < - r )

C = ker(Ad)

and

the invariant eigen distribution on ~logb*

of

de-

G°

B e . Then:

be a complete set of representatives f o r

G/~.

Then,

x e G °NG',

(~7)

~.(c~)=

*

~

b (c

Yi

)Q(~

Yi

).

l i) .

satisfies the required conditions, and (86) reduces to the relation

®6 ,g (b):l

for boB'.

We shall prove Theorem 29 by reduction to Lie algebra. the

®b*

This is natural since

have been constructed through such a procedure.

We recall the definition of the distributions define, for

T = T~

for

~ ~ S'.

Let us

~S', T

:

s(t)Tt~ .

w(c °,B °)\w(~c,bc) It is then clear that properties: equations

(i)

T

is the unique distribution on

$(u)T

= u~(~)To

for all

u c Is(gc)

7r-I ~teW(Zc,bc)~S(t)et~, ~ where, as usual, positive system

P . Note that

W(~c,bc)

under it, and the restriction of Suppose now that

T

obvious that

T 3(u)T

T~

having the following

to

b'

JCls(~c )

(ii) its restriction to

b'

is

7T is the product of the roots of the

itself operates on

to

b'

is a distribution on

invariant, and for some ideal restriction of

~

it is gempered, G°-invariant, and satisfies the differential

~

b,

such that

is stable

(a) T is tempered, G °-

of the elliptic type,

is invariant under

b'

is invariant under this action.

W(gc,bc).

If

~(J)T=0

UCls(gc),

(b) the it is

has property (a), while the radial component formula estab-

lished in I, Theorem 2.14 shows that

~(u)T

ary spectral theory we conclude that

T

has property (h) also.

can be written as

From element-

TI + . .. + T r

where T i

is a distribution not only possessing properties (a) and (b) but in addition satisfying the differential equations

~(u)T i =ub(~i)T i

270

for all

u c Is(~c )

and a

95 suitable

~i e ~'

see that

T i=ciTHi

Theorem 7.13.

(cf. argument at the beginning of I, Lemma 7.14). on

b'

for some constant

In other words,

T

ci e C .

Hence

From (b) we

T i=ciT~i

by I,

is a linear combination of the T*~i.

The

following is the result on which the proof of Theorem 29 is based. Recall, for any

CSA6 cg,

the Caftan decomposition

6= 61 + 6R

defined in

I, Section 7, n°3. T H E O P ~ 32.

Let

T

be a distribution on

G°-invariant, and for some ideal (b) the restriction of

CSA and l e t

W(~c,6c)

imaginary roots

(ii)

to

b'

~

such that

~.

(a) it is tempered,

of the elliptic type,

is invariaut under

W(£c,bc).

6= 6 i + 6 R be i t s Cartan d e c o m p o s i t i o n .

subgroup of

(i)

T

J~Is(£c )

Let

generated by the Weyl reflexions

Wi(~c,~c)

s

6

be a

be t h e

corresponding to the

Then

Wi(gc,6c)

leaves

61

and

T

to

the restriction of

The invariance under

6R

invariant, and fixes each element of oR

6'

is invariant under

Wi(~c,ae)

consequence of the invariance of T

.

asserted in (ii) is quite a delicate matter

Wi(~c,6c)

since it does not come from any global invariance properties;

process by which

b(J)T = 0 Let

Tlb'

is continued from

under b'

W(@c,bc)

to all of

rather, it is a

in conjunction with the ~'

Also (i) is obvious,

so we need only prove (ii). In view of the remarks made before stating this theorem there is no loss of generality in assuming that yeG c

for some

Y " bc = 0c . Let

such that

~(H)=TTa(H)T(H )

T=T~

(He a'). Let G',

Y

V T

If

is the centralizer of

is a

CSA

of t m = In,m], (Gc,ac).

Weyl reflexions WR(~c,6c)

Let

s

so that

Y(H)= (b(~a)~)(H) aI

in

G,

(He a')

we know that

and that the roots of WR(Gc , oc)

(mc,ac)

be the subgroup of

corresponding to the real roots

these

components

are of

ponent of the open subset of fix one of these, say of roots of

that

(cf. I, Theorem

the oR

+ E= oI + 6R.

form

61 + 6R

of

generated by the

(~c,ac).

Then

S+

6'(R),

+

where

where no real root of Let

aR

are precisely the real

acts simply transitively on the set of connected components of +

and

G

Write

aI = center(m),

W(£c, ac) ~

Select

' ~ a = % eP a H ~"

denote the continuous function on

6.8).

roots of

on

This we shall do.

P o = p o y - i ' ~a = R ~ e p J

restricts to m

#e~'.

6R

is a connected com-

(~c~ Oc)

vanishes.

We

denote the corresponding simple system

(mc,Oc).

There are uniquely determined constants ~(H) =

~

ct e C

such that

s(t)ctetV(H)

tew(~c, %)

271

(He 0'OE,

v=#°y-l).

96 This implies Y(H) =~a(v)

~

ctetV(H)

(H~ ~'QE).

t~W(~c,~ c) Now

Wl(~c,ec)

leaves

E

stable and the invmriance of

TIe'G E

under it is

equivalent to the conditions

(89)

c t = Cst

(teW(Bc,ac) , seWi(gc,ac) )

whicn in turn are equivalent to

(9o)

~(H s) : ~(~)

(H~ ~'0E, S~Wl(~ c,~c) ) .

We shall prove (89) by downward induction on cisely the S+

W(~e,bc)-invariance of

is nonempty. in

a

Pick

~ c S+

Tlb'

dim(~l).

For

So we may assume

and denote by

~

(resp.

(resp. ~R). Clearly a : ~I + ~R,~ and aR,~ + ~R" Let ~ be the centralizer of ~ in

the chamber and

~ l

(2, JR);

~oreover,

any automorphism of with

H' s ] R - H

(resp. -~);

~.

and

then

~ has a

~c;

CSA ~

dim(aR) > 0,

aR,~)

(resp. Y')

~R= aR,~ +JR. H'

z .~c= ~c, z . Y = Y

~.

Then

~ is reductive

that is not conjugate to H'TX',Y'

~ under for

~

in the root space corresponding to and we define

then

so that

the null space of

is one of the walls of

In fact, we choose a standard basis

X'

i~ ~ = e ~ - ~ ( ~ X ~ + a d Y ' ) } , of

~: b, (89) is pre-

z

~ by

~c:~c, sit

lies in the ~joint group

(Ye ~ ), z • (X'-Y')=iH'.

is a bijection of the set of roots of

(tc;~c)

~= a~ +JR. (X'-Y'). The map

-I

7~'=~oZ

with the set of roots of

(Zc,~;

as ~/ is pure imaginary on aI and real on aR, ~/' is pure imaginary on ai+IR. (X'-Y') and re~l on ~R,~"

Hence

(91)

~I: ~I +~{" (X'-Y'),

Further, if

7

is pure imaginary. W(~e,~c); then

~R: QR,~"

is pure imaginary, The map

and for any root

sT, ~WI(~c,~c),

s7,

yla : 0, ~ 7 ' I ( ~ R +JR. (X'-Y'))=0, ~ 7 ' _i R s~s' =z sz is an isomorphism of W(~c,~c) with

7

of

leaves

(gc,ac), ( s ) ' = s a~

stable, and

It is clear from (91 ) that dim(~l) > dim(~l). is applicable to ticular,

Y1 ~

~I~

and gives us the

is invariant under

s7,

,. If s ,=s~

is pure imaginary,

on

~.

Hence the induction hypothesis

Wl(~c,~c)-invariance of for all pure imaginary

is invariant under Wl(~c,ac). We now observe that ~ + E= ~I + ~R and that for t, t' {W(~c,~c), tv = t'v

ing

t,: s t

7

YI~7.

In par-

Thus

YI

is the hyperplane boundon

~

if and only if

(cf. remarks fol~o~i~ ~, (7.1~)). The Wi(~c,%)-invar~anoe o~ ~1%

then implies that ~ t ctetV

and ~'t CstetV

coincide on

~

for each

S { Wl(gc, aC), i.e.,

ct + Cs t : Cst + Css t

(t ~W(~c'~c)'

272

s~wI(~c'~c))

"

97

But

s

commutes with each element of

(92)

Hence~ writing

Wl(~c,~c).

dt, s = Ct - Cst

we obtain the relations

(93)

(t c w(~ c, %),s ~ wi(~c,%) ).

dt, s + dst, s = o To push the induction forward we must prove that

above. that

Suppose for some ds~t0,s 0 ~ 0.

t o C W ( ~ c , ~ e ) , SoC~(~c,~c),

Bat the root

~ c S+

dwt0,s0 ~ 0

We now remark that as

for all T

for all

dt0,s0~0.

t,s

as

From (93) we see

has been completely arbitrary in the above

argument, especially in the derivation of (93). to conclude that

dr, s = 0

So we can use (93)

repeatedly

w e WR(ge,Sc).

is tempered, the coefficients

ct

have the follow-

ing property:

But as

tv

is pure imaginary on

~I

and real on

°t¢°~(t~(H)~° Using (92) we see that if But

t v = s0tv

on

aR.

dt0,s 0 / 0 ~

V

dwt0,s0 ~ 0

this becomes

H~o~.

then either

tv

or

s0tv

is

~ 0 on

~.

Hence we get: dr,s0 ~ 0 ~ ( t v ) ( H )

As

aR~

for all

w c WR(~c,~c),

~ 0

WHo

we see that

~

.

(t0v)(H) ~ 0

for all

H

in

U w w a+R. Since this union is precisely the set of points of aR where no real r o o t of (gc,ec) vanishes, i t i s dense i n eR' Hence % v i s ~ 0 on ~R"

This gives

%~=0

=O

for

on

~cS +

~.

In p a r t i c ~ a r ,

(tO~)(%)=O,

contradicting the regularity of

~

S+,

v.

i.e.,

This concludes the

proof. We now have the following lemma on finite reflexion groups. L ~

~.

Let

F

finite reflexion group. function on

F

For each reflecting hyperplane in

which vanishes precisely there and let

these linear functions. (woW).

be a real finite dimensional vector space amd

Let

f

Then

Y : ~

~(w)eWf/~ w

wow extends to an entire function on

~W

be any linear function on

F . c

273

F

WCGL(F)

a

select a linear

be the product of F.

Write

s(w) = det(w)

98

Fix a reflecting hyperplane and let Let

w0

be the corresponding reflexion.

~

be a linear function which vanishes precisely on this hyperplane. Then, * _ eW0 h for any h ~ Fc, (eh )/~ is entire on F c. It follows from this that if u cF

is such that it lies on exactly one reflecting hyperplane,

C

a holomorphic function around that

u

to

u.

So if

'F

is the subset of all

C

lies on at most one reflecting hyperplane,

'Fc.

dimension

But

Fc\'F c

> 2.

ucF

C

such

extends holomorphically

Hence standard results from the theory of functions of several

Proposition ~4.

~

extends holomorphically to all of

F . C

Let notation and assumptions be as in Theorem 32.

be the product of all the pure imaginary roots MCa I

~

extends to

is a union of a finite number of linear subspaces of co-

complex variables imply that

If

~

~

in

P

is a compact set, there exists a constant

(9~)

(He

As in the proof of Theorem ~2 we fix a chamber H ~ a' N (M + a~).

7T~= ~ a/~ a,l" such that

CM > 0

IT(H)1 ! CMI~(H)I -I

sufficient to prove (941 for

a~

Let ~ % 1

and let

in

a' n (M+ QR)).

mR.

It is clearly

For the mnction

~= ~a"

(T]a')

we have the formula ~(H) =

By Theorem 32 ,

is

TIe'

tempered nature of

T

~ s(t)cte(tV)(H) t~W(~c,~ c)

Wi(gc , ec)-invariant.

(H~ a'0 (aI + a~)).

Hence~ taking into account the

we have

(t ~ w(~ c, ac), s { wi(~ c, ~c ))

et = Cst

(95) ¢ 0~(tv)(H) ~ 0

ct

For any For (95)

H c e

let

%

and

HR

and the fact that

fixed, we can rewrite

%

+

V

H~e R

be its projections on

Wl(gc,ac)

leaves

aI

in the following manner:

plete set of representatives for

aI

and

stable and let

Wl(~c,~c)\W(~c,ac)

aR mR

respectively. elementwise

tj (i 0

we have

for all eigenvalues

n®(h) with

of

bex~ a

Substituting these in the estimate for

I]

Ii- { ~(b)e-~(H) I

IQ(bexpH) I

-1~1/2 Cb ~A = a

Cb, a"

276

we get (i00) with

i01

Note that if gate to

a

LEMMA ~6. L

is any

ale ~

under

Gb~ e

is another

There exists a constant

CSG,

with

CSGA I

~

with

Cb>0

CSA's ~l,...,~rC8 o G b.

these by some element of and let

Cb= m a x l < i
0,

Let

]R

or

C)

and let leaves

qCm

q

be a

invariant

Q be the subgroup of the adjoint group of

Q=exp(adq)

the central series for

some

(over

be such that (i) ad H

m

HQ=H+q.

It is obvious that

that

H sm

is invertible.

q.

(0J.

=

Now,

of

g.

( n + n 0 ) A ~¢

( n + n 0)n ~¢= n0 ~ ¢ + n 0= nO

p~= ~ , .

defined by

Writing

As

(mlF,,a0).

u%F,0 ] ~F=mlF

Hence by the first remark, n o + s0 kn k e K ° such that q is standard.

Write

x-l=k-lan

(since

=

akCpF,08(pF,)=mlF ,

s=Ad(uk)1%.

roots for

and the

and the

This

for some

is st~ch that

Then

Let

nc N O .

Then

pk= ~ ' "

psa&gebra

G;

by the second remark above.

qX= PF

%,

is a

with H' { b, Y e no; by Lemma b n" m 0 + a0c For the second, let Zc D~

obtained by taking orthocomplements.

since c qn

HC~l.

q under

H = H' +Y

This gives

m 0 + s0

If DF

psalgebra of

is a subalgebra such that

He a0

and write

(qN p ~ ) + n 0 = ~¢.

We claim that

b

m 0 + a0cb n.

Z=H+Y,

K °.

is conjugate to

If

the same characteristic polynomial. that

is a minimal

K °.

For the first, select

the centralizer of

~

~ are conjugate under

such that

conjugating element can be chosen from

potents in

Then

x=k~n

with

s0uk= a0.

it is clear that

Consequently we can modify

284

u

where

Suppose

ksK,

and so we can find

such that ,,

c ~F )

aeAo, u

Clearly s.F

neN0, F,F'CT h e n 0,

asA0= and we have

in the analytic subpFUk= ~ ,

still.

is a simple system of

so that

s.F=F'.

~ut then,

i09

nuk F = nF, ,

as

F = F'.

s • A + = A+.

we must have

Hence

s=l,

For dimension reasons it is now clear that

giving us p~

~=

~,,

i.e.,

is minimal and that all

minimal psalgebras are conjugate. From the structure of the

~

and the conjugacy theorem proved above we ob-

tain immediately the following: Proposition 8: Then

mI

Let

q

be a psalgebra of

must be the centralizer of

ment of

a

in

m I.

a

in

9, m l = qO ~(q).

~.

Let

m

Then

(5)

ml

=

both sums being direct,

n

being as usual the nilradicag of

m +

%

q = m +

is called the split component of decomposition of

ct +

q; q = m +

n

a+n

qng

I •

is called the Lan~lands

q.

Parabolic subgroups and their Lan61ands decompositions.

3.

By a parabolic subgroup psalgebra of

9.

If

q

(psgrp)

of

is an one-one correspondence psgrp corresponding to

THEOREMs. K °.

P~

If

Q

G

we mean the normaliser in

is a psalgebra and

at once from (iv) of Proposition 3 that

under

a= pOcenter(ml).

be the orthogonal comple-

~;

We say

Q

q Q

G of a

its normalizer in

G,

it follows

is the Lie algebra of

Q.

Thus q ~ Q

corresponds to

q.

We write

PF

for the

these are called standard psgrps.

is a minimal psgrp.

is a psgrp,

Any minimal psgrp is conjugate to

~ a unique

FcZ

such that

PF; and the conjugating element can be chosen from

Q

P~

is conjugate to

K°.

This is immediate from Theorem 7. THEORI~M i0. Let

Denote by Q map

a

q

be a psalgebra of

MI=QOS(Q),

and let

the split component of

and is also the centralizer of m,n l~ mn

normalizes

m I.

Let

ml=qn@(q),

of

~ × N

Q

9

and

Q

the corresponding psgrp.

be as usual the nilradical of

q.

Then

in

G.

~ If

q O g I.

is the normalizer of N = exp n~

then

is an analytic diffeomorphism.

mI

Q= ~N, Moreover

in and the MI

N.

If

x ~ G,

If

x eQ

then

x e Ml~x

and normalizes

this we may assume

Q=PF

Then, as

8(ml)=ml,

So

and normalizes

ueQ

onto

a

u

normalizes ml,

for some

8(x-l)x=exp2_X m I.

As

q

and

8(q)~x e Q

we shall show that F c Z.

Write

normalizes a0 c ml,

285

x

x

normalizes

centralizes

x=uexpX ml ~ a d X

and

where

a. ueK,

normalizes

For X e ~.

ml ~ X e m

I.

we can find as in the proof of Theorem

ii0

7 an element

v

Q~U: ClO and

s=Ad(vu)ta o maps

implying

s= i.

centralizes E

in the analytic subgroup of

So

F

u~ and hence

a; y e M l,

We may assume

H e s. 8(s).

The So

where

~(b)

~(b)

erential everywhere.

6 ~ X e ~ Q ~ = 0.

implies that

nyeQ

x=expX

for

y e Q.

and normalizes M1

COROLLARY 11.

M0A 0

Then

~

Write

Let

Let

group of

el~

for

involution of

M1

~

'.

M0

Then

and

Since

~

G c.

Q

m 2.

a0

ml) , If

Let

p

X

If

centralizes

and

~i

is a reductive

mlnY= m 1 .

or

This

YeMlN.

For

Fc~

we write

%F

exp "F"

Then

,. ~,CSF~

~%FC~F

,. Also

G,

K.

Then

Z

and

m,_

M0A 0 =MI~.

A = exp a

is its split component.

=]BlmlXm I.

Then

then

m

a

in

G,

is a C a r t a n

is the Lie algebra of

Ad(~]C~c

Hence with

[X, ~] = 0.

Ad

(M)

,

the centralizer of is contained in the

k~'~K, l e p,

286

GI~

then

(Kn%)x(pnh)

x

cen-

onto

r~ . ~ ] = [(~n~) : (KnM1)°]

be the analytic subgroup of

be the ajjoint representation of

M,

(G,K,8,]B).

It follows from this that

This proves that M2

81

and the corresponding maximal compact sub-

x=kexpX and

P~ = MoAoN0.

and let notation be as in Theorem i0.

is connected.

a

a0 i n

the corresponding map of

m2= [ml~m1 ] and let

and

has bijective diff-

inherit the properties of

talc. If

centralizes

G

M=°MI,

is the centraliser of ~c

Q.

of

in

el~)~

being an ~almmio diffeomorphdsm. Let

for

F C F '.

be a psgrp of

(Mi,KnMl,81,~l)

Ml=(KFl~)xe~(~nml), < ~.

~

nyeM 1

be t h e c e n t r a l i s e r

(as well as

We know that

a ~k

~

An elementary

P F = ~ F N F C P F ,.

(as wellas

complex adjoint group of tralizes

M I.

Q

sES(~)~

[H~Y]=#(H)X

normalizes

such that

FCF' ~PFCPF

Suppose

Thus

~s Kn~\=K~

K~MI=KNM

in

, ~FCF

NF

is a reductive group of class 81

neN i.e.~

and

is the centraliser of

THEOREM i~.

y

= A +,

y eG

If we write

such that

we see that

normalises 3

m I,

PF

F~F'CE.

NFCPF,.

COROLLARY 1 2 . As

y

. A +

E c s*\ {0], n = Z

into

the reductive component of

Let

PFCPF , ~#FCPF

so t h a t

Xe Z

M1 x N

Xe ~

Then

for the reductive component of

nFCPF ,

of

So, by Proposition 4~

We shall call

a= % ; ~ = % .

then

showing that

m,n ~ mn

nF, s

So we need only check that it is one-one and onto.

S&ppose

q.

y~

such that =

We now prove that if

This proves the characterizations of

then writing

component of

A+\A~

is the set of all

calculation shows that the map

xe%~N~

of

mI n !

theft, as n ~ t

~t

q= pF ~ m l=mlF,

~

are stable under

y 6 Q~ 8(Q).

defined by

x, centralizes a.

for the set of restrictions to

8(n)=Zke_E~ (#)

G

into i t s e l f .

%

defined by

the analytic subgrottp of

G

iii

defined by

~i"

Then

finite center.

I~t

P(M2)

is a connected semisimple linear group and so has

ker(p)

is finite.

as we have seen in Section i, that follows from KAM I

Nl_~(KnMl)×exp(p,qml)

is closed. that

Thus

81= 8[MI

MI

eenter(ml)~] ~= Q. M2

m=m'.

with Lie algebra

So

L=A.

are both CM,

Let

m'

I,

is finite, which implies, MI

is of class

If

L

(~ O r a l ) + m 2 = m C m ' .

Since the only compact subgroup of

N

is

It

is the 81-stable

it is clear that

be the Lie algebra of

we see that

Z.

is a Cartan involution and

is the corresponding maximal compact subgroup.

split component of

and

So center P(M2)

M2

I c ~, so that I c M.

Since

KDM I

By dimensionality,

[i], K O Q C K O M

I, h e n c e = K D M I.

We th~s have the decompositions (6)

Q = MAN,

H1 :

MA.

The first of these is called the Lan61ands decomposition of split component of

Q;

and

dim(A)

Q;

A

is called the

is called the parabolic rank of

We shall generally abbreviate (6) by saying that

Q=MAN

Q (prk(Q)).

is a psgrp; it is then

to be understood that we are referring to the Langlands decomposition of shall also write

KM

THEOREM !4. gebra.

Then

for

Let

Q = MAN

be a psgrp of

~ = T + q, G = KQ= KMAN, If

map of

G

Since

onto

~=q+e(n),

then

X=(Y-Z)+(Z+8(Z)),

into

G

We

KNM I=KOM.

is proper and submersive. KxMp×AxN

Q.

G

and

q

the corresponding psal-

and the natural map of

M p = exp(mn p),

then

KxMXA

G = KMDAN,

XN

onto

G

and the natural

is an analytic diffeomorphism.

we have

X=Y+@(Z)

proving that

for any

$=l+q.

Xe~,

with

Yc q, Z e n ;

So the natural map of

K×Q

is submersive. In particular, KQ is open in G. As it is closed, o o K Q ~ G , ~ K Q D K G = G. The fact that the map is proper is trivial. The fact that KM=KOQ

implies that if x = k m a n = k ' m ' a ' n ' are two decompositions of an x in -i m, a = a ' ~ n = n ' for a suitable u c K M. In particular,

G,

then

k'=ku, m'=u

if

m,m' e Mp,

then

m=m'.

= ! +(ran p)+ a + n

The corresponding infinitesimal statement is that

is a direct sum.

The decomposition

G ~- K X M

xAxN

is now

clear. For any

xeG~

we have the decomposition

uniquely determined.

(7) where

a = aq(x),

log : A ~ a

x = k m a n , where

aeA, neN

are

We put n = n(~),

is the inverse of

are uniquely determined by

x.

iq(x) :

exp : a~A.

If we require

ly determined; we then write

287

log aQ(x)

Further the cosets m c MD,

both

k

and

kKM m

and KMm are unique-

112

(s) For H e a

we put

%(~) = ½ tr(adHl~)We also write i dQ(m) = Idet(Ad(m)In)l ~

(lO) Then

dQ

is a continuous homomorphism of dQ(me)

(ii) If

is

Q = PF

(ii F)

a standard

KF =KMF~ 4,

~'a2 ~ p

m(A21A I )

of a .

e oQ(l°gs)

:

dF = dPF~

be abelian subspaces and let

and

~l=~2=.

abelian subspace

and if

~; Q

ac D

A=expo,

we then call

Q.

We denote by

~(A)

A. = exp ~..

Let ~( ~21 ~i ) :

into ~2 that are induced by elements

~(~J~)=~(AIA) a

in

A = exp a A

is asubgrou~of

G/centralizer of

special also.

If

Q

aT(~) and

~

in

is special

as split component, we refer to

Note that it depends only on the set of all psgrps of

is the centralizer of P(A)

aF = ap F •

G.

An

is called special if it is the split component of a psal-

is a psgrp with

Weyl group of

in

~F = ~PF ~

be the set of all linear maps of

If

aeA).

(meM,

Associated psgrps.

is canonically isomorphic to Normalizer of

gebra of

into the positive reals and

we w r i t e

~F = DpF,

Weyl grouDs.

Let

psgrp~

MI

(meMl).

a

in

G,

then

G MI

A

and A = exp ~

~(AIA )

as the

and not on the choice of

with

A

Q •

as split component;

is of class

~

and the

are precisely those with Langlands decompositions of the form

if

psgrps MAN

where

M = °IV] I. Proposition 15.

Let ~ ' ~ 2 be two abelian subspaces of D.

Then ~(a2]~l) is a

finite set and all its elements come from K °. Let s0 be a maximal abelian subspace of p.

k. Since ~ k.m e K ° such that a.1I c s0'

we may assume that ~ , a 2 c s 0 . Then the proposition follows at once from L~MMA 16.

Let b c a 0 be a linear subspace and x e G c such that b x c a 0.

be the Weyl group of (~,a 0). Choose an element H 0 e

b

Then ~ s e a

~0

so

and inducing

Since

s0, 3

s.

y-lxeMc,

Then

Proposition 17 . m(a21 h )

Let s o p

Hx = s H V H c

Let

b.

such that if h is a root of (~,a0), A(H0): 0~Ib=0.

Let M c be the centralizer of H 0 in G c. of b in Oe a

such that

M c is connected and so is the centralizer aeh that

0=

so that for H e b,

Choose

normalizi

H x = H y = sH.

is finite V abelian subspaces ~ , a 2 c p.

be a special linear space and

A = exp ~.

By a root of

(~, a)

we

mean a linear function W e s*\[0} such that for some nonzero X in ~, [H,X] = ~ ( H ) X ~ / H e ~; the linear span of these X is denoted by ~a,~" Let A a be the

288

i13

set of all roots of (~,a).

Denote by s' the subset of s where the elements of A

are all nonzero, and by F s the (finite) set of connected components of s'. fer to the members of F s as the chambers of s.

For any chamber s+ we denote by

As(e +) the subset of A s of those roots which are >0 on a+.

(12) Let

n( s+) = ~ s M I (resp. ml)

M.

n( s+)

Let

( s+)~ ",~

be the centralizer of

the Lie algebra of

~

in

N( s+) = e ~ n( s+) G (resp. 9);

is a suhalgebra of nilpotents.

(13)

H = °MI;

decomposition;

finite

P(a+)e~(A),

t{~o(Sl" )

~(a+) its psalgebra, P( s+) = MAN( a+) its Langlands

s, t.en

such that

t" < =

m,

~( s+ ) = m + ~ + .(s+).

a+ ~ P(a +) is a bijeetion of T s onto ~(A).

If

and

Put

P( a+) = raN(s+), THEOPJ~4 18.

s We re-

In particular, ~(A) is

i) and P<s )are conjugate if and o a2.

~(Sl~ )

acts simply on

yif

F s.

We may assume s c s O where QO is our fixed maximal abelian subspaee of 9. Sinee a psalgebra whose split eomponent is ~ and hence whose reductive component is ml~ s= center(ml)N p.

Hence s has the following property: if H e sO and b ( H ) = O for all

roots b of (~;SO) for which b] s= O~ then HI a. and h + the union of %

and R2; where ~

and R 2 the set of all b c h positive system in f~.

Let A be the set of roots of (~sO)

is a positive system of roots of (ml,aO)

such that ~I s+eAs(s+) •

It is easily seen that h + is a

Let Z be the set of simple roots in h + and F the subset of

Z of simple roots vanishing on s. only if b is a combination Zo~Fm

It is then clear that b e A + vanishes on s if and ~(m

e 2Z; m >_0).

So s= SF~ p(s+)= DF; n(s +)= nF.

So we have the assertions concerning P(~+) and ~(s+).

and

t=Ad(x)[ a, it is obvious that

if

P( 0 for all ~ c E .

E = A s ( s +) for some a + e E

s

. But then

Q = P ( s +).

289

eZ~,,> 0).

This implies at once that

This completes the proof

In

114

m(Qla )

Note that

Q

Suppose that compositions of

Q

in general does not act transitively on

is a psgrp with

Q=MAN~

and its Lie algebra

q.

q=m+

~+n

F a.

as the Langlands de-

Then, in the notation of the above

proof;

(14)

~=m+~+

The elements of

~

E

~= ~

are called roots of

cannot be written as

B +7

linearly independent~

with

Z cS m

set

=h

let

(15)

h

~T

rafT=m+ ~+

Q.

~,7c E.

~ cE

a

N~

~ ~c,T

"b

~o(-~)

= ¢.

is said to be simple if it

As we saw above~ the members of

ISI =dim(~)-prk(G),

written uniquely as TcS

~ U ( - E ) = A a,

and every element of

where the

m

are integers

E

~ 0.

S

are

can be For any sub-

and define

"T

Zs,~,

nT:

~ 9s,b, bcEkAa,T

qT=mlT+nT

•

If

(16)

~r={i.ica, ~(H):0~ScT]

then

mlT

is the centralizer of

Langlands decomposition mlT.

Let

sition.

~

~T

in

m T + aT + nT,

9.

where

qT mT

be the corresponding psgrp and

is obviously a psalgebra with is the orthocomplement of

~=M~TN

T

~T in

its Langlands decompo-

Then

(17)

Q c % , ~ToM, AT=A, NTCN

It is easily seen on taking raining ence.

Q

Q

to be a standard psgrp

is of the above form

~

and that

T ~

PF

that every psgrp con-

is a bijective correspond-

From (17) or from the conjugacy of a psgrp with a standard one we get Proposition 19.

Q c ~.

Let

Q= MAN

Let

Q=MAN

and

Q= M A N

be psgrps of

G.

Suppose

Then

(18) Proposition 20. then

*Q=QN~

is a psgrp of

the set of all psgrps of

GCQ

~.

*M=M,

A=*AA,

If

Q=MAN Q ~ *Q

and the set of all psgrps of

is the Langlands decomposition of

(19)

be a psgrp.

The correspondence

*Q,

then

N= *N ~, *A : ~ FI A, *N = ~ N N.

290

is a psgrp

CQ,

is a bijection of ~.

If

*Q= *M*A*N

i15

Let

q=m+

if

that

b

s+n

be the psa&gebra of

is a Borel subalgebra of

Conversely, if

b

Q.

For dimension reasons it is clear

mc, b + ac + nc

is a Borel subalgebra of

~c

is a Borel subalgebra of ~e"

contained in

~c'

then

bD%

from which, for dimension reasons once ~ain, we conclude that

bn(I

such that

iZ(erl ..... rm)i ~ N(z)rl +'"

Fix an integer

N > i

.+r +l m

and let

%* = {~: ~c v*,lk(Crl ' .... rm)l O

log :A O ~ a0 aO

~

2~

on

inverts

A+ 0 = exp aO,

and

~ h c / ~ +.

is a fundamental domain for the action of

composition

Let

is the corresponding global lwasawa decomposition.

(~,aO) ; A +,

Weyl group of ~0

and employ our usual nota-

~= ~ + aO + n O.

so that,

are the root spaces.

aO.

From the polar de-

and the relations

u

% :

keK

u (c,e(%)) s se~

we get

(5)

a = K C~(A~)K Our main tool in the study of matrix coefficients of simple representations

of

G

is the behaviour of the differential equations satisfied by these.

therefore necessary to obtain the expressions for elements of operators on Let (resp.

~).

II

@

It is

as differential

KXAoXK. (resp. ~2) Then

be the orthogonal complement of

I i + 1 2 = (m0+a0)

=n0+e(n0). 304

Since

m0

(resp. aO)

n0~l=n0~

~=0 ,

in the

T

129

projections of and

I2"

nO

and

O(nO) on

~ and

D are injective, and map onto

11

Hence

(6)

dim(If) = dim(12) = dim(n0) = dim(0(n0) )

Let (7)

A~ = {h:h~A0, ~(logh)¢0 V Z c A ]

Note that

A O'-

LEMMA 2. f~

and g~

(8)

Us~~s.A~, Let

~

acting on A 0

ZsA+~ Xc gZ, Y=-OX,

on A'O by setting for all

in the obvious fashion. and let

fT,(h)= _(eh(log h) _e-h(log h) )-i

Then X - Y e Ii, X + Y e 12,

X~O.

Define the functions

hSAo,' gh(h) = -e -h(l°g h) fh(h)

and

Y = f~(h)(X-Y) (h-l) + g~(h)(X-Y)

(9)

This is a straightforward calculation. h-l) For any h s AS, ~ = I + a0 + I~

.Proposition I-

By (9), ~ = I + a0+6(n0) c I + a0+I~ h-l). reasons since

into

open map. and

Let

G

Let

a+= ~ K = ~o K.

~'~0

where

~i ~ 2 ¢ ~

For kl',k~s K, h'~ A0,

(k~l)

•

' k2~

, - (kl,h,k 2 ) - k l ~ 2 of

is open in G

(d~)= of

~

and

~/ is an

generated by

(i,t)

is given by

(~2) -1 (k~l) = ~l a ~2

(d*)(kl,h,k2)(~l®~®~2)

V k l,k 2sK~ heA~,

G+

be respectively the subalgebras of @

(i, aO). Then the complete differential

(zo)

h'

Then the map

is submersive; in p~rticula~

oo

•

The sum is direct for dimension

dim(l~h-l))=dim(0(n0) ) by (6).

Proposition 4. KxA6xK

the sum being direct.

and

a¢9]0.

we have the identity

(hk2)-I klklhh'k2k ~ =klhk 2 "k l'

from which the formula (i0) is immediate.

For any h ¢ A 0' we write ~®~0®~ into @ and

co

Dh= ( d ~ ) ( 1 , h , 1 ) .

305

ThUS Dh

i s a linear map o f

13o

-(h-1)a Dh(~l ®a® ~2 ) = ~l ~2

(lZ) Let

~0

be the subaAgebra of

®

generated by

,

(~

(l,u0).

~,

~2 ~

Since

a ~ ~0)

~ = ~ + a0 + ~(n0)

is a direct sum, the map A: 6 ® a ® { extends to a linear bijection of (z2)

(~ e e(mo),a c ~o,~ e ~)

~ ~a

8(~0)®N0®R

@.

onto

This implies that (direct sum)

o = O(no)¢ + ~o ~

Given

ge~

v(g)

we write

N0 ® ~

unique element of

(l~)

g - v ( g ) e e(nO)@,

We have

deg(v(g))Zdeg(g),

Proposition ~. the functions we can find (i) (ii)

Let

~

for the

Dh(t(g : h ) ) = g

seen.

defined by (8). ai~9] 0

deg({i)+deg(ai)+deg(~)0

L

v

and

V~

G

in a complete locally

are finite dimensional, c V ~

dim(V@) < C

dim(@)2Vte

So L = V @ C

dim(~(~)v)< ~.

be a simple representation of

Then all the

V~.

By

condition (iii) in the theorem this implies that From the definition of

is dense in

6G.

and

~

a con-

g(K).

The next theorem establishes a very close connexion between representations of

G

and those of THEOREM 14.

space

V.

®.

Let

Suppose

~

be a representation of

dim(V~) < ~ V

@e 8(K).

G

in a complete locally convex

Then all elements of

If

L

is a linear subspace of

CZ(L)

is invariant under

~(G);

~

are

~(K) U~(~),

invariant under

8-

~ = Z@ V@ (algebraic

finite, wea/CLy analytic (in particular differentiable), and sum).

E@V@

then

and

(32)

L = C~(L) n

Moreover, the correspondence

L ~ C~(L)

set of all linear subspaees of

~

is an order preserving bijeetion of the

that are stable under

if(K)U ~(@)

set of all closed linear subspaces of

V that are stable under

lar,

is irreducible under

Tr is simple if and only if

~

onto the

7(G).

In particu-

Tr(K)U~r(@)

(in the

algebraic sense). Since

V~nV~

is dense in

7T(8) leaves eeeh

F~rther

V@,

we mnst have

v~nv@=v~

The first group of assertions is now obvious.

Let

£0

(resp. £)

ordered set of all linear (resp. closed linear) subspaees of under

fr(K)U 7r(¢~) (resp. Tr(G)). Suppose

TF(X)V ¢ CZ(L) ~ x e G. L ~ C~(L)

of

Suppose Meg.

£0

Then

also~ we assert that

L ~ £0,

It follows from this that

into

£

and

~

v e L.

C£(L) ¢ £.

dim(V@)>0 V H , H ' ~ aO"

P

are

_> 0.

Then

So it suffices to show that

~++0c ~~0"

Now we

(H,H') > 0 V

But this is a standard fact. s0 £ ~

Clearly

be such that 4-- *

so .Zk+=-~+.

G4-

(~0) = O"

330

For

Hc a0

write

H*=-s0H, (expH)*=

155

L~b4A i~.

For all

(29)

l°0 VH' ~J~O~ in p~tic~r tion in (29) ~ select H0~-lu-lk)~/ksK.

~ ~0

be t h e fundamental dominant i n t e g r a l elements i n A. + p r o o f o f Lemma 18~ we see t h a t ~0 i s t h e set o f e l ! are

such that if

heCl(Ao) , the highest eigenvalue

for

~(logh-~

cj

be the

In other words,

(30)

the

A

e~(~(%(~)))=lf~(~)*ll

while all eigenvalues are

II~u(h~),ll<e ~(logh)

~(logh-H(hk))_>0.

Let

acts in a Hilbert space

e K,

~.

r>l

and regard

unitary V k

n0C~ep

Hence we may assume that

that are dominant and integral (relative to

We may ass[Ime that and

by

be as in the proof of Lemma 18.

~c rA, 7r~ is a representation of Gc.

G

~+°0"

Tr~ be the corresponding representation of the simply

connected covering group of

sentation of

l~h*+~0(~)

we may replace

G C G c,

we have

~j

~+ 0"

in

So (30) gives

To prove the second rela-

uh-lu-l=h~heA0.

I)/t ~o~'lu-lk)=-H0 (by) for some

~c"

where

Then H0(h*k) =

veX.

Hence from (30) we

get the following which in turn proves the second relation in (ix)):

GI)

~(Zog h+H0(~*k))>0

(~A,

h~ ~(AO), ~).

comT.mR~ 20. I p0(%(~))l ! %(log h)

(h ~ Cl(Ao), k ~ ~).

For, P 0 ( H * ) = P 0 ( H ) V H c a0.

ki=p0.

COROLLARY 21.

~

c >0

So we use (30) and (31) with

such that

[%(~)11 _< . 331

(h ~ Cl(Ao), k ~ ~).

156

from which the lemma follows at once. Proof of Proposition 17 . By Corollary 20,

-%(Xo(~))

-%(logh)

e

_> e

(he CI(Ao) , k 6 K).

Integrating~ we get the required result.

7-

The representation

YDWe fix JF=9/OF rank

FeZ

and write

are as in n°4.

r= [~ :~F ].

rF

of

D F.

mF

for the Weyl group of

Then we know that

JFDJ

and

(rufF,a0). J = ~ 0 JF

In this section we shall define a representation

associated with the actions of

~

and

~F

in

and

is a free J-module of FF

of

~F

9/0. We recall the resUlts of I,

Section 4 on finite reflexion groups. The form ]B(.,.)

gives rise to an isomorphism

bra of polynomial functions on then

(',.)

~0c"

9/0 generated by

write

for the space of harmonic elements of

a0,

(32)

(.~.)

uI = i ,

for ]{

of

such that (i) each

(iii) Ul,...,u r

uj

9/0,1R is the JR-sub-

is positive definite on

u 2 .....

9/0~IR"

(w= I~ol) (ii)

(ui,uj)=6ij ( l < i , j < w )

It follows from (iii) that

JF = JUl + "'" + JUr Let

J+

be the ideal in

then have a representation of

J

(direct sum).

of all elements without a constant term.

9/0 in

9/0/)/0J . Since 9

+

9/0 = 9]0J++ l~

sum, this representation can be realized in ]~I . We shall write ue910,

then

3

unique

(34)

We

9/0• We can then select a basis

uw

is homogeneous

form a basis for ] ~ N JF"

(33)

9/0 onto the alge-

u,v~ 9/0' (u,v)= (~(U)~)(0),

is a nonsingular symmetric bilinear form; if

algebra of I~[

a~

If we write, for

Pu : ij e J

uuj =

70

We

is a direct for it.

If

such that

E

Pu

l< w

u : ij

(l<j<w). i

--

--

Then (35)

70 (u)= (Pu : ij(0))lm0-mFVHe

~.

%

(resp. % )

is

~F(U) n = 0

A +,

be the product of all

be the product of all the vectors correis skew under

then

for

m

(resp. mF).

(resp.%).

JFJ+=~0J+NJF=~i~rJ+Ui

> m 0 - mF~

Finally~

Let ~

WF

be the degree of ~

has zero constant term~

(Pu : ij ~ JF )

~F(~) = 0.

n > m 0 - mF~

deg(~i)~m0-m F

. If

ue JF

In partieular~ if u e JF

so that

TF(H) n = 0

for

l0}

~eE\ F

Let (43)

v e ~ F.

Then, as we saw above,

wj

~

qv:ijeQ

l < i~< r ViVF(qv• ij )

such that

(nod ~F1F )

(l0

and

denoting the norm in

Cr,

C(0,v)>0

l@(mexp H;n)l _< C(~)H(mexp H)dF(mexpH)

(48)

] Cv(m exp H; 0)I 0 , am_>O

such

that

(55)

I®(m~(D) - e~(tlrF(~) + ..-+ tdrF(Hd))S(m) I a~

_< C cF(m) dF(m)e VmeMiF,tsQ

~.

-~(t I +

. . .

+ t d)

From (55) we obtain easily the second relation of (53).

We shall now rewrite (55) in a form suitable for the purposes we have in mind.

To this end, given any compact set

region corresponding to (56)

E

E~CI( a0) + , we define the "sectorial"

by S[E]= [exptH' :t> 0, H'e E]

The main result of this n ° can be formulated as follows : 337

162 +

Proposition 2~. Let H0eC£(a0) , H0#0 , and let F ~ Z be the set of simple roots of (g,a0) that vanish at H 0. Define h F as in Proposition 23 . Then ~ C, 6>0, a compact neighborhood E(H0) of H 0 in C~(a~), and an elelent ~ h F , such that, V h ~ S[E(H0)] ,

(~7)

J~(h)dF(h) - ~(h)J _< Ce -~ljloghIl-*~(l°gh)

Fix a, with 00, 6 >0,

l¢(h)-e(h)l ~ C e -~Hl°ghH -*p(logh)

such t h a t ~hs

S[E(H0)] ).

But Ed F is the first eo~K0onent of the vector ¢ while, by Proposition 23, the first component ~ of @ lies in h F. Proposition 29 is now clear. 9-

The fundamental estimate for

We can now prove the main theorem of this Section. The proof given by Harish-Chandra in [ 6 ] is different, for the Upper bound. The present proof is

338

163 of interest because the same method is used for the study of the ssymptotic behaviour of general eigenfunctions on THEOREM

"~o.

Let m0

G.

is now am arbitrary group of

be as in Lemma 22.

-%(logh)

(60)

G

Then 3 C > 0

-%(logh)

e

~(h) ~ C

~

e

We must establish the upper bound.

Fix

~0~c~(%),HoW°' and

(61)

such that

mo

(l+ Itloghll)

We use induction on

a split component, the result is immediate since +

use Proposition 29

~h~ CZ(A~)). dim(G).

dim(°G) 0

So by Lemma 22,

such that

l 0

%F

(h)

and so the induction hypothesis is applic-

such that

l0,m(0

and all matrix

Since

and all

for

r,r' i'~j' s

j,(~rs~s;h;ar,)l

we get

15f%(b;kfl~2;a)l

_< eI dim(%12dim(e2) 2

2

Ifij(~rs;h;~hs-la r,)l

r,r', i,j,s

_< cz aim(%)~dim(~2) 2 r,r'~i,j~s

344

I fij(~rs ;h ;~{sar ,)l

169

B~t 7sar,fijhrs=dim(@l)dim(@2)(Ui~rs)~((~sar,)(f~vj)). Writing C3=maXr, s C(hrs), C2=maX~,s,r,i~Zs r,l~, m=maXr,sm(hrs), we get sup

[ fi j ( ~r s ;h ;Tsar , )l ~ C2C3c( Ol )mdim( ~1)dim( ~2 ) l0, a,b~, s~(V).

if and only if

Moreover, the seminorms

II(I +~)rJ~blsll p

f already induce the t o p o l o g y of

~ ( G : V).

It is enough to consider the case

V= C;

the case of arbitrary

V

follows

from the estimates in the scalar case by replacing the vector-v~lued function with

ko f

where

kcV

. Let

~

described in Proposition 8.32.

be the function on

Then

tion 8.33, we may conclude that

that

f ~cP(G).

so that, by Proposi-

f

by

afb

and

$

Theorem 8 now by

i+~

we

So, if c'P(G) is the (Frechet) space of all f ~ C~(G) such

(l+~)r(afb) ~LP(G) V r ~ 0 ,

identity map

~rafbcLP(G) V r , a , b ,

Replacing

f

with the properties

a(~rf)b c LP(G) V r ~0, a,b e ~.

shows that suPG(~ -2/p ~rjfl)0, r>0, (x c G)

(~ is locally integrable and the corresponding distribution, denoted by

is of type (16)

3

p.

Let

(f ~ c~(o))

_~i'bi:r(f)

l~(x)l < C E(x)2/P'(l+~(x)) r Then

lO~

353

-r

we find the following result:

_< e ~(x)~/P(l+~(x)) -r ~r(f)

Substituting in (25) we get, with

If

(l+~(Yl~2))

lO

such that for {en : n e N(@)}

~0

c(8) -q

~ (7(~)en,7[(nq)en) n e N(8)

(TT(~q~)en,en)

n e N(8)

CI = E8 e g ( K ) dim(8)2c(@)-qO

being the so-called formal degree of

I(~-(~C~)en,en)l =

(nZl)

= d ( v ) -1

al(v(X)en,en)ledx

~.

In this case~

I~ (~qc~)(x)(w-(X)en'en)dX ~

d(~)-1/211~[t2

Hence

1%(~)t _ 0, ac~).

355

G

such that

Tk = T ~ k

c K.

is a&ready continuous with ref ~ suPG(F~-2/P(1 +~)rlafl)

180

By Theorem 4, we can select integers and an integer

r >0

IT(~')l _< Let

G@ (@eg(K))

mi_>0,

elements

a.le@ (lO,

llloghll_
0, ~b > 0 '

b e H,

and an open

such that

U b = b e x p n b is open in H and X ~ b e x p X is an analytic diffeomorphism of

~b

onto

Ub

(ii) w(be~ X) ~ Cb(n~% I~(X)I) %

LEMMA 7"

w

we can

Let

Z

(X~

be a finite set of one dimensional characters of

%)

H and

let

(3i)

Wz= n

J×-il

x{Z Suppose Then

wZ wZ

H'

is, as above, the set where

w>O,

and assume that

H'

is dense in H.

is admissible. is certainly analytic on

H',

while

H\H',

being the union of a finite

number of analytic sets, each of which is a proper subset of some connected

370

195

component of

H,

is of measure zero.

be empty and

ub

to be any sufficiently small neighborhood of

Fix

b c H.

If

Wz(b ) >0,

we take ~ 0.

to

Suppose

wz(b ) = 0 and ~ is the subset of Z of a l l X with X ( b ) = l . For X c ~ @ write ~X for the element of ~ such that × ( b e ~ Y ) = e x ~ i~X(~) (Y~ ~). can choose isfied

ub

t o be a n o p e n n e i g h b o r h o o d

such that

(i)

of (30) is

sat-

and s u c h t h a t

lexp(i%(Y) - l l V

0

of

we We

Xe %, X' C Z \ ~ .

%=2-r~,eZ\Zbl

~ ½1~(Y)l,

IX'(bexpY) - zl ~ ~l×'(b) - l l

I t is enough tO take

%= [~X: X e ~ } ,

~b=l

and

X ' ( b ) - i t where r = JZI .

The main result of this n° is then the following. Proposition 8. is admissible.

Let

Then

~0,w

be as at the beginning of this n ° .

~ P ( % , w , H ' : V)

Suppose

is complete, and

sP(%,w,H' : v) = ~(H' : v)

(32)

as topological vector spaces.

(~3)

In particular, if

~P(%,Wz,~'

as topological vector

estimating the

k

S:U

v) = ~ ( ~ '

:

is as in Lemma 7,

v)

spaces.

It is obvious that sion is continuous.

:

wZ

8~(H' : V)c~P(~0,w,H' : V)

and that the natural inclu-

We must therefore go the other way. -norm in terms of the vp

S:U

-norms.

It is a question of

Since

H

is compact, it

is enough to prove the following lemma. LEMMA 9"

Fix

following property:

b c H.

Then 3 an open neighborhood

given

u e ~, sub(V),

3

U

of

b

(i_ 0,

VB,o(%)

= j~ le~(atk)a~

=

j

ae

Itr(atue)li

for such

is unitary ~/

X £ ~.

b e rA

where all

is a weight of

G • Let

is already a representation of

acts in a Hilbert space, that

F ~ E.

the set of

with the following properties: are

>_ 0

and ~ c Z \ F

distinct from

c~ > 0

~,

where all c~ are ~ 0 and ~ e Z \ F

e;>0

+

Let of > 0

Z

H0 / 0

be an element in

vanishing at

H 0.

if and only if

P~S= [~i,...,~i].

C4(aO)

Then~ if

( Z ~ e Z d~ ~) (H0) >0.

~j(Ho)>O

for

_q i _< o "
0

for

(i) and (ii) of the lemraa. Let

tions of

F

is precisely the subset

are numbers

~ 0, ~ Z \

q<j = 0

cj

are all

(i 0;

imply

if

n. with J distinct from

~

dj• are all

where the ~

> 0

~,

and

can be written as

cj=0

(~,~> = 0,

n. are

~ ( Y j ) ~ = 0,

it is clear that

is a weight of

dl~ I + . . . + d ~

and the

for

q<j 0 and ~

Z q < j < _~

aj>0,

e ~ \ F+b~ >0.

or in

then

v I a0 =~8 c Z bs~

A . Hence

v=Z~6T

v(H0)>0 ~ Z B C ~ \ F

b~B

is enough to prove that

C~(Ao)

b~

where the

b~

are

are all ~'0 I a0

_> 0.

and

> 0 --

Since

VB, 0

VB,o(h)~0

as

is spherical and

h ~,

Then we can find a constant

~q<j0

h c>0

being in

G = K CZ(Ao)K, CZ(Ao).

and a sequence

it

S~ppose

[hn}

in

such that

(20)

h n ~ ~,

VB,o(hn) -->c > 0

~

n>l_

Passing to a suitable subsequence we may further assume the following : 3 subset

(21)

F ? Z

p crA

is bounded,

is a weight vector, of weight Let

B~Z\F=:> ~(loghn)~+~°

such that properties (i) and (ii) of Lemma 4 are satisfied.

select an orthonormal basis

2 ~ j ~N.

aij

Vl,...,v N pj

for the space of

(say), and that

be the matrix coefficients of

~p

~i = p. ~p

with respect to this

Let now K n

Then vB,o(hn ) = fK

= [k :k oK, h k

is elliptic and regttlar}

n

dk

and so, by

(20),

n

(23)

]"

dk>c>O

K

n

379

We

such that each v. 0 Then pj ~ p for

basis.

(22)

a

such that

~F~(loghn)

Choose

and

is either 0

bs>0"

Proof of Proposition ).

this is false.

where the

B/t, by the choice of the orderings, each

204

On the other hand, if value

i

since

k C Kn,

all eigenvalues of

h k c G[B' ]. Hence

~b(hnk)

Itr(~(hnk)) { i N

n

are of absolute

for s~ch

k.

In other

words,

k~K n~l

ajj(hnk)l_i)

ISjO

is a constant independent of

Lebesgue measure on (~=MLR)

,

and since

L R.

Since

f ~ T (Q)

f,

and

dL R

is of course a

is continuous from

C(G)

vLIM is tempered by Theorem 5, we are done.

For the results of this Section see Harish-Chandra

385

[ 8 ].

into

c(Ml)

12.

The invariant integral on

C(G)

We are now in a position to extend the theory of the invariant integral to C(G).

Throughout this section we fix a complete locally convex space

the functions considered will take values.

We note that if

G.

V

in which

are groups of

i

class

~

(i=1,2),

C ~ ( U I × U 2 : V) with

and

with

U. CG.

is an open set, the obvious isomorphism of

C~(UI :C (U 2 : V))

C(U I : C(U 2 : V))

induces an isomorphism of

as topological vector spaces.

C(U l × U 2 : V)

We leave the elementary

proof to the reader. i.

The case of a compact CSG

We assume in this n ° that contained in

K.

system of roots

Let P

b

for

G=°G

be the

and that

CSA

(~c~be),

and define

(feCc(G: V)). THEORE~ i. (i)

There exists an integer

~G E(bX)(l+d(bX))'qdx uniformly when

(ii) sup I'a(b)l

b

< ~

is a compact

L=B

B.

corresponding to 'IA P

q ~0

for all

'Ff

and

CSG

of

We fix a positive as before

such that b eB'~

the integrals converging

varies over compact subsets of

B'.

SG ~(bX)(l+d(bx))-qd~ < ~

beB' By Proposition i0.i0, for any the map

f~'Ff

of

C:(G)

into

feCe(G), ~(B')

'Ff,B= 'Ff

lies on

is continuous when

C:(G)

~(B'),

and

is equipped

with the topology coming from the seminorms

f~

~ Izfldx G[B']

But, by Theorem ii-5, these seminorms are continuous in the topology that inherits from

C(G).

So, ~ a continuous seminorm

v

on

C(G)

c~(G)

such that

f ~ Cc(G),

(1)

~up f'a(b)~ beB'

For each

(2)

beB'

we denote by

f(xbx-1)dx

~ ~(f)

G ~b

the positive Borel measure

%(0 = I 'a(b)l ~a f(xbx-1)dx

Theorem 9.11 is applicable and shows that the that for a suitable integer

q~O,

386

~b

(f ~ Ce(G))

are all tempered and in fact

211

(3)

sup b~B'

~ Z(l+d)

-q dBb < ~ ,

the integrals involved in (3) being uniformly convergent. for each compact

ECB',

Since infbcEI '~(b)I > 0

(i) and (ii) of Theorem i follow from (3) and the uni-

form convergence mentioned above. It is clear from Theorem i that the invariant measures on the regular elliptic conjugacy classes are tempered. THEOREM 2.

For any

f c C(G : V),

the integral

~G

(4)

f(xbx-1)dx

exists for all lies in If

beB'.

8~(B' : V), q

In particular, and

f ~ 'Ff

is an integer

continuous function

~ 0

f : G-V

'Ff

is well-defined.

is a continuous map of satisfying

Moreover,

C(G : V)

'Ff

into ~ ( B '

: V).

(ii), it is immediate that for any

such that

(5)

sup(~-l(z +o)qlfl s) < G

for all

s eh(V),

the integral (4) exists

assertion and shows that

'Ff

us write

f ~ 'Ff

¢

for the map

the topology as a subspace of ~(B'

: V)

tinuous linear map of sion by

¢.

Fix

in the topology of 8b

is as in (2),

b sB'.

is well defined on from

Proposition i0.i0 and Theorem 11.5,

pleteness of

V

C~(G : V)

¢

for all

C(G : V).

f e C(G : V).

~=(B' : V).

is continuous when

it follows that C(G : V)

B' into

This proves the first

C~(G : V)

Let

Then, by is given

By the density theorem 9.2 and the com~

can be extended uniquely to a con-

into

8~(B' : V).

Let us also denote this exten-

f eC(G : V), b £B',

and choose

fneC

C(G : V).

If

the fact that~

q~0

is such that

(G : V)

such that fn

SG ~ ( i + o ) -q dB b < = ,

where

V s eh(V)j

sup(~-z( 1 +~)qrfn- fls) " o

(n - ~)

G

implies that

V s e h(V)

This shows that

'Ff(b) : Zim ®f (b) = mr(b) n~

n

Theorem 2 follows immediately. 387

(b ~ B')

212

2.

In which L is arbitrary

We now treat the general case. is an arbitrary 8-stable The invariant integral

CSG. 'Ff

G

is an arbitrary group of class

We define

PI

and

dx*

Z

and L

as in Section i0, n°l.

is then well-defined for each

f e C~(G : V).

From

(10.19) we have the following relation which is noted for future use:

(6)

ID(h)l ½ = I 'Ai(h)h + (h) l

THEOREM ~.. Let notation be as above. (i)

S~* E(hx )(i +g(h x ))-q dx uniformly when

h

Then,

< ~ V

(hcL')

~

an integer

h c L',

q~0

such that

the integrals converging

varies over compact subsets of

L'

(ii) hSUp~ ~' I 'a I ( h ) < (h)l fG. ~(h = ) ( l + o ( h = ))-n d=* < ~ Moreover, for each

r>0~

~

a constant

-

(7)

C >0

such that,

V

h~L',

r

]'~I (hyA+(h) I ~

m(hx )(l+~(h X ))-(q+r)dx* ~ Cr(l+~(h)) -r • G*

We use Proposition i0.6. component.

For any

f ~ C(G),

Select a psgrp

Q

G

of

with

LR

as its split

f- 0

such that

(i +~(hX)) ~ c(i +~(h)) Let

write

Q=MLRN

h=hlh R

be as above a psgrp of

(hl e L l ~ h RcLR).

Since

G with

9.20,

3

n"= mn'm -I Cl>0

we see that

such that

LR

G = K M N L R,

h x = kmnhn-lm-lk -I = kmhn 'm-lk-I So writing

(x ~ G, h ~ L) as its split component and

we have, for

x=kmna~

(n ' = nh-ln -I )

h x = kmhm-ln"k-l~

and

n" e N.

l+c(mln0)>_ci(l+c(mi))~/mleMl,

By Proposition

n0~N.

Hence

(l+~(hX)) = l+~(hmn")_>Cl(l+~(hm)) NOW

hm=himhR

and we can write

hi=Ulh'U 2

F _> 2 ~(II%11+11"2LI), for %%fin ~(hR),

showing that

(13)

h ' c e x p ( [ i n p), Ul,U2CKM . So

~, ~2~ IR. .ence ~(h'h~)>_2-~(~(h') +

l + ~ ( h m ) _ > i + 2 - ~ ( l + ~ ( h R ) ).

the relation (12) with COROLLARY >.

where

~

e = c I • 2 -~. an integer

sup Im(h)1%i ( l + ~ h ) )

This gives, as

~(h)=~(hR),

In view of (6)~ we get

q>_0

such that for each integer

r>0,

r J[~G* ~(h x * ) ( l + ~ ( h x * ) ) - ( q + r ) d x *
0

and

rivative

< 0

bT~ T

respectively.

V'Fa~ B

is

C=

on

~!,b

and

~,i

that

To formulate it precisely,

the null space of

and we choose the open neighborhood

to be convex, bounded~ and star-like at

convex open subsets of

is of

We are then in a position to use I, Theorem

An application of this theorem to the integrals

5~

[~]

above is not only semiregular but that ~(b) = i

occur in (36) leads to the main result of this n ° . we introduce some notation.

if

~

in ~

b. of

0

b+ and bwill denote the 7~ 7 7, T i bT, ~ \ 7, and on which (-l)S~ is

whose union is Then, for any

ae C(G : V),

bexp b+T~T (resp. bexp b$~T)

and any

ve~

the de-

and extends contin-

+

uously to

(37)

CZ(bexp 67~m)

(resp. CZ(bexp b~, T))'7,

(V'Fa,B)~(b exp H)

=

lim

b± are well defined and continuous on

(38)

be~(~

In particular~

(V'Fa,B)(b expH)

(He 7)

~i~

exp 7-

On the other hand,

~)~L'(P%I)

so that

'F extends to a function of class C ~ on h exp I which~ as a,L 7,T usual~ will be denoted by the same symbol. We now apply I, Theorem 3.30. As

395

in

0.

220

before~ the reslmlting formula is valid not only for the special type of have been working with, but for all

f { C(G : V).

f

we

We obtain then the following

theorem. THEOREM II. as above.

Let

Let 'F.,B

b cB

be such that

and

'F.;L

Then~ for a suitable choice of

[~]~gI(2,~)

and let notation be

be defined respectively using

y

and a nonzero constant

c~

Pb

and

we have,

PI,I"

V

fcC(a:v), v~, ~7, (39) ('v'Ff,B)+( b expH) - ('v'Ff,B)-(b exp') = ce(6l#I-Sl)(H)( '(J)'Ff#L)(b exp')

From this we obtain the following result which will play an important role later. THEOREM 12.

Let B C K

be a

CSG

of

G

as before and

that for any noncompact CSG L of G~ 'F = 0 on L' f,L Under the hypothesis on

f ~C(G : V) be such

Then 'F is in C~(B : V). f,B

f, (39) shows that all the derivatives of

'Ff,B

co

extend continuously across Suppose now S+(b)

'F is of class C on b exp b . f~B ~/,T is an arbitrary element of B, not necessarily semiregular.

b

~.

So

is the set of roots of

is the nullspace in

b

of

(~c,bc)

~ ~ S+(b),

which are positive and singular, and

Cco

H ~ 'F (b exp H) (H ~ is of class f~B any point of b that belongs to at most one bB, b

in the neighborhood of

while all its derivatives

~y~ T

are bounded on

b~

then the above remark implies that the

bT,T)

function

If

. By I, Lemma 3.21, this function is of class

Cco on

bT, n- •

Going over to a different positive system of

(gc,bc) does not cause any diffi-

culty since its effect is to multiply

by a

5.

'F f,B

C

function.

The limit formula for f(1)

In this n ° we shall obtain the limit formula which expresses terms of the

'Ff,L.

THEOPd94 I).

G

Let

tive system of roots of We assume that

P\ PI

(40)

is now an arbitrary group of class L

be a 8-stable (~c,lc);

and

CSG PI'

of

G

with

in

~. CSA [ ;

P,

a posi-

the set of imaginary roots in

is stable under complex conjugation.

qG = q ~ = ½

f(1)

Let

{dim(G/K)-rk(G)+rk(K)] -81

Then where

qG

is an integer

61=iE ~. 2 C~,~pI

>_0. Then~

Define V

~fp=%ep

feC(G:V),

396

H~

and

'%=e

o'~]pO e 51

P.

221

(41) If

('~'Ff,L)(1) L

is fundamental, then

(42)

f(l)=

The constant

e(G)

PoP I

and

P\ PI

G=°G

and

L=BCK

= 0

(L not fundamental)

~

a constant

c(G) >0

(-l)

qGo ( G ) ( ' %

'Ff,L) (Z)

is independent of the choice of

such that

( f 6 C ( a : V)) P

and

is stable under complex conjugation. is a compact

independent of the choice of

CSG,

P=PI ~

integer

> 0

L

In particular,

+ (~)(7) 0)

of

small.

Let

if c(G)

P.

so that

(~Yp'Ff,L)(I)

~p'Ff~ L

is well defined,

extends qG

is an

by i, Lemma 8.9.

We proceed as in n ° 3

at

as long as

and (42) is valid with

Here we must recall Theorem i0 which guarantees that continuously to all of

PI

to descend to the Lie algebra.

where

T> 0

0

c= center(~).

in

and

~

GT, T=exp(9~,T)"

f eC:(G ,T : V).

For such an

g c C:(~7~T : V).

Let

hj

G/G ° , all chosen from

In what follows, they will be sufficiently

As usual~ it is sufficient to consider f,

let

If

g

be the function

X ~ f(exp X);

be a complete set of representatives for

G*= G/LR,

*

then~ defining

l<j<m

By Proposition i0.i~ as c R ( e x p H ) = s g n % ~ p

'Ff,L(expH ) = e-6I(H)

9~/~T=

is an open neighborhood (convex and starlike

(i_<j <m) K.

We write

gj

as in (24)~ we get

j (H~)d ~

|

G(H), G real, we get, V H e

n ((e c~(H)/2-e-~(H)/2)/c~(H))

O~¢P

~ iSj 0 such that IDI Z ' / 4 qmT defines a tempered distribution on L w . _

The proof of this is somewhat long and requires some preparation. tribution

T

has a tempered extension to

iant by averaging it over n~0

and elements

K.

ai~ ~

(lii!r),

Tf d~ i

(h~L, xcG).

Let For

which may be even assumed K-invar-

Proposition 9.16 now implies that for some integer

F

G[ x] V f eC~(G[L× ]).

G

The dis-

sup (~-l(z+~)nlaifl)

l0

0

Let

vc@.

such that for all

0,

in

IR.

m~m'~0

n 1]c~(bexpH) GcP

-(i - ~_(z)-I.

Also

Re 6(H) = 6(HR) ,

~(h)-!l

HR

Hence we get the following result :

being the

given

v c 9, we

such that -6(log hR)

i

M + h ~ L~L..

We estimate the deriv-

For the estimates of the derivatives of

IIDl- (h;v)l ! c(l+~(h))ml'l~(h)[-mlq(h)l-m'e

(~) V

~=

that come in, we use Lemma 7.

can find

is a finite set such

is a locally constant function with

values in some finite set; of course i atives of IDI -N using this formula. the

FCL I

then we can write

So, in order to obtain an estimate of the form (19) for

IOL-~(h;v)l,

it is enough to prove that 6 (log hR)

(22)

+

~=(h)-i _< e

(h ~

~ILR)

q of ~ assoWe shall now prove (22). We observe that we have a psalgebra + ciated with ~R in a natural fashion; |R is its split component and if is its Langlands decomposition,

q= m + I R + n

6(H) = ½ tr(adH)n Now we can choose Section 6); O F . But

k

k cK

such that

will map

IR

p F = P 0 I aF, a~cCZ(a~),

(23)

qk

onto

aF,

is the standard psalgebra ~F + + IR onto aF, and will take

k

(cf.

6 I IR to

and so, by Proposition 8.17,

~(e~H') -z _< e

Conjugating by

(He IR)

and remembering that

pF( ~' )

+ (~'~ ~F)

E(h)=E(hR) ,

we get (22) from (23).

This proves Lemma i0 and finishes the proof of Proposition 5. Conclusion of the proof of the necessity of (2) for O to be tempered

4.

We apply Proposition 5 to Then

'9"

(el L') is

extension to ~(~(h)#l sult:

let

L'(R)

an where

L

and use the notation of n°3 above.

9-finite function which has an analytic

L'(R)

for each real root. F

(9. We fix

analytic

is the subset of all h c L such that + Since LILR~L'(R) we have the following re-

be a complete set of representatives

for

L~/L~, .L

407

.5

and

b c F;

232

k.3 c

then we can find distinct elements on

IR

(-1)2I I

and exponential polynomials

fj

such that

'%(b exp H)e(b exp H) = ~. eXJ(HZ)fj(H~)

(24)

3 for all

+

Hc I

such that

+

H R ~ IR, b e x p H c L ' ;

in particular, if

H R e IR

and

b exp H I 6 L I . On the other hand~ by (i0.19), we can find a locally constant function

on

LI

whose values are in a finite set such that

ID(bexpH)l ½ = ~(bexpHi)e6(H)'1~(bexpH) for all

H c I

6 (~): 6 I 'R'

s~ch that

bexpH IcLI

(z)

+ H R c IR.

and

Writing

8

= 6 I |i

and

we have~ from (24) and (25)~

(26) ID(bexpH)12®(bexpH) : ¢(bexp HI) E e (xj+&(~))(H I) fj(HR)e 6(R)(HR) J H

being restricted as before. i

'

x

qmlDI2@

is tempered on

Now, by Proposition 5, for some integer

m>_0,

+

LIL R.

An elementary argument then shows that the

functions

HR 64 q ' (HR)m fj(HR)e

(27)

6(R)(HR)

(q'(HR) = q(expHR))

+

define tempered distributions on

L R.

By I, Lemma 7.6; this i m ~ e s

that the

linear functions a©pearing in the exponentials that make upfje 6 - have real + p a r t s ~ 0 on I R. B u t t h e n t h e formu]_a ( 2 6 ) s h o w s t h a t f o r some c > 0 and n ~ 0 , i

I D(b e~ H) I~I e(b e~ H)I_< (Z + II~ll)n for all

He I

such that

bexpH I eL I

R" H R c I+

and

pletes the proof that (2) is necessary for 5.

®

This proves (2) and com-

to be tempered.

Ei6endistributions with re6ular ei~envalues

For eigendistributions with regular eigenv~lues, the estimate (i) can be improved. CSA

c~

We recall that a homomorphism and a regular element

THEOREM ii.

Let

@

k e~

X :~

C

is said to be regular if 3 a

such that

be an invariant distribution on

is a regular homomorphism such that

z @ = X(z)® V z c 8.

and only if there exists a constant

C> 0

such that

408

G.

Then,

Suppose

X:~C

® is tempered if

233

I

(29)

Io(~)1 !cID(~)l -~ We fix a 6-stable

CSGL

(v x~a,)

and prove that for some constant

c > 0,

I

(h~T,')

le(h)l ! elD(h)l -~ We proceed exactly as in

n°4.

Let

'@p= ' /Ap -(@l L).

We select a regular k

I* C

such that

z®= #~/~(~)(x)®

(~ ~ 8)

Then, in view of the work of Section 4, we find

(~0)

(e-6O

vo

eS)'®p = v(X)'®p

kv/v ~ £ that are invariant under the Weyl group W(£c,lc).

get the following:

~ constants

e s , S e W ( g c , l c ) , such t h a t

'/&p(bexpH)®(bexpH)e 8(H) =~'c e (sk)(H)

(~l)

S

V H~

So, instead of (241 we

£

(32)

such that

x

bexpHlCL

cs

S

+

I, H R ~ IR;

and that moreover,

~ o ~(sXl II)~ (-l)~I I

The relation (26) now becomes 1

(~)

ID(bexpH)12®(bexpH) = E(bexpHi) Z) Cse(Sk)(H) S i

× R x that We argue from t h e t e ~ p e r e d n e s s o f IDI~® on LIL + with the h e l p o f I , Lemma 7 . 6 . H' ~ IR, whenever e s ~ 0 ,

Re(s:~)(~')fiO V But then from (33)

we have

sup w

ID(h)l{Io(h)l O ,

r>0

such

that (i)

If(x)I < C ~ ( x ) ( l + o ( x ) ) r

We shall choose a Hilbertian structure for l'I

for the c o r r e s p o n d i ~ norm.

are all analytic on

G.

U

such that

The elements of

The components of

(xcG)

f

T

is unitary and write

~(G : U : 7 ) ,

by Theorem 7.18,

in some basis of

U

are K-finite,

8-finite and tempered, and the s~ne properties are possessed by each derivative of these functions.

Consequently, for

f { G ( G : U : 7)

the weak inequality also (Theorem 9.13)-

complex-valued K-finite 8-finite function on and a basis

u I .... ,um

for

U,

and

a,b { ®, afb

It is also obvious that if

such that

G,

we can choose

satisfies g

is a

U,7,f { ~ ( G : U : 7),

g ( x ) = (f(x),ul) ~ x c G .

The central

question of this section is the determination of the asymptoti~ behaviour of the elements of Q ( G : U : T ) , i.e., the behaviour of f(x) when x ~ ~ if f c C ( G :U :T). + Since G = KCI( A0)K , this clearly reduces to the problem of determining how f(h)

behaves when

that

~(logh) ~

h eCl(A0)

case when there is a subset To the set

F

and

h ~.

Now, when

for all the simple roots F ~ E

such that

~.

h ~,

it is not necessary

So we shall have to consider the ~(log h) ~

is of course attached the standard

out that the differential equations satisfied by

psgrp f

for every PF=M~FNF;

~ e E \ F. and it turns

may be regarded as ~erturba-

tions of similar differential equations on MIF = M ~ F . It is therefore possible + f on A 0 by a solution fPF of the (unperturbed) differential

to approximate equations on that

MIF~

~(log h ) ~

the approximation being good as long as for each

~ e E \ F.

h~

in such a way

Since any psgrp is conjugate to a standard

410

235

one, this leads one to associate with each psgrp may be regarded as an approximation to fQ

is known as the constant term of

f f

Q=MA

N

an element

fQ

which

in suitable regions going to infinity; along

Q.

The determination of the

together with precise estimates for the differences

If -fQI

accurate description of the behaviour at infinity of

f.

fQ

give a reasonably

This i s the method

used by Ha~ish-Chandra in the case of both the discrete and the continuous spec-

[ 6 ] [ 9 ] Ill].

tra

Let US now turn to the precise statements of the main theorems. is as above.

If

Q=MAN

the homomorphism

is a psgrp of

dQ:MI~I~ +

maximal compact subgroup of

Then

TM

defined. (i) (ii)

N

(or

MI).

aeA

~

~ c>0

for each root

for

We know that

~

in

U.

MA

and define

~=KnM=KnM

G(M I :U : TM)

is a variable element, we say that

~(loga)~+~

Q = G,

we write

I

is a

Let

is a double representation of If

a If

G,

as in (6.11).

G(G : U : T)

~

such that for each root

of

Q

~

of

a~

is thus well

if

Q, ~ ( l o g a ) _ > c o ( a )

for all the

involved. the symbol

THE01KI~I~ i. unique element

Let fQ

a ~* G

means

a

f e G ( G :U : T). of

Let

G(MA : U : ~M)

(3)

varies freely on the split component of G. Q=MAN

be a psgrp of

such that for each

G.

Then ~ a

m e V~,

lim (dQ(ma)f(ma)- fQ(ma)) = 0 q fQ

is known as the constant term of

(4)

f

alon~

Q.

We write

BQ(H) = i ~ ~(H)

(H~ ~).

h a root of Q THEOREM 2. usualy, let

A0

Let

Q0 =MoAoN0

be a minimal psgrp of

be the set of all

aeA 0

with

:

~(~)

G

contained in

~(loga) 7 0 V r o o t s

Q.

As

R of Q0"

Write

~Q(H) Let

f ~ G(G • U : ~)

constants

(5)

C>0,

~>0

and let

fQ

i~

(~e ~0)

~ a root of Q

be the constant term of

and an integer

r>_0

such that

f

along

~vlhcCl(Ao) ,

ldQ(h)f(h) - fQ(h)l 0 , [ > 0 , [>_0

such that V

%in

(5) we get the following:

hcCl(Ao),

ld~(h)f(h) - f~(h) f _0, r>0_

sueh

e CI(A0) ,

d~ (h)fth) - (f~)~% (h) < C e -%(1°g h) (j_ + ~(h))r e-~F (l°g h) Combining the last t~Jo we see that with

~l=min(~,~),

~ C I > 0 , rl_>0

such that

rI -~Z~F(Zog h) IdF(h)f(h) - (f~)*%(h)l _< Cf-%(log h) (i + ~(h)) e

(Z2) VheCI(Ao).

(13)

Applying Theorem 2 to f we get, for some

IdF(h)f(h)- fQ(h) l _< C 2 e-%(logh) (l+~(h))

VhcCI(Ao).

Since

u=fQ-(fQ)*QI'

C 2 > 0 , ~2>0, r2_>0 ,

r 2 -~2BF(lOg h)

e

we see from (12) and (13) that for suitable

c3>o, ~3>o, r3_>o, V Cl(A~) (14)

lu(h)I < C3 e-*a(logh) (l+g(h)) r3 e -¢3BF(l°gh)

It follows at once from (14) that if

(15)

h c AO, a ~ AF,

lim u(ha) = 0

For fixed

h~AoJ Uh :a~u(ha )

is a tempered

~F-finite function on

therefore argue as in n°2 to deduce from (15) that u=0

on

4.

A0

by analytieity,

~u=0

on

So

u=0

We fix

~.

Given

F?Z,

as before by

For any

A F . We + on A0,

MI=K~0 ~.

~F(H) = min ~(H) ~ZkF t > 0 ~ Ao(F :t)

(16) Here

Uh:0.

Uniform estimates on the "sectors" Ao(F :t)

We continue with the above notation. ~F

then

(He s0).

is the "sector" defined as

Ao(F: t) = re: a ~ c1(Ao),~F(log a) zt 00(log a)} P0= PQ0"

The estimate (5) then yields at once the following.

414

we define

239

THEOP~M 7. that V h : A ~ ( F

Let :t)

(17)

f:G(G:U:T).

Fix

Then 3 C>0, ~ > 0 , r ~ 0

F ~Z.

such

(t>O)

If(h) - dPF(h)-ifpF(h)l !C :(h)i+t~(l +a(h)) r Indeed, ~ F ( l O g h ) :

~tP0(logh )

-P0(iog h) const.e

hcA:(F it)

ri (i +:(h))

THEOR~ 8.

for

Suppose

and

rI

G=°G.

with the following property:

Let

given

-~P0(log h) e i~(h) ~t.

and

f const.E(h)(l +a(h))

~l(h)dPF(h) -i !

f eG(G :U : T).

Then we can find

F ~ E, t > 0, 3 C = CF, t >0

such that

If(h)-dPF(h )-ifpF(h)l !C :(h) l+~t

(18)

Moreover, if we write, for any

~ e Z, F6 = Z\ [~}, then

~>0

(h a A:(F : t))

3

tO > 0

such that

C:(A~) = u A~(F~:to)

(19)

BeE

Since r ~0.

G= °G, e -¢#0

dominates

This gives (181 from (17).

subset of

CI(A~)

of all

a

(i +a) r

for any

¢ >0

c(a) = i.

and any S+ be the

If we put

72= max D0(log a) aeS +

Y2 are >0. Let 0 < t O 0

J(a~o)(x)l ! c,

such that

E = E -I and

such that

sup ff(r~y')l

~/

xcG)

~/

x~G)

y,y'cE So from (21) we see that for some

(24)

C=Ca, b > 0 ,

I(afb)(2)l !C ~(2) l+~

From (24) we argue as before that for each sup

r >0

(~-Z(z +~)rlafbl)
O We use the estimate (18) on Once again, as to obtain

G=°G,

for each

(i+~) r

by

e

-E0 0

(r_>0, ¢>0)

Hence, as f eL2(G :U) and as + (X=klhk2,klk 2 e K, he CI(A0))~ we conclude from (18) that

¢>0.

Z% (F t) dF(h)-2j(h) IfpF(h)12dh < ~

(25) Let

being fixed throughout what follows.

we use the domination of

~I+¢eL2(G)

dx= J(h)dkldhk 2

A+(F:t), t >0

(he C£(Ao) )

c >0

be a fixed positive number and let

h e A ~ ( F :t)

such that

~(logh)~c

V~eE;

E

be the subset of all

it is then immediate for some

cI > 0 ,

(h e E)

J(h) ~ eI dF(h)2 Hence

j

(26)

IfPF(h)12dh
c

A F = e x p aF, A E \ F = e x p V&eF,

is such that hlh 2 c E.

&(logh2)>c

V~eEkF,

So, for almost sdl such

If

h I eA~.\F

F

(resp. E\ F)

is such that

with the following property: and

BF(IOgh2)>T P0(logh2),

if h 2 6 A F then

bl,

]A IfPF(hh2lI2dh2 < ~ F(h :~) AF(~ : T) i s the subset of

~F(lOgh2) >T p0(!ogh2). ponential polynomial on

Write A F.

we can find a compact subset

AF of a l l

(~=dimGF).

of the unit sphere of

:T).

tl>0

ghl(exptH)=0Vt, on

~F(lOgh2)>c ghl

E, thence on

~F'

A F ( h : T)

A0,

ghl

integrable on

~)

and finally on

417

such that

AF(~ on

: T), (tl,m)

is a tempered exponential poly-

By analyticity,

proves everything.

that

having nonempty in-

t~t~-ilghl(exptH) 12 is integrable H e L 1.

and

is a tempered ex-

(both depending on

Ighl(.)l 2 is

Since

This contradicts the fact that

fPF = 0

Then

It is clear from the definition of L1

we see that for almost 8~Ii H e L l , nomial unless

h 2 e A F with

ghl(h2):fPF(hlh2).

terior (in the unit sphere), and a number

HeLl , t>tl~exptHeAF( ~

finally

where the members of

then ~ T = T ( h l ) > O

(2n where

a0 ~\F.

Let

ghl=0

~F = M~F=

on

AF~ hence

K~0~"

This

242

6.

Constant terms and cusp forms

Theorem 9 establishes

a very remarkable connexion between certain eigen-

fUnctions and cusp forms.

This is a special case of a much more general result

which associates cusp forms with any element of THEORI~M i0. above.

Let

psgrps

Q

Let

G

be an arbitrary group of class

fE~(G:U:7). of

G

Assume that

f~0

fQ#0.

Q=MAN

such that

with respect to

c.

Let

suoh that ~

~(f) ~(f).

If

Usual. (# ~)

of V

~

G e ~(f).

Then

G,

~

G

and

is a psgrp of

and since

~ = o(~),

Of course, since ~

~

of

Let

then

~

So, by (i0),_ (f~,a).Q=0

again implies that

~(f)

is a cusp form on F

CSGL

Q~,

is a psgrp of

acA.

be the set of all

be an element of

(m~M)

is certainly nonempty since

Since every psgrp (~ iv:) of

on

~(f)

minimal

and 3

#~,7# 0.

Qc~

(f~)*Ql=fQ=0.

~ and let notation be as

and let

is euspidal, i.e., ~ a 0-stable

*~ = Q0 (~).

We have

Then

(i) for each a~A, f ~ , ~ : m ~ ( F )

(ii)

C(G : U : T).

G ~

such that

be a minimal element of

fQ=0

i~!VaeA;

here

Q0~

Theorem -3 shows that

has a compact

~ ae~

CSG.

by minimality.

Let

and we know from Theorem 6 that on

is of the form

f~#0,

LR=~.

*Q=QOF

as

for some psgrp f~,~

for which

Q

is a cusp form

f~,~#0.

Theorem 9

The assertion (ii) now follows from

Proposition 6.23.

7.

A consequence of Theorem ~.

The case of ei~enfunctions

Let

X :~

We write

(28)

C

be a homomorphism.

~x(a :u : T) : {f :f ~c(Q :u :T), sf = x ( z ) f V z c8}.

We now examine more closely the constant terms Let ml=m+

Q=MAN

be a psgrp of

G; ~ = m +

fQ

~+n

for

f C~x(G : U : T).

the corresponding psalgebra;

s; %,52,~/, the subalgebras of C~ generated by (l,ml) , (l,m), (l,a) respect-

ively; 8(m), ~(ml) are the centers of 52, ~ ;

Select a e-stable

~ ( m l ) ~ 8 ( m ) ® ~I.

CSA I of ~ c o n t a i n i n g a.

Let

corresponding Weyl group.

We can then find

(29)

X(z) = XA(Z ) = I~B/I(z)(A)

The orbit

W(gc,le) " A

fc |* c

is uniquely determined.

W(ge,|e)

denote t h e

such that (z c 8).

Since the CSA's of gc containing

Sc are conjugate under the centralizer of ~ in ~c ~ it is clear that the subset c W(~ c,lc) • A I s of s does not depend on the choice of I,A. We now have the c e following theorem.

418

243 THEOP~ ii.

Let notation be as above.

Suppose

f e Cx(G : U : T).

Then the

subset E of ac of the restrictions sA Iae (s e W(~c,lc) ) is independent of the choice of I,A.

If E n i s * = ~ ,

then fQ=0.

distinct elements of EO is*.

Suppose E meets is* and i~,...~ih n are the

Let P(A) be the algebra of all functions on A of the

form a ~ p(log a) for some polynomial p on s. fke~(M :U : TM)®P(A ) such that (30) If

fQ(ma) = A

Then there are elements

i~(loga) ~ l0, r 0>_0

r0

(6~)

--(m)dQ(m) < cO ~l(m)(l +a(m)) For proving this select a minimal psgrp

Let

such that

a0=l°gA0"

We have~ for constants

Q0 =MoAoN0

-D0(log h ' )

h c A 0.

such that Q0~Q. p0 for

0Q0

r

(~ + ~ ( h ' ) )

0

(h' e C~(~O))

r0 (l+c(h')

+A0

Cl(Ao).

we have

-pO (log h) " - ( h ) = ~ ( h ' ) i toe

On the other hand, l e t

G

-1 s e ~ (= Weyl group of (g,~0)) such that h ' = h s

Then ~

Since D0(iogh' ) -D0(logh'S)>0 (65)

of

c0>0~ r0>0_ ~ writing

-~(h') < Coe Suppose

(me MI)

-DO (log h) =C

denote t h e subset of

r0 (l+o(h))

e

A0

of a l l

h

such that

c~(logh)>O for a l l positi~Qe roots ~ of (ml,aO). Then M 1 = ~Cl(+A0)~, Do(l°g n) e*O(log h ) while, for h e A0, e = dQ(h), *P being the p attached to (h'%)"

So, by (65), WhsC~(+AO ), r r0 ~(h)dQ(h ) < c0e-*O(l°gh) (l+~(h)) 0 -- operator norm.

If

Q, ,,e have Ml= 9 J 0 % a n d Ad(k) is ~nita~y for k ~ K ,

we have (671

cQ(m) = Im(Ad(m-l))nll Proposition 18.

Let

bI .....bqe@, ~ .....h c

(6s)

~= ('ql~2)~ ~ £ % ~

and

9(n), such that, V g s

Ig(h;m;n2a)l 0 v~S(~z)- Then, 3 C'=Cv,

r'=r' >0 v,q-

(70)

I{q(m)l < C'~l(m)(1 +a(m))r',

IYv, q(m)l _0

°~(Hj) hi c ~ ;

such that, V m ~ w ,

t>0, (81) If

~xp(-°~(t~°))8¢(hl;me(tt°~h2)

- ® (m) l 0 , ~l>0, -~i t

l¢(me(ttf);v) - exp(~(ttf))®(m;v)l O.

On the other hand, from the differential equations (63), =

+

From the estimates (70) we have, f o r suitable constants

429

c2>0 , ~2bO,

and

tf

254

-~2t

(t>o)

ITv(me(ttf))l 0, ~ 3 > 0 , -~3t

t>O

I~(v)¢(me(ttf)) - exp(~(ttf))~(v)®(m)l < c3e

Combining all of these and using the uniqueness of the limit in (77), we get

(82) If

®(re;v) = Y(v)e(m) H c a,

then

H e 8(ml).

(m c MI'V ~ 8(ml) )

The differential equations (82) then yield

(83)

O(ma) = exp(~(log a)) G(m) Let Us now consider the behaviour of

°A= exp°a, For

and let

a c A , let

and

aI

be the projections of

(80) we then obtain, for fixed O

when

C D be the split component of

oa

itive values at

¢(ma)

m c MI,

(m c MI~ a c A )

G. a

acA

and

Then

A= ° A C ~ OAxC~.

on

°A

and

as soon as all the roots of

a~.

Write

C . From Q

take pos-

a,

l¢(ma) - exp(Y(logOa))®(mal)[ < c4(i +a(al))r4(l +~(Oa))r4e-~4~Q (iog °a) where

c4>0 , ~4>0, r4>0

(= log o C )= are BQ(log

are constants independent of

mutually orthogonal,

BQ(loga)~ E0a(a)

(84)

for some ~0>0.

lira Aga ~

a.

~(a)>_max(q(al),~(°a)).

Since

°a

and

¢

Moreover,

So, using (83), we get

I®(ma)- e(m~)l = 0

(m~MZ).

We now turn to the question of obtaining estimates that are uniform in % . The estimates (80)

are

not

good

because

~Q

can become quite large.

We

therefore define

(85)

M[ = rm . ' m c % , ~ q ( m ) < l } . Clearly

+

Ml

is open in

a minimal psgrp of

G

~

contained in +

(86)

Q,

Moreover, i f %

=%%%

is

it is obvious that +

Ml = %(M[ n c ~ ( % ) ) %

The estimates (80) sine good on M1. Proposition 21. If

+

and Ml = ~ 7 ~ "

m ~ M I~

To go from

We can find a constant

there exists an

ac °A

such that

430

c >i

to

M1 we Use

with the following property.

255

(i)

8Q(loga)>O

(ii)

and

c(a)~c(l+c(m))

maeMi

Choose a minimal psgrp the positive chamber in can write

as

8

G,

contained in

kl,k 2 s ~ ,

heCl(+A0 ).

Q.

(ml~G0)o Let

y

Let If

+A 0

be

~ eM 1,

we

be the maximum of

runs over the simple roots in the set of roots of

Choose

a e °A

such that

tainly

ha e M 1+

and so

B(log a) =7 m a e M E.

for each of these simple roots

Now we can find a constant

b e °A, ~(b) 2 ~ c ~ E8 8(logb) 2,

On the other hand, for such stunt

of

relative to the roots of

m = k l h k 2 where

1 + 18(1ogh)l

for all

Q0 =MOAON0

A0

8,

c _ ~ > l independent of

~(a)2!c~d2c~(l+~))

h,

Cer-

such that

the sum being over the simple roots of Q.

l + 18(logh)l ~l+llBllc(h) 7 ~c2(l+c(h)).

2. This gives

cl > l

Q. B.

so that for some con-

Hence, with d = dim(°A);

~(a)!e(l+~(h))=c(l+c(m))

,

where

C = C l d c2 • We can now show that (80) we have~ since

8

~Q(m) < i

(87)

satisfies the weak inequsulity. Taking for

me~.

Choose

in

(meMO)

le(m)i2LCl~(ml(l+~(m)) rl

Suppose

~= 1

m s MI~

c>1~ a e ° A

as in Proposition 21.

Then

IO(m)l = le(rlma)l = lexp(-~(log a)) ®(ma) l We canreplace °~(Hj)

Y(loga)

(88) by

by

°Y(loga)

since

have only pure imaginary eigenvalues,

®(m')e°V ~ c '> 0

V

llexp°[(log a')lI0 , rl~0

such that, V

h ¢ CI(A~),

-BQ(Zogh) If(h;b)I i Cle

Z(h)(l+o(h)) r

so that, in view of Proposition 17, we have, for suitable constants

r2~0

and all

h e CI(A~),

-SQ(lOgh)

I%(h)f(h;b)l ~ C2e

433

% ( h ) ( 1 +c(h))

r2

C2>0,

258 Thus,

constants

C~>0,

r~O

such that for all

h c CI(A~), ro -~n(logh)

IdQ(h)f(h;z) Replacing

h

by

- r~(h;~Q(z))l

ha, a e A,

~ C3~l(h)(Z

(zf)Q

when a ~ ,

u = (zf)Q-k~Q(Z)fQ

on

MI

So

u=0

on

= 0

satisfies the limit relation

for each hcCt(Ao).

is a tempered ~-finite function on u(ha)=0.

(~(z)fQ)(ha)l

Since zf c~(G :U : m), we find from the definition of

that the difference

lira u(ha)=O

~e

we find from the last estimate that

~= Zim IdQ(ha)(zf)(ha) for each fixed h ~ CI(Ao).

+~(h))

CI(Ao) ,

by mM-sphericalness.

A hence

For fixed such

h,

a~u(ha)

and so, arguing as in n°2 we find that =0

on

A0

by analyticity, hence

=0

This proves Theorem 3-

As we mentioned at the beginning, we refer to Harish-Chandra's articles [ 6 ] [ 9 ] [ii ] for the questions treated here. see Varadarajan [ 2 ].

434

For a condensed treatment

l~.

The Discrete Series for

G

Everything that is needed for the determination of the discrete series is now at our disposal.

This section is devoted to an exposition of Harish-Chandra's

proof that the characters of the discrete series of G are precisely the distributions

sgn "~(b*)~

(-i) q constructed in Section 5 for regular i.

b~

B ~.

Discrete series for a separable unimodular group

In this n ° we define and recall briefly the well k n o w n ~ t s ~ y p r o p e r t i e s a f t l ~ discrete series for a second countable locally compact unimodular group denote by

Z(H)

H.

some Hilbert space

If

v

~(~),

(i)

is any irreducible unitary representation of and

~,~c~)~

fT:~,¢(x) : f

f9,9

on

H,

~(x) = (~(x)~,~)

~ ( x ) : g ( x - l ) e°nj.

class contragredient to S~ppose

~ e ~(H).

H

in

we write

is certainly a continuous function of g

We

the set of all equivalence classes of irreducible unitary re-

presentations of

function

H.

x

If

and

(x~H)

ft, = f ~ , 9

w cg(H),

we write

where for any ~*

for the

~. We say that

~

belongs to the discrete series of

H

if

there is an irreducible subrepresentation of the right regular representation of H

belonging to

w.

Since

H

is unimodular, the left and right regular repre-

sentations are equivalent and hence this is the same as requiring that an irreducible subrepresentation of the left regular representation of w.

The set of all such

series of

H.

involution

~

is denoted by

~2(H)

Using complex conjugation we see that

~ ~ w

THEOP/94 i.

of Let

S(H).

If

m c ~ c ~(H)

H

is compact,

and let

~(~)

H

belongs to

and is called the discrete ~2(H)

is stable under the

S(H) : S2(H ) . be the Hilbert space of

F.

Then the following statements are equivalent: (i) (ii)

~ ~ e2(i ) 3

nonzero

9,¢~(~)

(iii) f :~,¢ ~2(H) ~ , ,

such that

f

9{L2(H)

c ~(~).

It is enough to pro~e (ii)~(i), (ii)~(ii~) and (i)~(ii), as (iii)~(ii) is trivial. (i)~(ii).

We shall in fact prove that if r is the right regular representa-

tion of H and $ is any closed nonzero r-stable subspace of L2(H), ~ nonzero ~,9

435

260

in ,~ such that x ~ ( r ( x ) % ? )

is in L2(H).

g,hCCc(H), then x,~l(r(x)g,h)l l i e s

If

in L2(H); in fact, this function is bounded by the constant llgll

Ilhll,

and vanishes

outside the compact set (supp(h)) -I • supp(g). Next, let ~ e $, ~ e Cc(H ) and let

f ( y ) = (r(y)~,S) ( y e H ) . Moreover, if

Then f is bounded and so f T e L l ( H ) for a l l 7CCc(G ).

7CCo(H),

HxH But

L I~(~)11 ~(y) IdYO.

is closed, we must have

is closed so that

over, we have shown that 9

As

So

So

~so.

Theorem i justifies the name s~uare inte~rable for the representations s ~ ¢ 82(H). The rest of the n ° is now devoted to showing that the representations be-

longing to the discrete series possess properties remarkably similar to those of the representations of compact groups.

In particular they satisfy orthogonality

relations, possess a (formal) degree, and their characters can be obtained by integrating over conjugacy classes;

this last property needs to be formulated with

care, as we shall see presently. THEOREM 2. Then 3 a constant

Let

v ¢ ~ ~ 82(H)

d(~) > 0

(~)

and let

ZH I(W~)%*)1%

I]911= li*ll

9,*c$(v),

9,4 e ~2(G) s u c h that 9',*' ¢~(~'), then

(6)

~(~(~)%,)(~'(~)~',*

for all

$(~)

be the Hilbert space of

= d(~) -z

= 1.

•

,)conJdx

More generally,

if

7' e ~'¢

FO

Consider a

~c~(~), *~0.

A : ~f%~

r-stable subspace

~

So we write

(7)

c = c(~).

of

~(~),

L2(H )'i intertwines

c > O,

F

~)

maps

and

c-~

m'lw

if

Tr= ~r'

onto a closed

and has the property

r,

is unitary. Clearl~ i The unitarity of c-NA gives

c

may depend on

d(x - l ) = a d x

llf,,9112 = a-lc(,)]/9112 : c(9)11,112

437

4-

(~ ~ ~(~))

I1%,~112 = c(,)1191f 2

On the other hand, ~ a constant a > O such that fg,w(x-l) =f~,9~x) t ~conj . Hence, from (7), we get

(s)

if

The discussion in the preceding proof shows that

is everywhere defined on

that for some constant

82(H),

=

d ( . ) - Z ( % ~ , ) ( , , , ' ) c°nj

the map

F.

such that

while

262

Taking

~ = ~0

ac(~0)II4112.

to be a unit vector, we get from (8) the relation In other words, ~ a constant

e(W) =

such that

Hf%411 ~ = d-lll~li21i~ll £

(9) This proves (5). ~.

d>0

We write

Obviously

d

depends on

~

(%4 ~ ~(~))

and not on the choice of

~

in

d=d(~).

Write now, for

~'=~,

(zo)

~'=~,

and

%~',4,4'¢~(7),

z(~,~' : ~',4) =(f~,~,f~, 4,)

From (9) we g ~ t , V m , m ' , 4 , 4 ' e ~(~),

(ll)

Iz(~,~' ,4',4)1

For fixed

¢,~', I(-,. :~',4)

shows that on

$(~).

I(~'

with

~,

I(~,~' :~',4)

~(~)~

and

(y~ ~),

t(~',~) .i.

it follows that

~

and

T(4',~) ~'

T(~',~)

re-

commutes

Thus

Z(~,~' : 4 ' , 4 ) = t ( ~ ' , ~ ) ( ~ , ~ ' ) ~= ~'

be a unit vector; then, fixing

Hermitian bilinear form.

fore, but now

T(4',4)

thus proved (6).

S.

As before we find that

~ = d(~) -I.

intertwines

If m

~ ~ ~',

and

~',

t

is also a

t(4'4,) = (S¢',4) S= ~l

is called the formal de~ree of 7Hdx = i,

d(~)

depends on the choice of

(13)

for some scalar ~. ¢

hence must be

~.

When

H

(i)

Let

~ e ~ ¢ 82(H )

f~:q0,~ ( ~ , 4 ¢ ~ ) ) . If

Comparing

0.

We have

is compact and

is the actual degree of dx.

~.

{%]

is

For noncompact

(~ ~ e2(~))

and let

IA(~)

H.

be the closed linear span of

Then:

is an orthonormal basis of

orthonormal basis of

dx

Clearly

(6) are called the ortho6onalit ~ relations for

THEORI~M ~. aAl the

~(~).

the argument proceeds as be-

d(~) = d(~*)

The relations

for

The proof of Theorem 2 is complete.

normalized such that d(~)

we find that

I(~,~' :~',4) = ~ " (~,~,)(~,~,)conj V ~ , ~ ' , ~ , ~ '

this with (9) we see that

d(~)

~

Again, we find from (ii) that

a unique bounded operator Thus, finally~

H,

for a unique bounded operator

does not change on replacing

~(y)~'

hence is a scalar

(12) Let

is a Hermitian bilinear form and the bound (ii)

:4',~)= (T(4',4)%~')

Since

spectively by

! d-lH~ll I1~'11 114'1/ II0

be an element of

Cc(H )

such that

fHbmdX=l and supp(bm)CU m. Write Um(Y):(r(y)a,bm). If VeCc(H ), the function ( ~ , y ) ~ ( a ( ~ y ) - ~ ( y ) ) b m ( ~ ) ~ ( y ) l i e ~ i n TI ( H x H ) , and so by ~ b i n i ' s

theorem,

m

-
0. For each

b% c B ~T ,

such that

(27)

%(bD =

The map

there is associated uniquely a class

b*~(b*)

from

B*'

into

(-1)q ~(b*)% ~ 82(G )

= ~(b2) ~ s ¢ W(G,B) THEOREM ? .

('~'Ff,B)(I)Vf

Let

c(G) > 0

~ C(a).

is surjective; and, for bl,b 2 c B such t h a t

be t h e c o n s t a n t

b 2 = s[b 1 ]

such t h a t

Then the formal degrees

•

f(1) = (-1)qc(G) •

d(~(b*))

of the olasses

~(b ~)

are given by

(29) where

d(~(b*)) = o(O) lW(a,B)id(b*)~(X(b*))l d(b %)

i s t h e degree o f

choice of the positive

system

b ~.

The c o n s t a n t

c(G)

(b* c B*') does n o t depend on t h e

P.

The i d e a b e h i n d H a r i s h - C h a n d r a ' s

proofs

the invariant integral harmonic analysis in

o f t h e s e theorems i s as f o l l o w s .

Via

C(G) is reduced to that in C~(B).

Although there are serious difficulties in carrying out this reduction because 'Ff {C=(B) and because of the presence of the noncompact CSG's, if form,

f

is a cusp

'Ff c C~(B) and the invariant integrals over the noncompact CSG's vanish.

Thus the above mentioned reduction is possible. Since the harmonic analysis of . 'Ff is governed by the elements of B , the harmonic analysis of f is controlled by the eb..

Although this entire procedure is formally identical with the class-

ical one of H. Weyl used by him for compact groups, the actual situation is very much more profound and requires the entire machinery developed earlier. 443

e68

3.

Existence.

Let

G

Proof of Theorem 7

be an arbitrary group of class

left and right regular representations of we put

k(x,y)f= ~(x)r(y)f.

We recall that

a~a t

Proposition i0.

Let

k

~. G.

We write For

Z

and

(x~y) e G × G

is clearly a representation of

is the antiautomorphism of

®

r

for the

and f c L2(G), G×G

in L2(G).

such that

Xt= -X V

X c S.

and for

f e C(G).

a,beC~, bfatcL2(G)

and

Then

f

is a differentiable vector for

It is enough to prove weak differentiability. that for fixed

a,b e @

and

(30)

u e L2(G),

This comes down to proving

the integral

~G I(a~b)(~z)l [u(x)ldx

converges uniformly when is compact, and

r>0

y

and

z

vary over compact subsets of

is any integer~ we can find

C >0

--


0

be as in

as above

f(1) : (-1) q c(G)(¥,Ff)(1) ~T

Taking

{=~

for b * ~ ]\

in (37), we get (38); B* ',

the sum in (38) is only over

~-

Let

(4o)

(41)

~ e Z2(G).

-X-T

Then 3 b £ B

such that

x~0(~) = %/b(~)Clog b* + 8 )

We take

f=fTr:q),q)

because;

"~'(X(b*))= O.

Proposition i~.

Suppose not.

B

Let

~ s~

and let

in Proposition

~

12.

f(1)= ( % ~ ) = (-i) q c(G)

be a nonzero

(~ ~ .8) K-finite vector in ~7).

From ( 3 8 ) we have

S ~' ('Ff,b*~(X(b*))d(b*) b eB

446

Xw= XX(b.); i.e.;

271 Now, for any

z eS, z f = % (z)f,

'Fzf= 'b~/b(Z )'Ff

so that

and so, writing ~

'Fzf=X (z)'Ff.

On the other hand,

g= '~g/b(Z),

j~

"

.

( Fzf,b ) = ~ ~'Ff b*c°nOdb = JB 'Ff ~tb*c°nSdb As log b *c°nj = -log b*, ~tb*c°nj = ~t(-log b*)b *e°nj = ~(log b*)b *c°nj . Hence

( Fzf,b ) = 'Z~/b(Z)(log b*)('Ff,b ) In other words, (Xm(z)- ~(b.)(z))( v Ff,b * ) = 0

(42) Given

b EB

, we can certainly find

see from (42) that that

(~,m) = 0, 5-

( Ff,b )= 0.

Since

ze8

such that

b ~B

X,~(z)~X~fb.~(z ).

8o we

is arbitrary, we see from (41)

a contradiction.

The ortho~onalit~ relations for the ' ~

We shall now establish the orthogonality relations for the

'}

that play

the same role as the orthogonality relations satisfied by the characters in the case of compact groups.

We recall that

db

is the Haar measure on

B

such that

~B db=l. Proposition 14.

Let

(43)

d(

O(f) = Let

st~ch that each

~,w' £ g2(G), f c/A0(~').

rg~

and let

dim ~ ) ~ >

~[gTr(K).

being a constant.

Here;

~ [(K)

0.

~) -lf (i )

be the subset of

Then

if if g(K)

~'¢~* ~'=~* consisting of all

We select an orthonormal basis {e~ m : m ~ N(~)} N(~)

is a finite set of cardinality

We now use the notation of Theorem 9-15.

for

_< c dim(~)2~ e_>l Let

f~ m(X) =

(~(~)e~,m,e~,m) (m~(~)). Then f~,m~C(G), and llf~,ml[2=d(~) -~. Select r>_0 such that ~-(l+~)-r~L2(G) ~d q>_0 s~chthat Cl=Z~l~(~)lo(~)-qO, r~O, n(@)~c(dim(@))rv#cB(Kl).

is an operator of trace class; is a distribution on

~;

and

Then,

the map tr(~(g))=v(g)

(g ~ c~(~)). This is proved in the usual manner (cf. Theorem 9.15). Let notation be a s in the previous lemma.

L~vNJ~ 26.

(64)

t : S~t(B) = D

is a well defined distribution on

B,

n(#) F q # S d b ~B the series converging absolutely,

moreover invariant under the inner automorphisms of We take an orthonormal basis of

. . . . . H4,

and put

1111

as w e l l as

Then, writing

b

a=l-(4+...+H2

are

"a=e

W(K~,B)-invariant.

(62)~ ~ c > 0 , r_>0 varies over

such that

such that

thatV~

oCt(B),

~.

"aq

~ .~ = (l+ ll~(b*)l12)~,~.,

varies within a lattice in

a

and for a suitable constant

B

By

On the other hand, as ib*,

and to each ele-

[B : B ° ] elements of

n(#(b*))(l+ll~(b*)ll2) -q0,

fe~(b,

s>0

with the following property:

) (G-U-T),

and

for all

l0

Ilxiflt ~ < -~ (~f,f)

~vj Xe ~.

< ((pn-~

Taking

X=Xi,

we have,

-1)f,f)

0 Since

Ilnfllp0

F(k(b*) : H0)

~, 0 < a < l ,

will be at least

~

in

Finally, we have seen earlier that @(mexp t H 0 ) - 0

m e M I.

ing property: V U , T , b * c B

c'>0,

and its eigen values form a disc.rete subset of ]R when

b*c B

for each

is semisimple with real eigen values; c'(l+llk(b*)ll2) r'

so that, in particular, there will exist a

such that all the nonzero

as

By

We thus obtain the following result from Proposition

3 C>0,

r>0,

, fc~(b~)(G

and

~

with

:U : m), m C M l ,

466

0 < ~ 0,_

having the follow-

291

dQ(mexptH0) I f ( m exp tH0) I

(32)

_< C(1 + ~ Q ( m ) ) ~ ( m ) ( 1 In this estimate we now choose in

e0"

Then

m

a0 = a(~)+IR .H 0

+~(m))r(1 + t)rllT,k(b96)llrllfl12 e-~t

as follows.

Let

~(~)

is a direct sum, and if for +

H=H(~)+t~9 where ~(~)e~(~), H~CI(%)~%>_0 Let

h e CI~Ao) , H = l o g h

and take

the following: for suitable

h e Cl(%)

C>0~

H ~ a0

we write

and ~(H(~))_>0~/~e~\r~].

m = exp H(~) ; t = t . We then obtain easily 96 and all U;T,b ,f, as above~ and all

r_>0;

:

(;~) Let

be the null space of

If(h)l 0

and let

(34)

A~,~(t) = {h:

Cl(Ap,~(logh)

~t

%(logh)}

Then (33) implies at once the following: Proposition 8.

Let

~eE.

Then

~ C>0,

r~0

and a

~

with

having the following property: VU,T,b96 e B 96', f e G k (b96)( G : U : T ) , (t > 0

0 < ~ 0.

We now have the fundamental result of this n°: THEORKM 9" constants

There exists a

C=Ca,b>0

m>0

~ s= Sa~b>_0~

and; corresponding to arLv given

a,be~ 96 96~ with the following property: %/g~T~b c B

feSX(b96)(O:U:T ), and all xca,

(36)

I(a~)(x)f < c[l~,x(b*)l[rllf112~(~) l+~ I n f a c t t h e p r e c e d i n g d i s c u s s i o n has a l r e a d y t a k e n care o f t h e theorem f o r

a= b =l.

The same

~

serves a l s o f o r a r b i t r a r y

t e c h n i q u e as i n P r o p o s i t i o n

a~b s @.

We proceed by the same

2 t o t a k e care o f t h e d e r i v a t i v e s .

The d e t a i l s

are

elementary and are therefore omitted. Using the notation and technique of Proposition 4 we get at once TH~0R]~M i0.

There exists

to any given a ~ b c @ j 96T

~b*eB

a

~>0

with the following property: corresponding

we can find constants C = C a , b > 0 ~

, ~l,%e~(K),xcO, 467

S=Sa~b>0

such that,

292

(~7) 2.

The projections °E and E

restricted to Schwartz space

We recall (cf. Section 15, n°l) the definitions of °L2(G)

is the orthogonal direct sum of the

&(~).

°L2(G), &(e) (~e82(G)).

We therefore have the ortho-

gonal projections

We shall now prove the remarkable theorem that these define continuous projections on the Schwartz space itself. T H E O R ~ ll. of

C(G).

If

More precisely we shall prove

C(G) N °L2(G)

f~C(G), °Ef

and the maps

f~°Ef

C(G) N°L2(G)

and

and

O(G)

and

C(G) N&(w) (~ c 82(G))

lies in f ~ E f,

onto

(39)

C(G) N°L2(G);

E f

are closed subspaces lies in

C(G) N&(w);

are continuous projections of

C(G) N~(~)

respectively.

o~=

s

~

~(G)

onto

Moreover,

(f~c(~))

f

wcg2(G) the series converging absolutely.

Finally, for any

To prove Theorem ii we proceed as follows. Tr~ w

and let

subset of all

~cS(K)

that occur in

w

and let

~

and any f £ C(G),

~ c 82(G )

acts.

Let

select a

~ (K)

e ,~,m (mcN(~,~))

be the

be an or-

~(Tr)~. We know from Corollary 7.20 that

(41) Let

For each

~(Tr) be the Hilbert space on which

thonormal basis of

~ ~ 82(G )

IN(~,~)I = dim ,~(~)~ < w½ dim(~) 2 ZW

be the set of all

(~9,m) with

~c8

(K), mcN(w,~9).

We put

i

(42)

ac0:i,j(x) = d(~)2(Tr(x)e

j,e

Clearly,

a : i , j ~ °C(G), and for g cL2(G), ~ 8 2 ( G ) ,

(k3)

o~

=~

E

(g,a

i,j~Z Theorem ii Let

.

. a

..,

~:I,3 ) ~:z,8

=

E g

F

i,jcZ

i)

(g,%

(x~G,i,j

.

. a

.

~Z )

.

:1'3) ~:i'0

follows from the more precise Theorem 12 to be stated presently. ~>0

be as in Theorem i0 and let notation be as above.

the space of all r ®~

of

GXG.

then

g~C~(G)

g e L2(G)

Let

L2(G) ~

be

which are differentiable vectors for the representation

From the classical Sobolev estimates it follows that if g c L2(G) ~, and

~b~T.2(C)~a,b~¢.

Wep~t

468

293

(44)

+11 112

Then

z ~8

and (of. (2) and (3))

(v c ib*)

= 1 + II ll 2

TH~OPJ~ 12. given

U,v ~ ~,

For any 3

(46)

C>O

r

g ~ L2(G) ~,

let

and an integer

g~:i,j

m>O

= (g,atO:l, . j.)a to:3., . j.. Then,


> 0 V G e P .

>0, we must have

#

is of #=0.

Thus

296

Consider the invariant

eigendistribution

®6

defined by (5.102).

sl = l , s ,...,s be a complete set of ~#r~sentatives for r character exp H ~ e(Si6)~H)(H e b) of B. let b..2 be the 1

e8 =

(55)

2

Let

W(G,B) \ ~(%,be)

and

Then

~(si)~

l_c(#)>_l

On the other handj if we recall the classical estimate asserting that for any lattice in a finite dimensional Euclidean space the number of lattice points in a ball of radius

t

is

O(t n)

for

t ~+ ~

where

n

is the dimension of the

vector space, we can conclude with the help of (64) that the following is true:

474

299

a constant

c >0 s

such that s+~

(65)

~ (l+ll~(b*)iI2) s < c °(~) b*gB*',~ occurs in ~(b*) s

where

Z= dim(b). S~ppose now that

~G

~mg~m~L2(G)

(~mg ~ m ) ~ d x = ~ G g ( ~ m

~m)dx

define tempered distributions of involved converging absolutely.

~.

m~_0.

~mg ~m);

So this is valid ~

If

~Cc(G),

however both sides C(G),

the integrals

We then get

m

(66)

for all integral

(by definition of

*m

*m

(gmg~ ,a :i,j)=~(~ I :b )~/(~2 :b ) (g,a~=i,j)

For the

a:i, j

we have the estimates (49).

So, for

u,v e¢,

using (66), V

xeG~

(67)

I(ug~:~,J)(~)l 0.

But

dim(@)2=0(c(@) s')

for some

s'

(65), this comes down to proving the convergence, for some s+ls' + 7-

This is clearly the case for The analog~between

g ~ °C(G)

is finite for

m>0,

of

m

m>>l.

C (G)

for a compact

suggested by Theorem 20, is quite interesting. For

Cm

and so, in view of (64)and

let

475

G

and

°C(G) for arbitrary

G,

)oo

]]g!l(m) = (

(70)

~

]]~rg ~r!l~)~

m= 0,1,2,...

0 7,

while

G= (G1 ..... Gd) ,

is

is to

~ we get

: - f (~,~(t+~))e-~(~'!)d~

;(~) Hence, as

~-+~,

So~ writing

~nd ~

~

~

I!I=~,

IF(~)I _< (~l+ ... +~d) ~ e-~/e

m~

IGj(~+uT)lau

0 l~jZd ~ Zd B(~+ l%l)Se-Bmn(% ) ,~ (l+u)Se -~/2 d~ 0 since

min(t +uT) > min(t) + ~min(T ) . The integral above is

e~/2--~vSe-VC/2dv 0

Jtl )s+2n+le-rain(8,~)min(t)

satisfying the limit relation (12) (when

V(~)~0, ~ =+m).

Moreover,

[wl O , t¢ Q:

320

s(-Uo~)Q(oF(%~)

w = (2s)

F tj l<jO,

IF(t)l _ < A ( l + t ) r ,

The subspaees

~,v

(we £)

we

on B>0,

ta(t)[

as before.

~>0, r > 0 , s>_0,

and all t > 0 ,

_< S ( l + t ) S e -~t

S(F)

is the set of eigenvalues of

and Qw : V ~ VF, v

are the projections corresponding to the direct sum decomposition

V= Ev VF, v.

Let

Q(F) = max

(31) Finally,

I8(F)I

all eigenvalues of co

as

a(f) =

Let notation be as above. F

t ~+~.

are real.

S(F).

Suppose

Then ~ a unique element

Moreover, we have, for all

496

I~1

rain

denotes the number of elements of

Proposition 6.

F(t)~F

IIQ#[I

t>0

f

is semisimple and F ¢ VF~ 0

such that

321

(32)

IF(t)-0

IF (t)- 8 0F~l _< C(s :~)(A+B)Q(F)(I+t) s+l e -min(B'o(F))t

5 0

being the Kronecker delta.

Adding over all the

~

in

S(F)

we get the

estimate (32). 5.

Estimates without polynomial 6rowth assumptions

LEMMA 7" values in

V

Let notation be as above. s~ch that

f

is of class

Let

f,g,

CI,

g

be functions on

is continuous, and

~ f (t) -- g(t) Suppose ~ L,M>I, a > 0 ,

B>0

[o,z] with

(0_0 , sI = Sl(S,a,8,~)~0

such that

a

(t s c ) V

~cO

a

IF(~)I ~ C1L~B+lM( l + r~l)

sI

,

Iaj(%)l!C1

--+ 2 L~s M(Z+

I~1) sl

From now on we proceed as in the proof of (iii) of Proposition i. (~,...,T) eQ~,

we find that

following estimate for s2= s2(a,8,g,s ) ~ 0

F(7,...~T)~a limit

IF(t)-v I is valid:

3

v

as

~+~,

-~ rain(t)

e

Noting that and that the

C 2=C2(a,B,~,s )>0,

such that a

--+2

IF(~).-V I ~ C2L~8

M(l+ I~1)

s2 -~min(t)

e

~

(~eQ)

The estimate (36) is now clear. For the estimate of Proposition 3, see Harish-Chandra [ 6 ], Lemma 60. arguments and results of this appendix are quite similar to those in HarishChandra [ 9 ], Lemmas 42 through 63; see also Varadarajan [ 2 ] and Varadarajan [ i ].

499

Trombi-

The

19 •

Appendix

We shall discuss in this section certain representations lated to finite reflexion groups.

These are representations

that are closely reof the polynomial

al-

gebra of the vector spaces on which the reflexion groups act, and their spectral properties i.

are of very great importance The representations

for us.

7;7'

We work in the same context as in I~ Section 4, n°2. vector space of diminension Fc

is the complexification

Fc

and

F~

P

d,

i < d < ~.

of

F and

W

Fc

P.

S

F

is a real

is a finite reflexion group in

is its dual;

is the algebra of polyno~als on

gebra of W-invariant elements of

Thus

L

W

F.

acts naturally on both

!p= ~(w)

is the subal-

is the s$~nmetric algebra over

Fc, I S the

+

subalgebra

of W-invariant

elements of

without a constant term. Let

H

If

p eP~

S,

and

we say

be the space of harmonic elements of

and the map

v,u~vu

IS

p

the ideal in

is harmonic P.

Then

H

extends to a linear isomorphism of

(1)

if

IS

of elements + ~(uJp=0VueI S.

is graded and W-stable, Ip®H

with

P.

We have

dim(X) : lWt = w. From now on we write

of all

p

such that

an ideal in

P

I

for

p(x) = 0 •

Ip.

Since

For any

x e F c~ I x

P= IH~i®H,

denotes the idea& in I

PI x = I x H ~ l x ® H .

Thus

PI x is

with

(2)

dim(P/PIx)

We have a representation

of

P

in

in order to study this dependence these representations

P/PI x.

= w

(x c Fc)

This representation

depends on

and

x

it is convenient to get a realization of all

in a single w-dimensional

vector space.

We shall use

H

for this purpose. Let (3)

uI

be homogeneous qu:ij c I

elements in

H

forming a basis for

UUj =

We thus obtain a representation

(5)

uw

H.

If

u e

P,

3 unique

such that

(4)

wXw

= 1,u 2 .....

matrices

of elements of

~uj

~ l
0

Here

X

is an indeterminate and

is the stlbspaee of

H

H

of homogeneous ele-

m

ments of degree

m.

We know that if

of the homogeneous generators of (31)

P(H : X) :

I,

H ldeg(~ ).

is the ideal in

and is W'-skew.

skew elements, the isomorphism

l