Representations of Real and P-Adic Group

REPRESENTATIONS OF REAL AND P-ADIC GROUPS

LECTURE NOTES SERIES Institute for Mathematical Sciences, National University of Singapore Series Editors: Louis H.Y. Chen and Yeneng Sun Institute for Mathematical Sciences National University of Singapore

Published Vol. 1

Coding Theoty and Ctyptology edited by Harald Niederreiter

Vol. 2

Representationsof Real and pAdic Groups edited by Eng-Chye Tan & Chen-Bo Zhu

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

REPRESENTATIONS OF REAl INUHIOOC GROUPS D O

Editors

Eng-Chye Tan Chen-Bo Zhu National University of Singapore

SINGAPORE UNIVERSITY PRESS NATIONAL UNIVERSITY OF SINGAPORE

vp World Scientific

Published by Singapore University Press Yusof Ishak House, National University of Singapore 31 Lower Kent Ridge Road, Singapore 119078 and World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA once: Suite 202,1060Main Street, River Edge, NJ 07661 UK once: 57 Shelton Street, Covent Garden, London WC2H 9HE

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REPRESENTATIONS OF REAL AND p-ADIC GROUPS Copyright 0 2004 by Singapore University Press and World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereoJS may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Contents

Foreword

vii

Preface

ix

Three Uncertainty Principles for an Abelian Locally Compact Group Tomasz Przebinda

1

Lectures on Representations of padic Groups Gordan Savin

19

Lectures on Harmonic Analysis for Reductive padic Groups Stephen DeBacker

47

On Classification of Some Classes of Irreducible Representations of Classical Groups Mark0 TadiC

95

Dirac Operators in Representation Theory Jang-Song Huang and Pavle PandZiC

163

On Multiplicity Free Actions Chal Benson and Gail Ratcliig

221

Multiplicity-Free Spaces and Schur-Weyl-Howe Duality Roe Goodman

305

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Foreword

The Institute for Mathematical Sciences at the National University of Singapore was established on 1 July 2000 with funding from the Ministry of Education and the University. Its mission is to provide an international center of excellence in mathematical research and, in particular, to promote within Singapore and the region active research in the mathematical sciences and their applications. It seeks to serve as a focal point for scientists of diverse backgrounds to interact and collaborate in research through tutorials, workshops, seminars and informal discussions. The Institute organizes thematic programs of duration ranging from one to six months. The theme or themes of each program will be in accordance with the developing trends of the mathematical sciences and the needs and interests of the local scientific community. Generally, for each program there will be tutorial lectures on background material followed by workshops at the research level. As the tutorial lectures form a core component of a program, the lecture notes are usually made available to the participants for their immediate benefit during the period of the tutorial. The main objective of the Institute’s Lecture Notes Series is to bring these lectures to a wider audience. Occasionally, the Series may also include the proceedings of workshops and expository lectures organized by the Institute. The World Scientific Publishing Company and the Singapore University Press have kindly agreed to publish jointly the Lecture Notes Series. This volume on “Representations of Real and pAdic Groups” is the second of this Series. We hope that through regular publication of lecture notes the Institute will achieve, in part, its objective of promoting research in the mathematical sciences and their applications. Louis H. Y. Chen Yeneng Sun Series Editors

February 2004

vii

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Preface

A semester-long program on Representation Theory of Lie Groups was held at the Institute for Mathematical Sciences (IMS) at the National University of Singapore (NUS) from July 2002 to January 2003. This is the third research program of IMS since it started operations in July 2001. The goal of the program was to explore recent advances in representation theory of both real and padic groups. The program had three sub-themes: (i) representation of padic groups; (ii) unitary representations of real reductive groups; (iii) multiplicity-free actions and representations. As part of the program, tutorials related to the sub-themes were conducted by leading experts in the fields. These tutorials covered the fundamentals of representation theory as well as some of its recent developments and were meant for graduate students and researchers who would like to prepare themselves for original research in the fields. The current volume collects the expanded lecture notes of these tutorials. In the following, we give a brief indication of the ranges of topics represented in this volume. The first five articles are on the sub-themes (i) and (ii). The article by Tomasz Przebinda directs us to some recent developments in harmonic analysis on locally compact abelian groups, specifically three Uncertainty Principles. In Gordan Savin’s notes, he gives an elementary and elegant introduction, with exercises, to representations of padic reductive groups. The material represents real classics from the past quarter century. Stephen DeBacker’s article examines some distinguished classes of distributions on reductive padic groups and their Lie algebras and discusses one of the deepest connections between them: the Harish-Chandra-Howe local character expansion (and its ramifications). The lecture notes of Marko Tadib deal with the problem of classification of some important classes of irreducible representations of both padic and real classical groups (with more emphasis on padic groups), in particular the classification of the square-integrable representations modulo cuspidal data. Jing-Song Huang and Pavle Pandiib’s article examines the role Dirac operators play in repix

X

Preface

resentation theory and discusses various topics surrounding their proof of the Vogan’s conjecture on Dirac cohomology, an important development in the algebraic aspects of unitary representations. The last two articles in this volume are on the sub-theme of multiplicityfree actions. Here Chal Benson and Gail Ratcliff present a comprehensive treatment of these actions, covering topics such as the classification of linear multiplicity-free actions and eigenvalues of invariant differential operators. Finally Roe Goodman’s lecture notes explore the fundamental duality between the irreducible representations occuring in a linear group action and irreducible representations of the commuting algebra, which of course, is a recurrent theme in many parts of representation theory. Besides the local organizers (us), the other members of the Organizing Committee are Jian-Shu Li (Hong Kong University of Science and Technology and Co-Chairman), Jeffrey Adams (University of Maryland), Kyo Nishiyama (Kyoto University), Dipendra Prasad (Mehta Research Institute) and Gordan Savin (University of Utah). We are very much grateful to their invaluable services. We would also like to express our deep appreciation to Roger Howe (Yale University and Chairman, Scientific Advisory Board of the IMS) who helped to conceptualize scientific aspects of the program and was equally valuable in many other aspects throughout this program. Thanks also to all the participants of this program for their support and stimulating interactions during those few months! We would like to take this opportunity to thank Louis Chen, Director of IMS, for his leadership in creating an exciting environment for mathematical research in IMS and for his guidance throughout our program. The expertise and dedication of all IMS staff contributed essentially to the success of this program. Financial support to the program was generously provided by the IMS and by a grant from the Faculty of Science, NUS. Last but not least, we would like to record our appreciation to the Department of Mathematics, NUS for freeing some of our obligations during this period, and more importantly for providing all kinds of support to our work ever since we joined as faculty members more than a decade ago. Eng-Chye Tan and Chen-Bo Zhu National University of Singapore Singapore

Three Uncertainty Principles for an Abelian Locally Compact Group

Tomasz Przebinda Department of Mathematics University of Oklahoma Norman, OK 73019, USA E-mail: [email protected]

0. Introduction

The purpose of this article is to direct reader’s attention to some recent developments concerning three Uncertainty Principles: the classical Heisenberg-Weyl Uncertainty Principle, The Hirschman-Beckner Uncertainty Principle based on the notion of the entropy, and the Donoho-Stark Uncertainty Principle. We state all three in the first section with some further explanations and proofs in the following sections.

1. The main results Let S(R) denote the Schwartz space on R,[lo], and let f E S(R)be a real valued function. Then, by the Cauchy inequality,

Moreover,

Thus

7

T. Przebinda

2

If the equality holds in (l),then there is a real number a such that f’(z) = - 2 a z f ( z ) , so that

f(z) = *e--ax2-c,

(2)

where c E R. Since (1 f 112< 00, the number a must be positive. Let us write f as the inverse Fourier transform of the Fourier transform f of f :

f (z) =

s,

e2KixEf(E)dE.

Then

and therefore

Thus for

1) f

(12=

1,

s,

1f. (.)I2

dz.

s, lE!(4I2

dz 2

1

S’

(3)

A few more easy steps lead to the following theorem of Herman Weyl, (see

POI ). Theorem 1.1: (H. Weyl, 1931) Let f E S(R), with

(1 f

112=

1. Set

Then 1 0.524lr’ and the equality occurs if and only if

f(.)

= e-ax2-bz-c

,

where a > 0 , and b , c E @. In other words f is a constant multiple of a translation

Three Uncertainty Princaples

3

and a modulation f(z) + e i y o x f ( z ) of a Gaussian e-axc2

The above theorem states that a function and its Fourier transform cannot both be arbitrarily concentrated. In computer applications one deals often with finite cyclic groups, rather than with the real line. In this context there is no obvious or straightforward generalization of the theorem 1.1, because the quantities involved ( p ,c,...) do not seem to make sense. In order to circumvent this difficulty, Donoho and Stark, [7],have introduced a different, elementary, measure of uncertainty, which we explain below. Let A = {0,1,2,3, ..., N - 1) be a finite cyclic group of order ( A (= N . Let A denote the dual group of all the characters (i.e. group homomorphisms 6 : A 4 C"). It is customary, and sometimes convenient, to identify A with A by the formula

6 ( b ) = e*ab For a function f : A

-+

(a,b E A ) .

C define a Fourier transform of f by

f(i)=

c

(6 E A).

f (a)&(-a)

aEA

Theorem 1.2: (Donoho-Stark, 1989, [7])For any non-zero function f : A --f C, b P P f I . ISUPPfI

2 1-41.

The equality occurs if and only i f f is a constant multiple of a translation

f (a> f ( a + 4 +

and a modulation

f(a)

+

k4.f( a )

of the indicator function of a subgroup of A .

In applications one deals often with multi-dimensional signals, i.e. with functions defined on a finite product of finite cyclic groups. Hence it is natural to ask for a generalization of the theorem 1.2 to this more general context. This has been done by K. Smith. Theorem 1.3: (K. Smith, 1990, [19]) The above theorem 1.2 holds f o r any finite Abelian group A (a finite direct product of finite cyclic groups).

T. Przebinda

4

The theorem 1.3 also follows from theorem 1.10 below. In addition, a proof which uses no more than basic concepts from finite dimensional linear algebra over complex numbers and the structure of finite abelian groups is available in [12]. Clearly, the cardinality of the support is not the most precise measure of the concentration of a function. One possible improvement is based on the notion of entropy.

Definition 1.4: (based on Shannon, 1948, [18]) Let p be a non-negative measure on a measure space M . Let 4 : M + [ O , o o ) be a probability density function, i.e.

The entropy of

4 is defined as

where the log stands for the natural logarithm, whenever the integral converges. In his 1957 paper, [9],Hirschman has proven the following theorem and stated the following conjecture.

Theorem 1.5: Let f

E

S(R), with

H(lf 12) Conjecture 1.6: Let f

(Notice that since

;> 1, log(:)

112=

1. T h e n

+ H(lfI2)2 0.

S(R), with

H(lf I?

(4 (b)

E

11 f

11 f

(12=

1. T h e n

+ H(lfI2)2 log(;). > 0.)

The equality holds in (a) i f and only i f f is a constant multiple of a translation and a modulation of a Gaussian e - a x 2 ,a > 0.

As an application of his LP, LQ estimates for the Fourier transform, Beckner proved part (a) of Hirschman’s conjecture.

Theorem 1.7: (Beckner, 1975, [2], [3]) Part (a) of Hirschman’s Conjecture is true. Theorem 1.8: (Ozaydm-Przebinda, 2000, [IS])Part (b) of Hirschman’s Conjecture is true.

Three Uncertainty Principles

5

Let A be a finite Abelian group and let a be a Haar measure on A. Thus a is a positive constant multiple of the counting measure on A. Let ii be the dual Haar measure on the dual group in the sense that for a function f : A 4 @, the Fourier transform and the inverse Fourier transform are given by

a,

JA

(4)

The following theorem is known. For the idea of a proof, various particular cases and generalizations see [9], [14], [13] and [6].

Theorem 1.9: Let f E L2(A,cr),with

(1 f

((2=

1. Then

H(lf 17 + H(l.FI2) L 0. Theorem 1.10: (Ozaydm-Przebinda, 2000, [16]) The equality holds in the above theorem 1.9 i f and only i f f is a constant multiple of a translation and a modulation of the indicator function of a subgroup of A .

Corollary 1.11: (DeBrunner-Ozaydm-Przebinda, 2001, [17]) The discretization of the minimizers f o r the entropy inequality o n R does n o t give minimizers f o r the entropy inequality o n any finite cyclic group A . In other words, it is certain that no discretization of a signal defined on the real line and best concentrated in the time-frequency plane, will lead to a signal defined on a finite cyclic group which is also best concentrated in the corresponding finite time-frequency plane. In a mathematically natural search for the most general theorem, which would generalize all the cases considered above, we arrive at the notion of a locally compact Abelian group, (see [S]).The integration and the notion of the Fourier transform are both well established for such groups. Let A be a locally compact Abelian group. As was explained to the author by Michael Cowling, a result of Ahern and Jeweet [l],together with [8],9.8, imply that A is isomorphic to the direct product of a finite number of copies of R and an Abelian locally compact group B , which contains an open compact subgroup:

A = R " x B. (5) Let be the Pontryagin dual of A . Then = R"x B ,where also contains an open compact subgroup. Let Q be a Haar measure on A and let i 3 be the

a

a

T.Pmebinda

6

A,

Haar measure on dual to (Y,so that the Fourier transform and the inverse Fourier transform are given by the formulas (4). Let V ( L 2 ( A , a )be ) the group of unitary (norm preserving) operators on the Hilbert space L 2 ( A ,a ) , and let G V ( L 2 ( A , a )be ) the group generated by all the translations, all modulations and by the multiplications by complex numbers of absolute value 1. This is the Heisenberg group attached to the Abelian group A .

Theorem 1.12: (Ozaydln-Przebinda, 2000, [16]) For any function f E L 2 ( A , a ) ,with 11 f 112= 1, such that (*)

(I f

Ill< CCJ

and

II f^ Ill
,

(c E

C),

6E B.

u C € C

((S.PPfC)

+ c).

(9)

T. Przebinda

10

Hence,

IsuIPfl =

c

IS'1LPPfCl.

CEC

By the Fourier inversion formula on C we have

(6 E B, c E C ) . Hence, SUPP f c

c_ P ( W P f)

(c E C ) ,

(12)

I lsuppfl

(c E C).

(13)

and therefore

IsuPPfcl Suppose c E C is such that

ICI. IsuPPfcl

< Isumfl.

Then, by (13),

PI.ISUPPfCl. IsuPPfcI < I s W P f l . IsuPPfl

=

14.

Hence, ISUPPfCl.

IS'LlPPfCl

< IBI,

which contradicts the fact that f c # 0. (Here we use the assumption that we have the support inequality for the finite cyclic groups of the form ( Z / 2 Z ) N . )Thus

ICI. ISUPPfCl 2 lsuppfl (cE C). (14) Clearly, (10) and (14) imply (b). Further (b) and (13) imply that for all CE

c,

1S.uPP.fcl. 1SUPP.f

1

I--lS1L?)Pfl. - PI

ISUPP.fCl

1

I * ISUPPfl = IAl/lCl = PI. ICI Thus (a) follows. Part (c) follows from (a) and (b). Part (d) follows from (12) and (c). 5 -1WPf

Theorem 2.5: L e t A = ( Z / 2 Z ) N .Suppose f : A

4

CC is a m i n i m i z e r , i.e.

IsUPPfl. IS'1LPPfl = 14.

T h e n , u p t o a constant multiple, a modulation and a translation, f i s t h e indicator f u n c t i o n of a subgroup of A .

Three Uncertainty Principles

11

Proof: We proceed via the induction on N . It is easy to check that the statement holds for N = 2 . Let f : A --f C be a minimizer. Suppose there is a proper subgroup B A , such that supp f B. Then there is a subgroup C A such that A = B @ C. (In order to have some standard linear algebra a t the disposal, it might be easier here to view A a vector space of dimension N over the field 2/22 of two elements.) In these terms

s

f = (flB)€3 d, where 6 is the Dirac delta a t zero on C. Hence

where 8 is a constant multiple of the function IIc. We see from the above two equations that is a minimizer on B . Thus, by the inductive assumption f l ~ , and hence f has the desired form. Suppose there is a proper subgroup B 5 A and an element c E A such that s u p p f C B + c. Let g(a) = f ( u c), a E A. Then g is a minimizer on A and suppg 2 B. Hence, by the previous argument, g and hence f has the desired form. Let A = B @ C as in Lemma 2.4, with ICI = 2 . By the previous two cases we may assume that

f l ~

+

fc

#0

( C E

C).

By Lemma 2.4 (a), each fc is a minimizer on B. Therefore, by the inductive assumption, for each c E C, supp fc is a coset of a subgroup of By Lemma 2.4 (d) these cosets do not depend on c E C. Thus there is a subgroup D C B and an element 6 E B such that

B.

sup.fC= D

+

60

(C

E C).

Replacing f by an appropriate translation of f we may assume that each jcis a constant on its support. Thus there is a function h : C 4 C such that fC

=

8 h(c)

(c E

C).

Therefore

We see from (15) that h is a minimizer for C. But

T. Prtebinda

12

and our assumption (fc # 0 for all c E C) implies that supp h = C . Hence, J s u p p i J= 1. Thus s u p p i = {&}, for some & E C. Therefore, by (15), suppf

=

(D

+ 60) +

20

=D

+ + (&I

20).

Thus supp f is a coset of a proper subgroup of A. Since f^ is a minimizer, our previous argument implies that and hence f, has the desired property.

f,

0

3. A few words on the notion of the entropy In this section we recall a few basic facts, which indicate that the notion of the entropy is quite natural and useful. For more details we refer the reader to [4] and [18]. Let X be a discrete random variable with the probability distribution function p ( z ) = P ( X = z). Set

H ( X ) = H(P) = - C P ( 4 l O g z ( P ( z ) ) .

(16)

2

Example 3.1: Suppose X has a uniform distribution over 23 = 8 outcomes. Then

c 8

H ( X )= -

x=1

1

1

-logz(-) 8 8 = 3.

This agrees with the number of bits needed to describe X in binary: c(0) = 000, c(1) = 001, c(2) = 010, c(3) = 011, c(4) = 100,

~ ( 5= ) 101, ~ ( 6 = ) 110, ~ ( 7= ) 111.

Example 3.2: Consider a horse race, with eight horses taking part. Suppose the probability of winning the race is distributed as follows: P ( 0 ) = 1/2, P(1) = 1/4, P ( 2 ) = 1/8, P(3) = 1/16, p(4) = p ( 5 ) = p ( 6 ) = p(7) = 1/64. Then H ( X ) = 2. We may encode the horses in binary as follows: c(0) = 0, c(1) = 10, c(2) = 110, c(3) = 1110, c(4) = 111100, c(5) = 111101, ~ ( 6 = ) 111110, ~ ( 7 = ) 111111.

Let l(c(z)) be the length of the code word c(z). Then l(c(0)) = 1, l(c(1)) = 2, l(c(2)) = 3, l(c(3)) = 4, l(c(5)) = ... = l(c(7)) = 6.

Three Uncertainty Principles

13

Hence, the expected value of l ( c ( X ) )is

Thus

H ( X )= E(l(c(X)). This last equality is not a coincidence, as we shall explain below. For details see [4].

A source code for a discrete random variable X is a mapping c from the range of X to (0,I} u {o, I } u~(0,1}3 u ... .

If a codeword c ( x ) belongs t o (0,l } k ,let I ( c ( x ) ) = k denote the length of C ( Z ) . The code is called instantenous if no codeword is a prefix to any other code word. For such codes one can recognize the separate words ~ ( x l )c,( x 2 ) , ... by looking a t the string c ( z ~ ) c (.... z ~For ) instance in our second example, the string 010110111111 is made of the codewords 0, 10, 110,111111.

Theorem 3.3: ( [ 1 8 ]see , also [ 4 ] )For an instantenous code c we have, E ( l ( c ( X ) )2 H ( X ) . Moreover, there is an instantenous code c such that

A theorem of Shannon, which we quote below, gives a characterization of the entropy as a measure of uncertainty of the outcome of an experiment. Let P = { P I , p2,p3, ...,p,} be a finite probability sequence, i.e. p j >_ 0, &p.j = 1. Theorem 3.4: (Shannon, 1948, [18])Suppose H i s a function defined o n finite probability sequences such that (4

(b) H ( $, $, $, ...,

H ( p 1 ,p2,p3, ...,p,) is continuous,

i) is monotonically increasing as a function of n,

(c) if Q j = { q j l , q j 2 , q j 3...}, , j = 1 , 2 , 3 , ..., n, are probability sequences, then H(Uj”=i~ j Q j )= H ( P ) C j ” = l ~ j H ( Q j ) .

+

T. Przebinda

14

Then, up to a constant multiple, n

H ( P )= - X P j W P j ) . j=1

Suppose M is a finite set and p is a positive multiple of the counting measure on M. For a function f : M -+ C and for 1 5 p < 03 we have the LP norm o f f defined by

II f IIp=

(1 If (.iIPWzi) M

lip.

There is a simple and explicit connection between the entropy (see Definition 1.4) and the LP norms, expressed in the lemma below, which may be verified by a straightforward computation (left to the reader). See [20] for more explanations.

Lemma 3.5: Let p

= p(t)=

i, 0 < t 5 1. Then for any function f : M

---f

c, (a) In particular, for f with

(b) and for f with

(b)

4. The entropy inequality for a finite Abelian group In this section we give a proof of theorem 1.10 for a finite Abelian group A. For the general case we refer the reader t o [16]. The proof is based on the following four basic theorems.

Holder’s Inequality 1111:

Plancherel’s Formula 181:

I1 P l12=11 f

112

.

Riesz-Thorin-Young Inequality (1211, Chapter 9, (1.11)): 1 (5 5 t 5 1). I1 lll/(~-t)5ll f JJl/t

f

Three Uncertainty Principles

15

Hopf's Maximum Principle ([ll],Theorem 3.1.6'): Let D 2 CC be an open unit disc, and let u : D -+ R be a harmonic function which extends to a continuous function on the closure B of D , u : D -+ R.Suppose z is a point o n the bounday of D such that u ( z ) 2 u ( z ' ) for all z' E D, and the directional derivative of u at z along the radius which ends at z , is zero. Then u(z) = u(z')

Let f : A

-+

f o r all z' E D.

CC be a minimizer. Consider the following function -

where = 0 outside the support of f , and similarly for f-. A straightIf1 forward application of Holder's inequality and the Riesz-Thorin Theorem shows that for 5 x 5 1, y E R,p = and q defined by the equation 1 ; = 1 (with q = 00 if p = l), we have

i

+

511 l p + i 2 y

l i p . 11 lf122fi2Y

IIp=ll

f

112

. II f

112=

1.

The function F is analytic in the open strip (17) and continuous in the closed strip. A straightforward calculation shows that

Hence, by the Plancherel formula,

Since f is a minimizer, the right hand side of the equation (19) is zero. In particular Re F ( z ) is a real valued harmonic function on the interior of the disc of radius centered a t z = which achieves the maximum at z = !j and has derivative equal t o zero a t this point. Hence the Hopf's Maximum

a

i,

T. Przebinda

16

Principle implies that R e F ( z ) = 1 on the disc. Hence, F ( z ) = 1 on the disc. In particular,

The formula (20) may be rewritten a s

Since,

the equation (21) implies that

Hence,

Thus for ii E s u p p f

Therefore

Im=II f I l l

(6 E

S W f ) ,

(23)

f* Ill

.( E

SzLppf).

(24)

and similarly

l.f(.)I

=lI

The statement (23) implies that the function I f 1 is constant on its support. /~. Since 11 f 112= 1, the constant is equal to & ( s z ~ p p f ) - ~Hence,

H(lfI2)= l o g ( & ( w P f ) ) . Similarly

H(lfI2)= log(+Wf)). Since f is a minimizer,

Three Uncertainty Principles

17

Therefore

a(suppf) &!(suppf) '

=

1.

(25)

We may assume that 0 E suppf^ and 0 E suppf. Then (23) implies

Therefore there is X E C such that f = Xlfl. Hence (23) may be rewritten as

Therefore SUPP

where for a subset S

C A , S'

f^ c

(-SUPP

f)',

(27)

= {Li E A; 21s = 1). Similarly (24) implies

W P f

c (SVP.f)+

(28)

By dualizing (27) and (28) we deduce

But, as is well known and easy to check,

By combining (25), (29) and (30) we see that the inclusions (29) are equalities. In particular suppf is a subgroup of A and f is invariant under the translations by this subgroup. Thus f is a constant multiple of the indicator function of a subgroup of A , as claimed.

Acknowledgement Part of these notes was written during the author's visit t o the Institute for Mathematical Sciences (IMS) at the National University of Singapore. The author thanks IMS for its support and hospitality.

T. Przebinda

18

References 1. P. Ahern and R. Jewett, “Factorization of locally compact Abelian groups”, Illinois Jour. Math., 9 (1965), 23&235. 2. W. Beckner, “Inequalities in Fourier Analysis”, Annals of Math., 102 (1975), 159-182. 3. W. Beckner, “Pitt’s Inequality and the Uncertainty Principle”, Proceedings of the A M S , 123 (1995), 1897-1905. 4. T. Cover and J. Thomas, Elements of information theory, John Wiley & Sons, Inc., New York, 1991. 5. I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1992. 6. A. Dembo, T . M. Cover and J. A. Thomas, “Information Theoretic Inequalities”, E E E Transactions o n Information Theory, 37 (1991), 1501-1518. 7. D. L. Donoho and P. B. Stark, “Uncertainty Principles and Signal Recovery”, S I A M Journal of Applied Mathematics, 49 (1989), 906-931. 8. E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Springer Verlag, 1963. 9. I. I. Hirschman, Jr., “A Note on Entropy”, Amer. Jour. Math., 79 (1957), 152-156. 10. L. Hormander, The Analysis of Linear Partial Differential Operators, I , Springer Verlag, 1983. 11. L. Hormander, Notions of Convezity, Birkhauser, 1994. 12. E. Matusiak, M. Ozaydin and T. Przebinda, “The Donoho - Stark Uncertainty Principle for a Finite Abelian Group”, preprint, available at http://crystal.ou.edu/Ntprzebin/papers.html.

13. H. Maassen, “A discrete entropic uncertainty relation” Quantum Probability and Applications, 5 (1988), 263-266. 14. H. Maassen and J. Uffink, “Generalized Entropic Uncertainty Relations”, Phys. Rev. Lett. , 60 (1988), 1103-1106. 15. M. Ozaydin and T. Przebinda, “Platonic Orthonormal Wavelets”, Applied and Computational Harmonic Analysis, 4 (1997), 351-365. 16. M. Ozaydin and T. Przebinda, “An Entropy-based Uncertainty Principle for a Locally Compact Abelian Group”, preprint, available at http://crystal.ou.edu/-tprzebin/papers.html. 17. T. Przebinda, V. DeBrunner and M.Ozaydin, “The Optimal Transform for the Discrete Hirschman Uncertainty Principle”, I E E E Transactions on Information Theory, 47 (2001), 2086-2090. 18. C. E. Shannon, “A Mathematical Theory of Communication”, The Bell Syst e m Technical Journal”, 27 (1948), 379-656. 19. K. T. Smith, “The Uncertainty Principle on Groups”, S I A M Journal of Applied Mathematics, 50 (1990), 876-882. 20. M. Wickerhauser, Adapted wavelet analysis from theory to software, Cambridge University Press, 1990. 21. A. Zygmund, Trigonometric series, second edition, volumes I and I1 combined, A K Peters, Ltd., Wellesley, MA, 1994.

Lectures on Representations of p-adic Groups

Gordan Savin Department of Mathematics University of Utah Salt Lake City, UT 8.4112, USA E-mail: [email protected]. edu

1. Introduction This text contains an elementary introduction, with exercises, t o representations of p-adic reductive groups. The text is intended t o be more accessible then the standard references, such as the paper of Bernstein and Zelevinsky [I],and (unpublished) notes of Casselman. The material presented here has been known for the past quarter century, and has undergone many modifications in that period. Thus, it would be hard (if not impossible) to acknowledge properly contributions of many people who have worked on the subject. However, it is safe to say that Bernstein, Casselman, HarishChandra, Howe and Jacquet (in alphabetical order) are some of the most prominent contributors to the subject. The text is organized as follows. Sections 2-4 contain definitions and preliminary results on p-adic fields, structure of GL,(F) over a p a d i c field F , and smooth representations. Then, in order to keep the exposition as simplf! as possible, we restrict ourselves to GL2(F). However, the topics and their proofs are chosen so that they easily generalize to GL,(F) and other reductive groups. In Section 5 and 6 we introduce induced and cuspidal representations, and prove that irreducible smooth representations are admissible. Sections 7 and 8 are devoted t o describing the composition factors of induced representations. In section 9 we discuss unitarizable representations, and construct the complementary series for GL2(F).Sections 10 and 11 are devoted to two 19

G. Savin

20

examples. In section 10 we show that the Steinberg representation is square integrable, and in section 11 we construct one cuspidal representation. Finally, in Sections 12 and 13, we go back to GL,(F) and describe the composition factors of (regular) induced representations. This result, due t o Rodier [4], is based on the combinatorics of the root system, and the reduction to the special case of GL2(F) (obtained in Section 7). As such, it gives a good introduction to further, more advanced topics.

Acknowledgments: This material was presented during a week long tutorial in the Institute for Mathematical Sciences at the National University of Singapore. The author would like to thank the people associated with the institute, especially Eng-Chye Tan and Chen-Bo Zhu, for their hospitality and support. Thanks are also due to Jian Shu Li, Allen Moy, and Marko TadiC for useful comments and suggestions at various stages. 2. The field Q, This is a crash course on the field of p a d i c numbers. Let Zdenote, as usual, the ring of integers. Define the p a d i c absolute value on Z as follows. Let x # 0 be an integer. Write x = p"m where m is relatively prime to p. Then

1x1,

=

1

-.

Pa We also put 101, = 0. This absolute value satisfies the usual properties: the multiplicativity, lxlp . lyl, = lxyJp, and the triangle inequality. In fact we have an even stronger property:

1%

+YIP

5 max{l4,1

IYlP).

It follows that d(x,y) = Iz - yl, defines a metric on Z.The ring of p a d i c integers Z,is the completion of Zwith respect to this metric. To understand this ring we can proceed as follows. Let B ( z , r ) be the (closed) ball of radius T 2 0, centered at x. Note that Z can be written as a union of p disjoint balls of radius l/p, centered a t 0,1,. . . ,p - 1: 1

1

1

P

P

P

Z = B(0,-) u B(1, -) u . . . u B(p - 1, -). Note that B ( i ,$) is simply i+pZ, a coset of the maximal ideal (p). Similarly, Z can be written as a union of pn disjoint balls of radius l/pn, centered a t all reminders modulo pn. There are two consequences of this: First, Z is totally bounded, so its completion Z, is compact. Second, the completion

Lectures on Representations of p-adic Groups

21

of Z is the union of completion of individual balls of radius l / p n , which implies that Z/pnZ = Z p / p n Z p .

Exercise. Let x be an integer relatively prime to p . Show that the multiplicative inverse of x exists in Z,.Hint: for every n, there exist integers yn and zn such that x y , pnz, = 1.

+

The ring Z,has ( p ) as unique maximal ideal. The field of fractions Q, is obtained by adjoining l / p . It follows that Qp

= U--m_ 1, and f must be a multiple of e K . The lemma is 0

We can now finish the proof of proposition. Since V is generated by e K , Proposition 4.2 implies that there exists an irreducible quotient V' of V. Let P be the projection from V onto V'. Since V' is cuspidal, by Proposition 6.3 there is a splitting s : V' -+ V such that P o s = I d v f . But s0

PE

HOrnGO

(V,V )= c Idv '

so s o P must be equal to I d v . It follows that V = V', and the proposition is proved. 12. The root system for G L n ( F ) The purpose of this section is to introduce the root system for GL,. It is a combinatorial object which provides us with a language to study representations of GL,(F). (For more information on root systems see [a]). Let IR be the field of real numbers, and consider the space R" with the standard basis e l , . . . , en. With respect to this basis, we shall identify every element IC in IRn with an n-tuple ( ~ 1 ,. .. ,IC,). The group S, acts on R" by permuting the entries of ( X I , . . . ,x,). This action preserves the inner product n

i= 1

Let

a

Note that the group S, preserves R. In fact, it is an irreducible representation. We shall now describe the action of S, by means of Euclidean reflections. Let

9 = {ei - e j I i # j } ,

G. Savin

42

be a finite set in R. Its elements are called roots. Any root a defines a reflection w by W(X)

= 2 - ( a , x ). a.

Exercise. Let a = ei - e j . Show that the corresponding w is simply the permutation of i and j . It follows from the exercise that the reflections w define the representation of S, on 0. Clearly, the fixed points of the reflection corresponding to the root a = ei - ej is the hyperplane { x i = x j } , defined by the equation xi = x j . Consider the open subset obtained by removing all hyperplanes {Xi = X j }

R"

\ (U{Xi

=R

= Zj}).

The connected components of R" are called chambers. We shall point out a particular chamber given by

c+= {.

E

R I 2 1 > . . . > x,}.

Next, note that the entries of any x in R" are pair-wise different, so there exists a unique permutation which puts them in a decreasing order. This means that R" is a disjoint union of w(C+) as w runs thru S,. In particular, every chamber C is equal to

c = {x E R I Xil

> ... > X i n }

for some permutation ( i l l . .. ,in). The choice of C+ gives us a set of generators of S, as follows. Note that the closure of C+ is obtained by adding parts (called walls) of hyperplanes {zi = zi+~}The . corresponding roots

{ e l - e2,. . . , en-1

-

en}

are called simple. Note that the simple roots form a basis of R. Let wibe the (simple) reflection defined by ei - ei+l. We claim that the reflections wi generate S,. Of course, since wi is nothing but the permutation (i, i 1); this is a well known fact. We shall give a proof based on the fact that S, acts simply transitively on the set of chambers. Let w be an element in S,. Pick x+ in C+ and x in w(C+)such that the segment between them avoids the singularities of R \ R". Since those singularities form a codimension 2 subvariety, a generic choice of x+ and x will do. The number of hyperplanes

+

Lectures on Representations of p-adic Groups

43

intersected by the segment is clearly independent of the particular choices, is the length l ( w ) of w. The chambers containing the segment form a gallery

c+, c1,. . . Cqw,= W ( C + ) . Any two consecutive chambers in this gallery share a wall. In particular, there exists a simple reflection wil such that C1 = wil (C+).Next, note that the walls of C1 correspond to reflections wil (the shared wall) and wil wjw;' for all j # il. In particular, there exists i 2 such that C2 = w i l w i z w ~ l ( C 1 ) , which implies C2 = wi2wil(C+).Continuing in this fashion, we can obtain wil l . . . , wit(wlsuch that

. . . . . Wil (C+)= W ( C + ) . Exercise. Make a picture of the root system for GL3(F). The roots form a regular hexagon. Let w be the element in W such that w(C+) = C-. Express w as a product of simple reflections by picking a gallery between C+ and C - . 13. Induced representations for G L , ( F ) Let e l , e2,. . . ,en be the standard basis of F". Let V, be the span of e l , . . . , e,. Consider a partial flag

K, c K, c . . . c K* and let P be its stabilizer in G. Then P is called a parabolic subgroup of G, and it admits the Levi decomposition P = M N , where M is isomorphic to the product of GL(V,/V,-1), and U is the unipotent radical of P . Note that the minimal parabolic subgroup B corresponds to the full flag. If

(7, E )

is a smooth representation of M , then we can induce it to G

by

Indg(7) = { f : G 4 T I f ( u m g ) = 6b'2T(m)f(g)}". The character bp of M is defined so that for every locally constant, compactly supported function h on U we have

I,

h(mum-l) du = S p ( m )

I,

h ( u ) du.

G. Savin

44

As a particular case of interest, consider the minimal parabolic subgroup x of T is given by

B = T U . Note that any smooth character X

( ) .-.

= Xl(a1) . . . . xn(an)

an

for some smooth characters x 1 . . . xn of F X. Assume that x is regular, which means that x # xw for any w in W . In the case of GLn(F) this simply means that xi # x j if i # j. As in the case of GL2(F),the Bruhat-Tits decomposition can be used to show that

@P

Indg(X), =

X " .

WEW

It turns out that V , # 0 for any irreducible subquotient, just as in the case of G L z ( F ) . In particular, each irreducible subquotient is completely determined by V N .Let S be the set of all { i , j } such that x i / x j = I . I*'. Let

R,

=R

\ (US{Zi

= Zj})

where {xi = xj} is the hyperplane in R defined, of course, by the given equation. A result of Rodier states that the irreducible subquotients corresponds to the,connected components of 0,. We shall here give a special case of that result, describing V, for the unique irreducible submodule V . (Since any irreducible subquotient of Indg ( x )is a submodule of Indg (x") for some w in W , the general case easily follows.) Let 0; be the connected component containing the positive chamber Cf. Define W , to be the set of all Weyl group elements w such that

W(C+)c Rxf. Proposition 13.1: Let x be a regular character. Let V be the unique irreducible submodule of Indg ( x ) (normalized induction). Then

v, =

@

P X " .

W€W,

Proof: The proof of this statement is combinatorial, and based on the special case of G L z ( F ) ,proved in Proposition 7.3. Let Pi be the parabolic subgroup corresponding to the flag with V , omitted. Note that its Levi factor Mi has exactly one factor isomorphic t o GL2, and let wi (= w) be

45

Lectures o n Representations of p-adic Groups

the permutation matrix in that factor. Assume that is in W,. Then, by Proposition 7.3

xa/xi+l

# I I*', *

so wi

I n d 2 (x)E In@ (X""). Inducing both representations further up t o G, this gives Indg(X) is a summand of VN.For a general element Inds(XWt). In particular, w of length m in W, we have to consider a gallery x w t

c+,C', . . . , c,

=W(C+)

which is contained in 0: (since 0: is convex), and repeat the above argument m-times. This shows that W€W,

The opposite inclusion will follow from the following lemma: Lemma 13.2: A s s u m e that xa/xI = I . ' 1 for some a < j . If x" is a summand of VN then C+ and w ( C + ) are on the same side of the hyperplane {xa = 2 3 ) .

To be specific, assume that xa/xI= I . I. Let w' be the permutation (i + l , j ) ,and XI = xw' be the character of T obtained by permuting xj and xz+l.Then X:/X:+~ = 1 . I. Inducing in stages (first from B t o Pa),and using the second part of Proposition 7.3 (GLz(F)-reducibility) we get an inclusion 1 4 % (77)

G I c IndB(X 1,

where 77 is a character of M,, which on the GLz(F)-factor is given by composing the determinant with the character

xzl.11/2 = Xz+ll . [-I/? A more general form of the Bruhat-Tits decomposition implies that Indgt(7])N =

@ W(X')W W€W%

+

where W, is the set of all w such that w(i) < w(i 1). This, however, is equivalent to wwI(i) < ww'(j), which happens if and only if C+ and ww'(C+) are on the same side of the hyperplane { z a= xI}. Summarizing, we have 1ndgt(7)N =

@ W(C+)C{Z,>Z~)

61/2Xw.

G. Savin

46

x

be a regular character, and let R1,. . .R, be the connected components of 0,. T h e n I n d g ( X ) has m irreducible subquotients Vl, . . . , V, so that

Corollary 13.3: Let

(K)N =

@ P w ( C + )c a

Exercise. Show that I n d g ( 6 ' / 2 ) has 2"-l special case of GL3( F ).

X " .

subquotients. Hint: Try first the

References 1. J. Bernstein and A. Zelevinsky, Representations of the group G L , ( F ) , where F is a non-archimedean local field, Russian Math. Surveys 31 (1976), 1-68. 2. J. Humphreys, Introduction to Lie Algebras and representation theory, Graduate Texts in Mathematics 9, Springer-Verlag, 1978. 3. N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta Functions, Second Edition, Graduate Texts in Mathematics 58, Springer-Verlag, 1984. 4. F. Rodier, De'composition d e la se'rie principale des groupes re'ductifs p adiques, 408-424, Lecture Notes in Mathematics 880, Springer-Verlag, 1981. 5 . M. Tadid, Representations of classical p-adic groups, 129-204, Pitman Res. Notes Math. Ser. 311, Longman, Harlow 1994.

Lectures on Harmonic Analysis for Reductive p-adic Groups

Stephen DeBacker Department of Mathematics Harvard University Cambridge, M A 02138, U S A E-mail: [email protected]

1. Introduction In his paper T h e Characters of reductive p-adic groups [14] Harish-Chandra outlines his philosophy about harmonic analysis on reductive p a d i c groups. According t o this philosophy, there are two distinguished classes of distributions on the group: orbital integrals and characters. Similarly, there are two classes of distributions on the Lie algebra which are interesting: orbital integrals and their Fourier transforms. The real meat of his philosophy states that we ought to treat orbital integrals on both the group and its Lie algebra in the %&meway” and similarly, we should think of characters and the Fourier transform of orbital integrals in the same way. This philosophy has many manifestations (see, for example, Robert Kottwitz’s excellent article [12]). In this series of lectures, we will examine the various distributions discussed above and discuss one of the deepest connections between them: the Harish-Chandra-Howe local character expansion. Because of requests on the part of participants, I will spend much time reviewing the basics of p a d i c fields and discussing some of the uses of Moy-Prasad filtrations in the representation theory of reductive p a d i c groups. As such, at a small cost in terms of generality, I will concentrate on those techniques which use this theory. These lectures and the notes are meant as an informal and elementary introduction to the material. For complete, rigorous proofs, please see the references. Very little of the material in this set of lectures is original. I have borrowed heavily from the work and lectures of Harish-Chandra, Roger Howe, 47

S. DeBacker

48

Robert Kottwitz, Allen Moy, Gopal Prasad, Paul Sally, Jr., and J.-L. Waldspurger, among others. I thank the organizers of this conference, in particular, Eng-Chye Tan and Chen-Bo Zhu, for inviting me and allowing me to present this series of tutorials. I thank Jeff Adler, Amritanshu Prasad, Joe Rabinoff, Loren Spice, and Chen-Bo Zhu for their helpful comments on earlier drafts of these notes.

2 . Basics 2.1. A n introduction to the p-adics The usual introductory mathematical analysis course proceeds roughly as follows: The class agrees to agree that the set of natural numbers

{1,2,3,…} is very natural and therefore a good place to begin the course. In order to form a group with respect to addition, the additive identity and additive inverses are tossed in to the mix to give us the integers

z:= {. . . , -4,

-3, -2, -1,o, 1 , 2 , 3 , .. .}.

This set does not form a group with respect to multiplication; it is therefore enlarged to form Q,the field of rational numbers. Everything so far has been very natural. At this point, the incompleteness of the rationals is demonstrated by proving that the square root of two is not rational. To compensate for this, the fact that the rationals are ordered (that is, there is a notion of nonpositive and nonnegative) is invoked to define the absolute value, 1.1, of any rational number q: 141 =

q

ifq20

-q

ifqo, measdx(tj) = measd,(Kj).

3. Moy-Prasad filtrations In this section we describe the Moy-Prasad filtration lattices of g and subgroups of G. 3.1. The apartment of T

Recall that T is the group consisting of diagonal matrices in G. We write t E T as t = ( t l , t 2 , . . . , tn) with t j E k X for 1 < j 5 n. Let a c X * ( T )= Hom(T,kX) denote the set of roots of G with respect to T (that is, the nontrivial eigencharacters for the action of T on 8). More explicitly = {aij 11

< i # j < n and a i j ( t )= t i / t j } .

With respect t o our Bore1 subgroup B we let A denote the set of simple 11 5 i 5 ( n - 1)).We let denote the set of roots; that is, A := {ai(i+l) positive roots in @.We fix a Chevalley basis

{ Z , H ~ , X IP r E A and y E

a}.

Here

(2)kl = bkl and 1

ifk=Z=i

-1

if k = 1 = (i

0

otherwise,

+ 1)

and (Xaij)kZ

bzkajl.

We see that the center of g is k . Z and the Hp together with Z form a basis for t, the Lie algebra of T .

S. DeBacker

54

For example, for GL2(k) we have

Let Z(G) 5 T denote the center of G. For 1 5 k 5 n define Xk E = Hom(kx,T ) by setting

X,(T)

(&(S))aj

=

{

s

ifi=j=k

#k

1

if z = j

0

otherwise,

for s E k x . With respect to our choice of a Chevalley basis, we can identify A = d ( T ) ,the apartmentb corresponding to T , with the real vector space (X*(T) @ R ) / ( X * ( Z ( G ) @ ) N. Let 20 denote the origin in A. The apartment of T is an (n- 1)-dimensional Euclidean space spanned by the set (20 xk 1 1 5 k 5 n}. Note that these spanning vectors satisfy the relation

+

3.2. A simplicia1 structure for the apartment of T We let

9 := { b + n ( 6 E ch and n E Z} denote the set of affine roots of G with respect to T and v. For II,= b+n E Q we let = 6 E a. If II,= S + n E 9 and CXi @ra E X , ( T ) @ R , then we define

4

Here ( , ) denotes the usual pairing between characters and cocharacters. Note that if v~ E X,(T) and vz E X,(Z(G)), then + ( v ~ vz) = II,(vT). Consequently, we can and shall think of an affine root as a function on d(T). For II,E 9, we define

+

= (2 E

d(T) I II,(x) = 0).

In these notes, we are always looking at the reduced building and apartments.

Lectures on Harmonic Analysis for Reductive p-adic Groups

55

The H+ are hyperplanes in d ( T ) and they provide us with a simplicia1 decomposition of d(T). For example, for GL2(k) the apartment d(T) is one-dimensional and it is spanned by the vectors xo XI = 20 - x 2 . See Figure 1.

+

XI

Fig. 1. The standard apartment for GL2(k).

For GL3(k) the apartment A ( ? )is two dimensional, and it is spanned by the vectors 20 XI, 20 XZ, and 20 = xo - (XI &). See, for example Figure 2.

+

Fig. 2.

+

+ x,

The standard apartment for GLs(k)

+

S. DeBacker

56

The maximal facets in this simplicia1 decomposition of A(T) are called alcoves or chambers of A(T).For GL2(k), the alcoves are open line segments. For GL3(k), an alcove is the interior of an equilateral triangle. For GL4(k), an alcove is the interior of a regular tetrahedron.

3.3. Some subgroups o f G Note that for all a E

a, the root group

is naturally isomorphic to k as an additive group. From (l),k, regarded as an additive group, has a natural filtration indexed by Zu {fm}. Namely,

woo.R := {0} c . . . c w'. R

c W .R c R =WO R c w *

- *~R

c... c k.

14

Similarly, U, has a natural filtration indexed by ZU {ho} Z {$ E @ = a } U {kco}. The problem is: how do we decide which )I E {y!~ E 0 = a } corresponds to which subgroup of U,? The solution to our problem lies with our choice of a Chevalley basis. The choice of this basis determines the subgroup KO = GLn(R). We define

14

U,+o:= U, n KO. This, along with the requirement U,+l c U,+O , completely determines a natural indexing of subgroups in U, by the set {I) E @ I = a}. For example, for GLB(k), we have

4

Similarly, we have a filtration of gar the a-eigenspace in g, by setting B ~ + O:= M n ( R ) n ga

Note that for all $ E 0 we have an isomorphism of the corresponding additive subgroups g i and U, via the map X H (1 X). The maximal compact open subgroup

+

T o = { ( t l , t z ,. . . , t,) € T J t i € R X } of T also has a filtration indexed by the integers. Because of future considerations, we define, for r E RZO,

T, := {t E To I v(1 - ~ ( t )2)T for all x E X * ( T ) } .

Lectures on Harmonic Analysis for Reductive p-adic Groups

57

We also define

Tr+:= {t E TOI v(1 - ~ ( t )>)T for all

It is a matter of definition to see that Tr i < T 5 (i + l), then

x E X*(T)}.

Trr,, and, for i E Z ~ Oif ,

=

Tr = {(ti&, . . . ,tn) E To I t j E 1 + ZZ(~+') . R for 1 < j 5 TL}. We can define filtrations o f t = Lie(") in a similar fashion. For example, for GL3(k), we have

t5.2 =

(,gR

W'.R

0

0

).

w6.R

3.4. The Moy-Prasad filtrations Suppose r E R and x E d(T). In [27,28], Moy and Prasad define filtration lattices of g according to the formulae

and

For some people, it is easier to process these definitions when they are presented in the following form.

and

Similarly, if r

2 0, then they define

and Gz,r+ := (TT+ , U+){.LE*I + ( z ) > r ) .

These lattices (resp., subgroups) are referred to as the Moy-Prasad filtration lattices of g (resp., subgroups of G).

S. DeBacker

58

It follows immediately from the definitions that if X E 82,. and Y E g X + , then X . Y E g x , ( r + s ) . The following exercise is highly recommended; we have previously considered these statements for the congruence and Iwahori filtrations.

Exercise 3.4.1: Suppose s E R and r E R>o. (1) Up to a constant we have [ g x , s ] = [gz,(-s)+]. ( 2 ) If g E G x , r , then ggx,sg-l = g X + and g g ~ ~ , ~ += g-l (3) If 0 < r 5 s 5 2r, then X H (1 X ) induces an abelian group isomorphism of g x , r / g x , s and G x , r / G x + *

+

3.4.1. The special case r = 0 We first attempt to understand these definitions when r = 0. In this case, if x is a point in A(T), then G,,o is called the parahoric subgroup attached to x and Gx,o+is called the pro-unipotent radical of Gx,o.Let F be a facet in A and let x,y E F . It follows from the definitions of both the simplicia1 structure of A and the Moy-Prasad filtrations that GX,o= GY,o,gx,o = gY,o, Gx,o+= G,,o+, and gx,O+ = gy,0+. Thus, it is enough to understand these filtrations on a facet by facet basis, and it is natural t o label the filtrations using facets rather than points. For GL2(k), Figures 3 and 4 describe the subgroups GF,Oand G F , ~for + F a facet in the apartment of T .

Fig. 3.

Parahoric subgroups GF,Ofor GLz(lc)

Note that as we move to the right, that is, in the positive direction as defined by the spherical Weyl chamber corresponding t o B , the positive root spaces “expand” and the negative root spaces (‘shrink’’.This simple observation will play a significant role in what is to come. It is highly recommended that the reader complete the following exercise.

Lectures o n Harmonic Analysis f o r Reductive p-adic Groups

59

Fig. 4. The subgroups GF,O+for G L z ( k )

Exercise 3.4.2: Make diagrams similar to those above, but for the filtration lattices ~ F , Oand g ~ , o + Now, . do the same for GL3(lc). The origin xo is the facet in A defined by the intersection of the hyperplanes H,+o for 0 E @. We see from the above examples (and definitions) that G,,,o = KOand G,,90+ = K1. For GL,(lc), up to conjugation, the facets are indexed by

e

P ( n ) :=

{ ( p i , p z , . . . ,pe) 1 p1

2 p2 2 . . . 2 pe 2 1 and

c p i = n}, i= 1

the set of ordered partitions of n. We briefly describe how this works. We let CObe the alcove in A with vertices { q = xo = xo Cy’l x i , 2r2 = 211 - i 1 , v 3 = wz - x 2 , . . . ,w, = in}. To p E P ( n ) we attach the facet Fp of dimension C with vertices q ,w ~ ( ~ ~ + l ) , . . ., ~ ( ~ , , + ~ , , + . . . + ~ , , - , + 1 ) . The parahoric subgroup G F , , ~can be described as K1 . Q p ( R )where Q p is the “standard” parabolic subgroup of G containing B which corresponds to the partition p = ( P I ,p 2 , . . . ,pg). The pro-unipotent radical GF,,o+ is then the inverse image in G F , , ~of the unipotent radical of the image of GF,,o in GL,(f) Ko/K1. Note that if p , v E P ( n ) with p < v in the usual partial ordering of ordered partitions, then Q p IIQv and similarly for the associated parahoric subgroups. More generally, note that if Fl and F2 are facets in A(T) with F1 c where denotes the closure of F2, then it follows that

+

GFl,O+ c GF2,0+c GF2,O c GF1,O. Moreover, as the examples and exercises above show, Gp2,O/GF1,O+ is a parabolic subgroup of the connected reductive group GF~,O/GF~,O+. The unipotent radical of GFz,o/GFl,o+is GF2,0+/GFl,0+ and GF,,o/GFl,o+has ~ ,of. Levi component isomorphic to G F,o/GF2

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As a tangent, we also observe that if W = NG(T)/T (N&!')nKo)/To denotes the Weyl group, then there exist IWI alcoves in d(T) which contain in their closure. This is because these alcoves correspond to the Bore1 subgroups in GL,(f) = Ko/K1 = G,,,o/G,,,o+ which contain "T(f)".

ICO

3.4.2. The Moy-Prasad filtrations for arbitrary r For arbitrary r , there does not exist such a nice description of what is happening. We first consider GL2(k). In Figure 5 we have identified the T

= (1 - a)(.)

( ;$ ) .... . .. ....

.... . .... ..

( :;)

r ....

....

....

....

. . .... . .. .

.. '..

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

(; ;) ...........

.

:

:

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

(; ;)

..........

... .

....

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

.

.

.

( ;;) ...

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

..... .....

.

.

"1)

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

.... ....

(;

( P2 P P

...

.

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

.,..::.::.....

...........

... .

1

(8;)

..-.

........

... ....

..... ..... .:

r = ( a + O)(z)

(2 - a)(.)

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

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

(; E)

=

( Pz R P-') R

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

Pi')

'"..... :

..

....'" '.. ..

Fig. 5.

apartment of T with the horizontal axis. The vertical axis measures r . Given a pair ( z , r ) E d(T)x R, we wish to describe the lattice gz,r. Note that the plane has been divided into open convex polygons. The diagonal dotted where $ E 9.These measure lines are the graphs of the equations r = $(). where the root subgroup filtrations "change". The horizontal dotted lines are the graphs of the equations r = n where n E %. These measure where the toral subgroup filtrations "change". Each of the convex polygons is labeled by a lattice: if (5,r ) belongs to a convex polygon, then B,,~ is the lattice so

61

Lectures on Harmonic Analysis for Reductive p-adic Groups

identified. For purposes of assigning lattices to every point in the figure, the convex polygons are “closed at the top”. We note that gr,r = gx,r+ unless the point (x,T ) lies on a dotted line. Finally, note that if we fix T and move to the right, then the positive root space “expands” while the negative root space “shrinks”. Fortunately, even though life is not so nice for arbitrary T , there exists a wonderful result of Moy and PrasadC.Define

0 := { x

E

d(T) I x is the barycenter of a facet}.

An element of 0 is called an optimal pointd.

Lemma 3.4.3: (Moy and Prasad) Suppose z exist points x,y E 0 such that Bx,r C B z , r

E

d(T) and

T

E

R. There

C By,r

and there exist points x‘, y‘ E 0 such that Bzl,r+

C

&,r+

C By’,r+.

A similar pair of statements can be made for the Moy-Prasad filtration subgroups. For example, for GL3(k), the optimal points, up to the action of G (see f + x3, and 3F13+y2+F11. below), are the points

x,

9

Exercise 3.4.4: Check that, up to conjugacy, the points listed above are the optimal points for GL3(k). Describe their associated filtration lattices g x , r and gz,r+.

4. G-domains Recall that the ultimate goal of these notes is to relate certain invariant distributions on G with certain invariant distributions on g. Since the distributions in question are invariant, we will need sets in G and g which are invariant. Such a set can never be compact, so the most we can hope for is t o have sets which are invariant, open, and closed. A set with these properties is called a G-domain. As we show in this section, it is possible to use the Moy-Prasad filtrations to define G-domains which have very nice properties. In particular, we will ‘You will not find the result stated exactly as follows, but it is easy to derive this formulation. dBeware, 0 plays many roles in these notes. However, there is no possibility of confusion.

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define a family of G-domains which form a “neighborhood basis” of n/,the set of nilpotent elements. By using the Cartan decomposition of G (see [33]), in [18] (Lemma 2.4) Howe demonstrates that, for all integers i, the GL,(k)-orbit t of fr, is contained in & N.From, for example, [13] (Lemma 12.2), it is also true that for any compact set w c g, there exists a lattice C c g such that G~ c C N. We analyze these statements from the perspective of MoyPrasad filtration lattices. After doing this, we look at the situation on the group.

+

+

4.1. S o m e comments o n the Bruhat-Tits building of G Before we can continue, we must recall some facts about B, the reduced Bruhat-Tits building of G. To each maximal split torus T’ in G we can attach an apartment A(T’)exactly as we did above. The building of G can then be thought of as the “gluing” together of all these various apartments. In Figure 6 we present a picture of B for GL2(k). An apartment in B is

Fig. 6. A picture of the building of GLz(lc)

the image of any continuous injective map from R which maps integers t o vertices. Just as any two maximal split tori in G are conjugate, there is a natural action of G on B(G) with respect t o which any apartment can be carried into any other. Moreover, a version of the two body problem can be solved: given any two points in the building, there is an apartment which contains both of them. Combining these two facts we have: for any two points x,y in B , there exists a g E G such that gx,gy E d(T). (Here gx denotes the

Lectures on Harmonic Analysis f o r Reductive p-adic Groups

63

image of x under the action of g on B(G).) Finally, the action of G on B is semisimple; that is, for h E GI either there is a point in B which h fixes, or there is a line (i.e., a one-dimensional subspace of some apartment) on which h acts by nontrivial translation. Suppose x E t? and r E R.We define the Moy-Prasad filtration lattices gx,T and gx,r+ as follows. Choose g E G so that gz E A ( T ) .

gx,r := 9-l.&X,r . g and gx,T+ := g-'

' ggX,T+

. 9'

One can check that these definitions make sense. For r 2 0 we define the subgroups Gx,Tand G,,,+ in a similar fashion. Finally, we note that B is endowed with a nontrivial invariant metric, denoted dist. Moreover, with respect to dist, B has nonpositive sectional curvature. 4.2. G-domains f o r the Lie algebra

We begin with a generalization of the result of Howe discussed above; the proof of this result (from [l])nicely illustrates the benefit of working with the Moy-Prasad filtration lattices.

Lemma 4.2.1: Let x,y E

B, and let r

+

E

R. Then g X y Tc

+N.

+

Proof: Since gx,T c N if and only if &X,T c g g y , T N for g E GI we may assume that x and y are elements of A(T). Choose v' E X,(T)@R such that x = y+v'(working modulo X*(Z(G))@ R). Let B' be a Bore1 subgroup determined by 17.That is, B' has a Levi decomposition B' = TN' such that for all roots a E a$,, the set of roots which are positive with respect to N ' , we have ( a , q 2 0. Let B' = TN' denote the parabolic opposite B = TN' and let g = ii' t n' denote the associated Lie algebras. We have

+ + gx,r

= tr @

c

g$

t$€* I Il(z)>rl

In fact, a kind of converse to the above lemma is true. Namely, if X E g belongs to gx,r+ N for all x E B, then there is a point y E B such that

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64

X E gY,,. This result along with the above lemma gives us the following theorem.

Theorem 4.2.2:

In order t o simplify our notation] we define

From its definition] g, is open and invariant. From the above theorem we have that g, is closed. Consequently, g, is a G-domain. We now consider some examples. The set go is usually referred t o as the set of compact or integral elements of g. The set go+ is usually called the set of topologically nilpotent elements in g; it consists of those X E g such that powers of X tend to zero in the p a d i c topology. These G-domains have very natural interpretations; namely, g, = { X E g 1 v ( e ) 2

T

for all eigenvalues e of X }.

(If E is a finite extension of k, then there exists a unique extension of v to E.) For GL,(k) and X E g, the value of .(ex) must lie in the set { k / n I k E Z}. Of course, just as we have Moy-Prasad lattices of the form g X , , + , we also can define gr+ := UxEB g X , , + . These G-domains satisfy the obvious analogues of the results discussed above. From the previous paragraph, we have that g, # gr+ implies that r = k / n for some k E Z.

Exercise 4.2.3: Define the subspace D, of CT(g) by

where the sum is interpreted as follows. A function f belongs t o D, if and only if f can be written as a finite sum f = fi with fi E C c ( g / g x Z , , ) for some xi E 23. We can define D,+ in a similar way. Show that the Fourier transform gives us bijective maps from D,+ to CT(g-,) and from D, to

xi

CT(g(-r)+).

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65

4.3. G-domains in G . We now turn our attention to the group-side of things. Unlike the Lie algebra, an element of the group can only belong to a Moy-Prasad filtration subgroup if it first belongs to a parahoric. It turns out that the interesting part of the story here rests in proving the equality

u

n

G ~= , ~G , , ~.u

XED

XED

where U denotes the set of unipotent elements in G. Once this result is known to be true, it is very easy to establish the equality Gr =

n

Gx,r . u

Z€B

where T 2 0 and G, = UxEDGx,,.As above, it follows immediately that G, is a G-domain. We show how to prove the equality when r = 0.

Lemma 4.3.1: Go =

n

G, .U.

XED

Proof: We first show that the right-hand side is a subset of the left-hand side. We will argue by contradiction. Suppose that g E Gx,o ‘ U does not belong to Go. Since the action of G on B is semisimple, we either have that there is a point x in B which g fixes or a line e in an apartment A’ of B on which g acts by nontrivial translation. In the first case, we can write g = h . u with h E G,,o and u E U.Since u is unipotent, it must live in G,,o for some y E B. But, from a result of Eugene Kushnirsky this implies that u E Gx,o [lo] (Lemma 4.5.1). In the latter case, there exists a facet F’ in A’ such that F’ n e is open in .! For all z,y E F’ we have G,,o = G,,o. By hypothesis, there exist elements h E Gx,O and u E U such that g = uh. Since u is unipotent, there exists w E B which is fixed by u.We have that for all y E F’ n e

nzEB

dist(w, y) Thus, we have , for Figure 7.

= dist(w, uy) = dist(w, uhy) = dist(w, gy).

IC,

y E F’

n e a picture something like that described in

66

S. DeBacker

Fig. 7

However, since I3 has nonpositive sectional curvature, the line segment from x to w must be shorter than the line segments from y and gy to w. Similarly, the segments from gy to w must be shorter than the segments from x and gx to w.Consequently, we have dist(z, w) < dist(gy, w)

< dist(x, w),

a contradiction. We now show that the left-hand side is a subset of the right-hand side. We need to show that for x, y E B,we have G, c U.G,. As in the proof for the Lie algebra, we may assume that x and y both belong to d ( T ) . Let B' = T N ' be a Bore1 subgroup of G so that the (spherical) chamber in d ( T ) determined by N' is invariant under translation by the vector (y -x). Let N' be the unipotent radical of the parabolic opposite B = T N ' . From [3] we can write G = N' . N ' . N' . T . With some work, it follows that we can write

Gx,O= N : . N:. N: .To where NL = N' n G,,o and NJ = N' n G,,o. Because of the way in which N' was chosen, we have NL c G,,o. Thus, if g E G,, then there exist n1, n2 E N,, E E F,, and t E TOsuch that

Lectures o n Harmonic Analysis f o r Reductive p-adic Groups

67

4.4. A neighborhood basis f o r the nilpotent cone

We remark that T

and

u =n G T . T

It follows that the G-domains we have defined provide us with a neighborhood basis of the nilpotent cone (resp., unipotent variety) consisting of open, invariant, closed neighborhoods. 5. Nilpatent orbital integrals as distributions

A distribution on g is any element of the linear dual of CF(g). The aim of this section is to convince ourselves tha.t integrating against a nilpotent orbit defines a distribution on g. We follow the argument of Ranga-Rao [29]. 5.1. Orbital integrals Fix an element Y E g. Let Oy = GY := {gYg-' I g E G} denote the G-orbit of Y. We identify the orbit of Y with the homogeneous space G/CG(Y). Thus, the tangent space t o the orbit of Y a t the point Y is identified with

BICB(Y).

We define an alternating bilinear form ( , ) y on g by (A,B ) = tr(Y . [A,B)) for A , B E g. Fix an element B E g. A calculation shows that (A,B ) = 0 for all A E g if and only if B E C,(Y). Since we have a similar statement when we switch the roles of A and B , it follows that ( , ) y induces a nondegenerate alternating form on g/C,(Y). Thus, the dimension of the orbit is even, say 2m. Similarly, for each g Y = gYg-l E G Y ,we have a nondegenerate alternating symmetric form on Tan,y(Oy). Consequently, there exists a nondegenerate, invariant two-form w for Oy and from this we can form a nonzero, invariant volume form w A w A . . . Aw ( mtimes) on oy . Let X I , . . . , Xam be coordinates for g/C,(Y). For each 1 5 i 5 2m, fix a one-form d X i and associated measure IdXiI normalized so that for all f E Cp(g/C,(Y)), we

x:!,

have f(X) = f ( - X ) , where the Fourier transform is taken with respect to the measure IdXiI. There exists a locally convergent power series f ( X 1 ,X 2 , . . . , X z m ) so that (locally)

n:rl

w = f ( X l , X 2 , .. . ,X2,)dXl

A

dX2 A ... A dXzm.

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68

Define 2m

IwI := If(X1,x2,.. ., X2m)I

IdXZl. i=l

Thus, we have an invariant measure on O y . (Since GL,(k) is unimodular, it follows from standard measure theory results that CG(Y) is unimodular as well.) Exercise 5.1.1: Check that this definition is invariant of the various choices we have made.

5.2. A framing of the problem

If Y is a semisimple element of g, then O y is closed in g. Consequently, if f E C r ( g ) , then the restriction of f to O y is an element of Cr(Oy). So it makes sense to define the distribution O y : Cp(g) -+ C by setting O Y ( f ):=

/

f ( V )dg*.

G/CG(Y)

for f E C r (9).Here dg* denotes the invariant measure on G / C G ( Y defined ) above. What if Y is not semisimple? In particular, what if Y is nilpotent? In this case, we need to do some work to show that for arbitrary f E C r ( g ) the integral

makes sense. 5 . 3 . The basic notation associated t o nilpotent orbital integrals

We begin by recalling the familiar parameterization of nilpotent orbits in g. We then discuss various properties of nilpotent orbits which will be important in the sequel. From basic linear algebra we know that the nilpotent orbits in g can be parameterized by P ( n ) ,the set of ordered partitions of n. To p E P(n) we associate the nilpotent element X, E N n gro,O having Jordan canonical form corresponding t o p. That is, if p = ( p l , p 2 , . . . , p k ) , then in the ith

Lectures on Harmonic Analysis for Reductive p-adic Groups

69

block of size pi x pi we put the matrix with ones on the superdiagonal and zeroes elsewhere. For example, for p = (2,2,1) E P(5) we have

x,=

01000 00000 00010

!:c:j

.

Note that the nilpotent orbits in g z o , ~ / g s o , ~E++ Mn(f) are also indexed by P ( n ) and the map taking X, to the image of X, in M,(f) gives a bijective correspondence between 0(0), the set of nilpotent orbits in 8, and the set of nilpotent orbits in M, (f). We let 0, denote the G-orbit of X,. Note that if p < p' in the usual partial order on P ( n ) ,then 0, c 0,).

+

Exercise 5.3.1: Show that if 0 is a nilpotent orbit such that 0 n (X, # 0, then 0, is contained in the padic closure of 0. Moreover, G 0, n (X, + gzOtr+)= X, for all non-negative T .

gzo,o+)

W ~ Y +

Note that if X E N, and t E k x , then t X E N and in fact, t X and X are GL,(k) conjugate. (In general, it follows from Jacobson-Morosov that t 2 X and X are conjugate, but in GL, we can do better.) So, N and each nilpotent orbit are closed with respect to scaling. We can associate a parabolic subgroup P ( p ) of G to p as follows: For all positive integers j , the matrix X i acts on V = k", and we define V, = ker(X;) c V . Define P ( p ) = {g E G ( g . V, c V, for all positive j ). Then X, lies in the nilradical n(p) of the Lie algebra of P ( p ) , and the P(p)-orbit of X, is dense in n(p) and equal to 0, n n(p). We need most of the basic facts about sl2(k)-triples. For a good reference, see [4].We can complete X, to an slz(lc)-triple (Y,, H,,X,). Explicitly, the ith block of H, is given by the element diag((pi l),(pi - 3), . . . , (1 - pi)) and the ith block of Y, is given by certain entries on the super-subdiagonal. We let A, E X,(G) denote the associated one-parameter subgroup, that is, the ith block of A,(t) looks like diag(t(pi-I), t(pi-'), . . . , t ( ' - , z ) ) . For i E Z,we define g(2) = {X E g

xi

1 x,(t)X= tix}.

We have g = g ( i ) , For j E Z,define g ( 2 j ) := Ci2jg(i). We let p, denote the parabolic subalgebra g(> 0) with Levi subalgebra m, = g(0) and nilradical n, (= g ( 2 1)).We let P, denote the corresponding parabolic

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70

subgroup with Levi decomposition M,N,. The P,-orbit of X , is equal t o M g X , g ( 2 3) and hffiX, is an open and dense subset of g(2). Finally, it is true that CG(X,) c Pp.

+

5.4. A sketch of the proof that nilpotent orbital integrals

define distributions Fix p E P(n).Let f E CT(g). The Iwasawa decomposition (see [33]) tells us that we can write G = G,o,oP,. Fix a Haar measure dk on K and a left Haar measure dip on Pp so that

for all h E CT(G), the space of complex-valued, compactly supported, locally constant functions on G. Because it will be important, we recall that for all po = mono E Pp = M,N, we have de(p0pp;') = (det (rnolnP)l-l dep. Define f E CF(g) by f ( X ) := JGzo,o f ( ' X ) d k . Let us assume for the moment that

/

(* )

f("x)dep*

P/CC(X,)

converges. (Here dep* is the quotient measure.) In this case, we have

(*) =

/

/

f(kpx)

dkdep*

=

/

f(gx,)dg*,

G/Cc(XP)

P/CG(X@L)

and so we can define the invariant distribution 0, by r

f E CT(g). We now sketch the proof of why (*) makes sense. Let dZ be a Haar measure on g(> 2). We'd like to define a function FO on g(>2) so that

for

-1

For po = mono as above, we have d(poZp,') = Idet (rnoJg(z,)) dZ.Consequently, it is sufficient to produce a nonzero function FOwith the property FO(P0Z) = det (pole(')). The map X H (Y H t r ( Y . X ) ) induces an isomorphism of g(1) with g(-1)*. Just as at the beginning of this section, for Z E g(2), we can define

Lectures o n Harmonic Analysis for Reductive p-adic Groups

71

an alternating bilinear form ( , )z on g(-l). When 2 = X,,the form is also nondegenerate. Consequently, both g( 1) and g ( -1) are even dimensional (of dimension 2n’). Let az be the matrix which represents ( , )z with respect to some fixed basis of g(-1). From [22](Theorem 6.4) there is a polynomial (Pfaffian) Pf of degree n’ on g ( 2 ) so that det(az) = (Pf(Z))2. Moreover, 2 since d e t ( q m z ) )= det (m-llO(-l)) .det(az), the polynomial Pf transforms in the manner we desire. For 2 E g ( 2 2 ) , set F o ( 2 ) = Pf(Z2) where 2 2 denotes the image of Z under the projection map from g ( > 2 ) t o g ( 2 ) .

Example 5.4.1: For the partition ( 3 , 2 ) E P ( 5 ) , we have that the dimension of g(-1) is four. The space g ( 2 ) is spanned by the Chevalley basis elements X,,,, X,,, and X,,,.If 2 = z X a l z yX,,, t X a q 5 ,then

+

+

FO(2)= zy. 5.5. An important calculation

For j

> 0 we calculate 0 , ( [ X ,

+ tj]).

Proposition 5.5.1:

+ 41) = 4(( 1-2j).dim(Og)/2)

0,([X,

Proof: Suppose that m > j is very large. Let 0: := KmX, = {‘X, Km}. It follows from Exercise 5.3.1 that

Ik E

+

c?p([Xp e j ] ) = [Kj : Km ’ CK, ( X p ) ] ’ 0 p ( [ o r ] ) .

Now, by definition, we have 0, ([O,”]) = m e w g *(0,”) = meaSldX1I. IdXz I ...IdXz, I ((ern = [el-, -

+ C,(X,)

[Ce,l-,,

: t,

+ c,(X,) ) / C ,( X J )

+ C0(X,)]-1/2

(X,): C L (X,>1’/2

[t(l-,)

: t,]l/2

On the other hand,

After putting the two pieces together and doing a bit of calculation, we arrive a t

The proposition follows immediately.

0

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5.6. Nilpotent orbital integrals are homogeneous

Suppose t E I c x . For f E Cr(g) we define the dilation t via f , ( X ) = f (tx). For p E P ( n ) we have

ft

E C r ( g ) o f f by

Consequently, since nilpotent orbits are closed with respect to scaling and the invariant measure on a nilpotent orbit is unique up to a constant, O,(ft) and O,(f) must differ by a constant. A calculation shows that

For example, from Proposition 5.5.1 we have

+

oP([m(-2j+l)xpe l + ] )

= 1.

6. The Fourier transforms of invariant distributions 6.1. Basics

A distribution on g is any element of the linear dual of CF(g) (no topological restrictions). We denote the subspace of invariant distributions on g by J ( 8 ) " . Suppose T E J ( g ) . We define the Fourier transform T E J ( g ) of T by

F ( f ) := T(j) for f E CF(g). It is a remarkable fact that 5? is represented by a locally integrable function; that is, there is a function T E L:,,(g) such that for all f E Cy(g),

T ( f )=

I

T ( X ) . f ( X )d X .

Unfortunately, describing this function is beyond our abilities in all but the simplest situations. Example 6.1.1: Consider the trivial nilpotent orbital integral U ( l , ~ , . . . ,El ) J ( g ) . For f E Cr(g) we have

eAccording to Howe, he chose the letter J because and J follows the letter I in the alphabet.

I (for invariant) was already taken,

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73

and we also have

l)(f)=

6 ( 1 , 1 ,...,

We therefore conclude that

i

f ( X ).6(1, 1,...,l ) ( X ) d X .

6(1,1,...,1) = 1

We let g""."' denote the set of regular semisimple elements in g; this is a dense open subset of g. In our situation, gr's's'consists of those elements of g having distinct eigenvalues. The main idea of this section is t o present a very elegant description of ? ( X ) when T E J ( g ) and X E g""'"~. (From this description, it will follow that T is represented by a locally constant function on gr.".'..)The existence (of a form) of this expression was conjectured by Paul Sally, Jr. and proved by Reid Huntsinger [21]. The proof follows an argument of Harish-Chandra [13]. Originally, Huntsinger and Sally were only interested in studying the behavior of the Fourier transform of a nilpotent orbital integral. We shall temporarily restrict our attention to this situation. Suppose p E P(n).For f E C r ( g ) we have = OJf)

f ( X ). 6 , ( X ) dX = =

J'

f(gx,)dg*

G/CG(X+)

J'

= G/CG(XF)

/

f ( X ) . A(tr(gX, . X ) )dX dg*

g

So, if we could justify the equality in quotation marks, we'd have

6JX)

=

1

G/CG

(x,

A(tr(gX, . X ) ) dg'

= O,(Y w A(tr(Y. X ) ) ) .

This is nearly correct; we now describe what is true.

Theorem 6.1.2: (Reid Huntsinger) Let K be a n y compact open subgroup of G and let dk denote the normalized Haar measure on K . For all X E gr'S'S' we have

6 , ( X ) = O,(Y

H

A(tr(Y. ' X ) ) d k ) .

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S. DeBacker

Remark 6.1.3: For reasons which will become clear later, we remark that the function 6, is canonical. On the face of things, it depends on two choices: the additive character A and the choice of a measure on 0,. However, our choice of the measure on 0, was not arbitrary; it depended on A. As one can verify, this dependence makes 6, canonical. Remark 6.1.4: In our situation, since every nilpotent orbit is Richardson, there is a very nice description, due to Howe, of the function 6,. Namely, 6, can be related to the character of the representation obtained by inducing the trivial representation on P ( p ) up to G.

6.2. A m o r e general statement More generally, following a suggestion of Bob Kottwitz, Reid Huntsinger proved the following statement.

Theorem 6.2.1: (Reid Huntsinger) Fix r E R . If T E J(gr), t h e n T i s represented o n gr.s'"' by

Here, f o r Y E g, q x ( Y ):= J,(A(tr(Y."))) dlc. (As before, K is a compact open subgroup, and dlc is the normalized Haar measure o n K . )

At the heart of the proof of this theorem lies the statement that the map := qx .[gr] is locally constant. Since from g""'"'to Cm(g) sending X to qx,,. the verification of this statement requires some fairly detailed analysis, we shall skip the proof. This statement immediately implies that the function X H T ( Q X ,=~T) ( q x )from gr'S'S'to C is locally constant. ) . need to show Suppose f E C ~ ( g r . " ' " .We

From the previous paragraph, there exists a finite collection { w i } E 1 of compact open disjoint subsets of gr.".'. such that both X t+ T ( q x ) and X ++ f ( X ) are constant on wi and the support of f is contained in u w i .

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Using Fubini’s theorem and the invariance of TI we then havef

m

= T(Y

H

/ / /f(X)

0

= T(Y

H

A(tr(kX.Y)) d k dX)

K

p Y )d k )

= T(f).

7. Characters

In this section we define the character of an admissible representation and discuss some properties of characters. 7.1. Admissible representations

Fix an admissible representation (7rl V ) of G. Recall, from Gordan Savin’s lectures [33], that (7rl V ) is a representation for which (1) for all 21 E V there exists a compact open subgroup K such that 21

E

vK:= (21 E v I n(k)w = 21 for all IC E K )

and (2) for all compact open subgroups K of G we have dime V K < co. The second condition is equivalent to saying that for all compact open subgroups K of G and all irreducible representations 0 of K , the multiplicity of u in 7r is finite.

Example 7.1.1: Suppose a is a cuspidal representation of GL,(f) % Ko/K1. Inflate a to a representation of KO and extend this inflation to a representation (T of Z(G) . KOwhere Z ( G ) denotes the center of G. The ‘This part of the proof supports David Vogan’s adage: “The p-adics: if you can add, you can integrate.”

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representation (xu,Vu) obtained by (compact) induction of r~ from Z(G).Ko up to G is an (irreducible) admissible representation.

Example 7.1.2: Suppose m > 0 and is a character of T/T,. Inflate jj to a character x of T . Since the unipotent radical N of B is normal in B , we may extend x to a character of B . The representation (7rx,Vx) obtained by (compact) parabolic induction of x from B to G is an admissible representation of G. 7.2. T h e character distribution We define the character distribution, O,, associated to the representation ( 7 r , V ) as follows. For f E CF(G), we define ~ ( fE)End(V) by 7r(f)v=

s,

f ( g ) . 4 9 ) vd9

for w E V. Since f is compactly supported and locally constant and ( T , V) is admissible, this integral is really just a finite sum. In fact, if K is a compact open subgroup of G such that f ( k l g k 2 ) = f ( g ) for all k l , k2 E K and g E G, then we have

4f).

= 7r([KI*f

* [Klk = 7 W 1 ) 7 r ( f ) 7 r ( [ K I ) V .

(Here * denotes the usual convolution operation.) Thus, ~ ( fis) a map from V to VK. If eK := measd,(K)-l..rr([K]), then e K is the projection operator from V to V K ,and we have

v = VK @ (1

-

eK)V.

Consequently, since dima:(VK) < m, it follows that ~ ( fis) a finite-rank operator. We can therefore define the character distribution 0,: C r ( G ) 4 C

by sending f to tr(7r(f)). Just as the Fourier transform of an invariant distribution on g is represented on g'.",". by a function in Coo(g'.s.".), so too is the character distribution [15]. We abuse notation and denote this function by 0, E Coo(Gr.S,S.). (Gr.".".denotes the open, dense subset of G consisting of regular semisimple elements, that is, those elements of G whose eigenvalues are distinct.) We call this function the character of 7 r .

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Remark 7.2.1: Fix y E G‘.”.”. and m > 0 large enough so that yKm G‘,”.”..Define fm = rneasdg(Km)-’ . [yK,]. We then have

C

Since the function 0, is locally constant on Gr.‘.’., the right-hand side becomes Q,(y) for all m sufficiently large. Thus, “in the limit” the function 0, looks like a character.

7.3. A calculation Fix j 2 1. Suppose that 7 is an irreducible representation of Kj / K2j. Since Kj / K2j is abelian, 7 is a character. Let denote the corresponding character of Kj; we regard r as an element of CF(G) in the obvious way. Let resKj n denote the restriction of T to K j . Suppose that resKJ n = @,Ezm(a,~ ) a where m(o,T ) denotes the multiplicity of o in resK,

T.

We have

=m ( ~ - lT , ) . measdg(Kj).

In other words, the character picks out the multiplicity of (up to a constant).

7-l

in resK,

T

7.4. Depth

The depth of a representation was introduced in the fundamental papers of Allen Moy and Gopal Prasad [28,27]. Essentially, the depth of a representation is a rational number which measures the first occurrence of fixed

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vectors with respect to all filtration subgroups of G that arise naturally from Bruhat-Tits theory. Recall from $3.4.2 that 0 denotes the set of optimal points. Up to conjugation, the set of optimal points is finite, and so the subset { r E R I g x , r # gx,r+ for some IC E 0 ) is discrete (and in fact, is a subset of Q). For an admissible representation ( T , V ) ,we define p ( ~ )the , depth of ( T , V ) , by

1

p ( ~ := ) min{r E Q>o -

there is an z E 0 such that VGz3r+ # 0).

Proposition 7.4.1: p ( ~ is) the unique rational number satisfying the following statement. If (2, r ) E d(T) x R>o - with V G z 3 ~ #+{0}, then r 2 p ( ~ ) . Proof: Suppose ( x , r ) E d(T) x lR>o - with VGz3r+ # (0). From the group version of Lemma 3.4.3 there exists y E 0 such that GX,,+ 3 GY,r+.Consequently, {O> # c V~Y,V+. 0 v

~

~

~

T

+

All the various things you would want to be true about the depth of a representation are true. For example, if o is an irreducible representation of a parabolic subgroup of G, and T is an irreducible subquotient of the induced representation, then p ( n ) = p ( o ) .

Example 7.4.2: Any representation with Iwahori (that is, Bo) fixed vectors has depth zero. Example 7.4.3:The representation ( T ~V,g )defined in Example 7.1.1 does not have Iwahori fixed vectors, but it does have depth zero. Example 7.4.4: The representation depth (m - 1).

(T,,

V,) defined in Example 7.1.2 has

7 . 5 . Elementary Kirillov theory Fix an irreducible representation ( T , V ) . For nilpotent real groups, Kirillov [24] established that the representations were parameterized by coadjoint orbits in the linear dual of the Lie algebra. In our context, the term Kirillov theory is used to describe the connection between representations of compact open subgroups occurring in T and coadjoint orbits in the linear dual of g. Via our nondegenerate trace form, we have identified g with its dual. The type of result discussed below was first studied, I believe, by Howe [17]. Fix r , s E R>o - and z,y E B.

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The map g H (g - 1) induces an isomorphism of the abelian groups G,,,+ /G,,(zr)+ and B , , ~ + /g,,(2r)+. Moreover, every character of gz,r+/gr,(2r)+ is of the form Y H R(tr(X . Y ) for some E gz,-2r/gz,-r. (That is, the Pontrjagin dual of G,,,+/G,,(zr)+ gz,T+/gI,(2T)+is isomorphic t o Bz,-2rlBz,-r.) Fix a character d of G,,,+/G,,J~~)+and a character 7 of Gy,s+/Gy,(2s)+ Let Xu gI,+ E ~ ~ , - 2 ~ / grepresent ~ , - ~ 5 and X, E represent 7.

+

+

Proposition 7.5.1: If u and r both occur in T , then there exists a g E G such that

"XT

+ gy,--s) n (Xu+ g z , - r )

# 8.

Proof: Let V, c V (resp. V, c V ) denote a one-dimensional subspace of V on which resGz,r+T (resp. resGV,*+ T ) acts by u (resp. 7 ) . Since ( T , V ) is irreducible, there exists a g E G such that the image of T(g)Vuunder the projection of V onto V, is nonzero. This implies that for all h E G,,,+ n gGy,,+, we have

a ( h )= T ( g - ' h g ) . Thus, for all H E B,,~+ n ggy,s+, we have R(tr(X, . H ) ) = A(tr(gX, . H ) ) . This implies that for all H E B,,~+ n ggy,s+,we have tr((X, - gX,) . H ) E P. Consequently, (Xu - gX7) E (g,,.+ n g g y + + ) * = g2,-r ggy,--s. The proposition follows. 0

+

7.6. Understanding the distribution r e s c r ( G p ( . r r ) +0 ),

) be an irreducible admissible representation. From our discussion of depth, we know that we can find an x E A ( T ) such that VG=,p(n)+ # (0). Thus, the trivial representation of Gz,p(n)+ occurs in T ; the associated coset in g is gz,-p(a). Now fix s > p ( ~ ) y, E B, and ? E Gy,s/Gy,s+such that the character T of Gv,s occurs in r. Let X , E gy,(--s)/gy,(--s)+ be the coset corresponding to ?. From our discussion of Kirillov theory, there exists a g E G such that 9 g I , - p ( n ) n ( X , + B~,(-~)+) # 0. However, from Lemma 4.2.1, we have Let

( T ,V

-

+

9

Bz,-p(a)

c By,-p(n)

+N c

By,(--s)+

+N .

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Consequently, we can assume that X , is nilpotent! We now use this fact to say something about the distribution we define k E rescr(Gp(n)+)0,. For a function h E Cp(gp(s)+), CT(Gp(,)+)by h ( g ) = h(g - 1). We define the distribution O,g on g by O,,,(f) = O,(.fpc,)) where fp(,) = f . [gp(,)+]. We let Q,denote the Fourier transform of @ .,, From Exercise 4.2.3, we know that the Fourier transform of an elebelongs to Cr(gp(,)+). Suppose z E B and s > P(T). If ment of Lp(,)

f E C ( g r , - s / g z , - p ( x then ) ) , we have & ( f ) = Q,(f). It follows from our discussion above and a few lines of calculation that

&(f) # 0 implies

supp(f) n (g2,(-s)+

+N)# 0.

This last statement is equivalent to the statement

6,(f)# o implies

supp(f) n g(-’)+

# 0.

(3)

7.7. The Harish- Chandra-Howe local character expansion Considerations similar to those above led Roger Howe to make his finiteness conjectures [19] (which we will discuss later) and establish the following remarkable connection between the character of an irreducible representation of G and the Fourier transforms of nilpotent orbital integrals.

Theorem 7.7.1: (Harish-Chandra-Howe local character expansion) If T is an irreducible admissible representation of G , then there exist constants c P ( r ) indexed by p E P(n) such that O,(1

+X ) =

c

CP(7r).

B,(X)

(4)

PEP(,)

for all X E gT.‘.’. suficiently near zero. Remark 7.7.2: Howe [16] proved this result without any restrictions on the characteristic of k. Later, under the assumption that the characteristic of k is zero, Harish-Chandra [13] generalized the proof t o all connected reductive groups. The above theorem and its analogues (local expansions about any semisimple point) play a crucial role in Harish-Chandra’s proof that characters are locally integrable on G just on Gr.’.’.). (The question of integrability is still open for arbitrary groups in positive characteristic. From work of Rodier [30] and Lemaire [25], it is known t o be true for GL,(k) when k has positive characteristic.)

(a

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One of the difficulties with this theorem is that it provides no indication of where equation (4) ought to hold. The conjecture of Hales, Moy, and Prasad [28] is a precise statement about this issue:

Statement 7.7.3: Equation (4) ought to be valid on gif;y+. For integral-depth representations this was proved by Waldspurger [35]. With some restrictions on k, it was proved for arbitrary depth representations in [9]. 8. An introduction to Howe’s conjectures and homogeneity In his paper, Two conjectures about reductive p-adic groups, Howe proposed two remarkable finiteness conjectures that now bear his name. Howe’s conjecture for the Lie algebra looks like: dim@rescc:(g/L)J ( w ) < co. Here w c g is any compactly generatedg, invariant, and closed subset of g, J ( w ) denotes the space of invariant distributions supported on w , and L is any lattice in g. For T E J ( w ) , (g/L) T denotes the restriction of T to Cc(g/L). Howe’s conjecture for the group looks very similar: dim@resCc(G/K)J ( w ) < co. Here K is a compact open subgroup of GI and w is a closed, compactly generated, invariant subset of G, and J ( w ) denotes the space of invariant distributions which are supported on w . Of course, although we still refer to these statements as Howe’s conjectures, both are known to be true. Howe’s conjecture for the Lie algebra was proved by Howe [16] in the 1970s for GL, and later by HarishChandra 1131 for general groups (but only in the characteristic zero setting). Waldspurger [37] has also given a proof. Howe’s conjecture for the group was proved in the 1980s by Clozel [6,7] in the characteristic zero setting. In the 199Os, a characteristic free proof of Howe’s conjecture for the group was given by Barbasch and Moy in their beautiful paper [2]. gThat is, w can be realized as the closure of G C for some compact subset C of g.

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8.1. Homogeneity Beginning with Waldspurger’s paper [36], much energy has been spent trying to produce optimal versions of Howe’s conjectures. By optimal, we mean that we’d like to choose w and L in such a way so that we can not only describe the dimension of resCc:(g/L)J ( w ) , but we can also find a good basis for this space in terms of distributions with which we are very familiar, namely, nilpotent orbital integrals. These optimal versions of Howe’s conjectures are referred to as homogeneity results. This is an appropriate name, because, according to Webster’s Ninth New Collegiate Dictionary, the word homogeneity means “the state of having identical distribution functions or values”. We restrict our attention to the Lie algebra. Fix T E R.The role of w in Howe’s conjecture will be played by the G-domain gr+ (which is compactly generated). Guided by Harish-Chandra’s philosophy and Theorem 7.7.1, it appears that we want something like the following: For all T E J ( g r + )and for all f E C?(g(+.))

PEP(,)

This last displayed equation is equivalent to requiring

c

T(f”)=

CPL(T).

WP).

PcP(n)

From Exercise 4.2.3 we have f^ E Dr+.So, the role of C?(g/L) will be played by Dr+. Putting it all together, we arrive at the homogeneity statement resoV+J ( g T + ) = r w r + J ( N ) .

+

The fact that gr+ c gz,r+ N for all x E B gives some feeling as to why this homogeneity statement ought to be true. 8 . 2 . From Howe’s conjectures to the Harish- Chandra-Howe local character expansion The homogeneity result discussed in the previous section is actually not strong enough to prove Theorem 7.7.1. As in [16,13,9,35]we need something a bit stronger, but, unfortunately, more complicated. For x E B and s 5 r , define Jz,s,r+

I

:= {T E J(9) for

f

E C(Bs,s/Bs,r+),

if supp(f) n (g,+) = 8, then T ( f )= 0).

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83

I

c JX+,,+.

Note that J(gr+)

Remark 8.2.1: Although this definition seems a bit unnatural, it arises naturally from our understanding of resC?(G,(,,+) 0,. In fact, from Equa-

tion (3), we have that

6,

E

JX,,,,+if -r > p(.).

The following is the stronger homogeneity statement that needs to be proved in order t o recover Statement 7.7.3:

-

resDT+J,+ = resDT+J ( N ) where

9. Proving homogeneity results

We recall that for

T

E IR we

have

and XEB

We want to show r e s ~ J~( g+r + )

=res~?+ J(N).

(5)

We decided that knowing this, or, in truth, a stronger (but more complicated) statement would tell us that the Harish-Chandra-Howe local character expansion was valid on g;T$+. As we discussed before, g r = gr+ unless T = for some k E Z. Consequently, we only need to verify Equation (5) for r E { Moreover, by taking advantage of the fact that nilpotent orbital integrals are homogeneous, we can further restrict our attention to T of the form with 0 5 k < n. Indeed, pick m E Z such that m r E [0,1).For T E J ( g ) and t E k x , define Tt by Tt(f) = T ( f t ) .Note that

i}.

+

D,+ if and only if

f

E

E

J(g,+) if and only if Tm ,

frn-,,,

E

D(,+,)+

and

T

J(g(,+,)+).

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Consequently, if we know

then for T E J

and f E Dr+ we have

and so we may assume T E [0,1). For the remainder of this section we discuss how to prove Equation (5). We first prove it for GLl(k), we next prove it for GLz(k), and finally we discuss a way to go about giving a general proof. 9.1. A proof f o r GLl(lc)

From the reasoning above, we need to show resCc(k/63)J ( @ )= resC,(k/63) J ( N ) . The right-hand side is one-dimensional and spanned by the distribution f H f(0) for f E C c ( k / 6 3 ) . Moreover, the right-hand side is a vector subspace of the left-hand side. Since GLl(k) is abelian, all distributions are invariant. Note that J(63) consists of linear maps from C T ( k ) to C which are supported on 63. If f E Cc(k/63),then we can write

f=

c

CX.[X+63]

X€k/63 where the c~ are complex numbers which are almost always zero. For T E J ( P ) , we have

Vf)= T(c,s.[@I) = WJI). f(0). Consequently, the left-hand side is also one-dimensional and so the equality is established.

9.2. A proof f o r GLz(lc) This is where things begin to become interesting. Thanks to the remarks a t the beginning of this section, we only need t o verify two statements: reso,,+

J(gO+)

= resDo+ J ( N )

(6)

85

Lectures o n Harmonic Analysis f o r Reductive p-adic Groups

and

We begin by considering the first statement, Equation (6). We note that Do+ may be thought of as an invariant version of Cc(g/f?l), but, for a good way to think about the space Do+. our purposes, this is

9.2.1. Descent and recovery Fix T E J(go+). We wish to show that resDo+T is completely determined by resc(eo/ei)+C(bo/b i / z T . Fix f E Do+.We will demonstrate that T ( f )is completely determined by resC(~o/el)+C(bo/bl/2) T . We write f = f i with f i E CC(g/gzi,o+) for some xi E B. Since T is linear, without loss of generality we may assume for some z E B. We can write that f E Cc(g/gr,o+)

Xi

f=

c

c x . [X+85,0+1

X€0/B,,o+

with the cx complex constants which are almost always zero. Again, since T is linear, without loss of generality we may assume that f = [ X gz,O+]. Now, T ( f )= 0 if the support o f f does not intersect go+. Consequently, since go+ c g2,0+ N , we must have

+

+

(X+Br,O+)nN#@. loss of generality, X E N .

So, without Up t o conjugacy, we have two choices for gz,o+; it is either f?, or b1/2. In what follows, the reader is encouraged t o consult Figure 8 to get a more geometric understanding of what is happening. We first deal with the e l case. Since xo is the only point x where gz,O+= e l , we may suppose that X E N n (gzn,--m gz,,(-m)+) for some m > 0. In other words, X E N n (Lm\ e l p m ) . Since we are free t o conjugate by G,,,o = K O ,we may assume that

with u E R X .For any point y in the chamber CO(that is, a point between xo and x' in Figure (8)),we have that X is "closer" to the origin with respect t o the y filtration than it is in the z filtration. For exampIe, X E g y o , ( 1 p m ) + where yo is the barycenter of CO.The problem is: a t the point y we require local constancy not with respect t o gzo,o+ but with respect to gy,o+ = b1/2.

86

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87

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In order to recover the proper type of local constancy, we take advantage of the invariance of T . We write

1

=-

+

. T ( [ X hip]).

4 We have succeeded in writing T ( f )in terms of T evaluated a t a function f’ E Do+ which is supported closer to the origin with respect to some other point in the building. We now examine the b1/2 case. In this case, we are looking a t the coset X + gy,O+ where X E and y is any point in CO.Since we are free t o conjugate by Gy,o= Bo, we may assume that X is either

or

with m

> 0 and u E R X. In the former case, we can write T ( [ X+ By,o+I)

=

c

+ (:

:)

+ BZl,O+l)

@€P/P2 where

21

= d i a g ( w - ’ i l ) ~ ~= Z’is the other vertex of

GO.We have

+

From Figure 8 it is clear that we have expressed T evaluated a t [ X gy,O+] in terms of T evaluated at f ‘ where f ’ E Do+ has support closer t o the origin with respect to the z1 filtration than [ X gy,O+]had with respect to the y filtration.

+

Exercise 9.2.1: Do the analogous analysis for the latter case.

To summarize, the point of descent and recovery is as follows. We begin with a simple function f E C((gZ+ \ gZ,s+)/gZ,o+)for some z E B. From this function, we find a point y E B and a function f’ E C(gy,s+/gy,O+)so that T ( f )= T ( f ’ ) .After a finite number of steps, we will have shown that T (f ) is completely determined by reSC(eo/&1)+C(bo/bl,2) T.

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9.2.2. Counting

We now know the following facts: (1) Thanks t o Harish-Chandra [13], the dimension of the complex vector space reso,+ J ( N ) is equal t o the cardinality of P ( n ) , which, in this case, is 2. (2) From $4.4, we have J ( N ) c J(go+) and so resoo+ J ( N ) c resoo+ J(flo+ 1. (3) From the previous section, we know that dim@reso,,+J ( 8 0 + ) = dim@reSC(eo/el)+C(bo/bl/2)J(80+ ).

Consequently, we need only show that

Since

for any x E ??B, we have that for T E J T is completely determined by

Since

we are done.

Exercise 9.2.2: Using the above proof as a template, prove Equation (7). 9.3. The general approach

In general, the proof is very much like that produced above, only more complicated. 9.3.1. Descent and recovery

Suppose T E J(gr+).The main point is to show that resDr+ T

=0

if and only if r e s p T r+

=0

where XEB

That is, we can find a very small space of functions from which we can choose a dual basis for reso?+ J ( g r + ) .

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Suppose f E D,+ . As before, without loss of generality we may assume that f = [ X + g x , r + ] for some X E N and some z E B. Under some hypotheses, we can use slz(f)-theory to find a direction in the building in which to move so that, for some y E B in that direction, the support o f f is nearer the origin with respect to y than it was with respect to z. Moreover, if necessary, we can use the basics of slz(f)-representationtheory and the invariance of T to ‘Lbeef-up”the coset gX,,+ so that we arrive a t a function f‘ E C,(g/gY,,+) with the properties: (1) T ( f ’ )= T(f) and (2) the support of f ’ is “closer” to the origin with respect t o y than the support of f with respect to 2.

Remark 9.3.1: Invoking sLz(f)-theoryrequires some restrictions. The theory only works well if the highest weights for the representations one wishes to consider are less than ( p - 3) where p is the characteristic of f.

9.3.2. Counting

This is the real key. In the end, you need some type of correspondence between “suitable” elements of D,‘+ and O(O),the set of nilpotent orbits in 0. When r is zero, the situation is easy to understand. As discussed before, the conjugacy classes of facets in B are in one-to-one correspondence with the elements of P ( n ) .Similarly, the nilpotent orbits are in one-to-one correspondence with the elements of P(n).For p E P ( n ) ,let Fp be the facet described in $3.4.1.The image of x,, in &7,,0/gF,,0+ is distinguished nilpotent (that is, it does not lie in a proper Levi subalgebra of g~,,o/g~,,~+). We let [X,, gF,,o+] E D:+ denote the characteristic function of the coset X , gF,,o+. This is the correspondence between “suitable” elements of D:+ and O(0) alluded to above. Indeed, for T E J ( g O + ) the , restriction of T t o D:+ is zero if and only if

+

+

is zero for all 1-1 E P ( n ) . When r # 0, life is much more complicated; but something beautiful is true. We refer the reader to [lo].

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10. A few comments on the cfi(7r)s Let ( T , V ) denote an irreducible representation of G. We recall that for all regular semisimple X in some neighborhood of zero we have the HarishChandra-Howe local character expansion:

0,(1+ X ) =

c

CJT)

.6,(X).

PEP(,)

In his paper [16], Howe proves that for irreducible supercuspidal representations (that is, those representations that do not occur as subrepresentations of parabolically induced representations) the coefficients occurring in the Harish-Chandra-Howe local character expansion are all integers. In fact, given a bit of additional information, the same proof shows that this is true for all smooth irreducible representations. We first recall what we know about the C ~ ( T ) S ,and we then prove this result of Howe.

10.1. The coefficient c(1,1,...,1)(7r) Suppose that ( T ,V ) is a discrete series representation (that is, the matrix coefficients of ( T , V )are square-integrable mod the center of G). In this case, by using Rogawski [32] one may extend a result of Harish-Chandra [13] to show that

Here deg(T) denotes the formal degree of n,St is the Steinberg representation (see, for example, [5]), and C denotes the semisimple rank of G. On the other hand, if ( T , V) is a tempered representation (that is, it occurs in the Plancherel formula) which is not in the discrete series, then, using results of Kazhdan [23], Huntsinger [20] showed that c(1,1,...,1)(~)= 0 (see also the paper of Schneider and Stuhler 1341). 10.2. The leading coefficient According to Mceglin and Waldspurger [26], the set {P E

v.1

I C P ( 4 # 01

has a unique maximal element; call it ps. Moreover, according to Mceglin and Waldspurger [26] and Rodier [31], the coefficient C ~ , ( T ) is an integer which is equal to the dimension of the degenerate Whittaker model corresponding to ps.

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Groups

Example 10.2.1: Suppose p = ( n ) E P(n). Recall that X(n) is regular nilpotent. Let B be the Bore1 subgroup consisting of lower triangular matrices in G. Let N denote the unipotent radical of B ;that is, the subgroup of B formed by the set of lower triangular matrices with ones on the diagonal. Define the character 6 of by f i H h(tr(X(,) . (fi - 1)).For an irreducible smooth representation ( T , V) of G, let V ( 6 )denote the subspace of V generated by the set

m

{ ( ~ ( f i-) 6(fi))w I w E V and

fi E

m}.

Define V(n)= V/V(6).Then according t o the results quoted above, we have qn)(T ) = dim@(V(..))= dim@HomG (V, I n d z 8). 10.3.

The remaining coeficients

Not much is known about the remaining coefficients. Following an argument of Roger Howe 1161, we show that they are all integers. We begin by making Remark 6.1.4 precise. If p' E P ( n ) ,then opt

(x)=

@ I ~ ~ G1

WW')

(1

+ x)

for all X E go+. Here Ind$(,,) 1 is the representation of G obtained by inducing the trivial representation of P(p') up t o G. Suppose j > 0, p E P ( n ) ,and T~ is the character of Kj represented by the coset m(1-2j)XP t(l-j) E t l - 2 j / t l - j . From 57.3 we have

+

@,(T~)

= measdg(Kj) . m ( ~ ; l , r )

(8)

and

O I nPb') d ~ l ( ~ P ) = measd,(Kj) . rn(T;',Ind&,)

1).

(9)

From the discussion above, Exercise 5.3.1, and Equation (2) we have

dpt(7,)

= opt(pP)= measdx(t,)

=

.

([-m(1-2j) .

measdx(tj) . m(7;1,1nd&,) measdX(tj)

1)

{o

x P + t (1-j)l)

if p < p', if p = p', and otherwise.

Thus, if j is sufficiently large, then c ~ ( T.m(T;',Ind&) )

m ( ~ i ' , r=) PlP'

1).

92

S. DeBacker

We now proceed by induction. From Rodier [31] we have t h a t c ( ~is) always a n integer. Suppose p E P ( n ) .By induction, if p' > p, then cp(w) E Z.We have c , , / ( w ) . m ( ~ ~ l , I n d ~ (1) , , E)

c p ( w ) . m ( ~ i l , I n d & ) 1) = m(~;',w)-

Z.

PL'>P

Since m ( ~ ; ' Ind&) , 1) = 0,([-a(1-2j) .X ,

+ t2,1-j)])

= 1, we are done.

Acknowledgements The author was partially supported by National Science Foundation Grant

No. 0200542. This paper is based on lectures delivered at the Institute for Mathematical Sciences at the National University of Singapore in 2002. The author thanks the Institute for its support.

References 1. J. Adler and S. DeBacker, Some applications of Bruhat-Tits theory t o harmonic analysis o n the Lie algebra of a reductive p-adic group, Mich. Math. J., 50 (2002), no. 2, pp. 263-286. 2. D. Barbasch and A. Moy, A new proof of the Howe conjecture, J. Amer. Math. SOC.13 (ZOOO), no. 3, pp. 639-650. 3. A. Bore1 and J. Tits, Groupes rdductifs, Inst. Hautes Etudes Sci. Publ. Math., NO. 27, 1965, pp. 55-150. 4. R. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, Reprint of the 1985 original, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. 5. W. Casselman, Introduction t o the theory of admissible representations of p-adic groups, to appear. 6 . L. Clozel, Orbital integrals o n p-adic groups: a proof of the Howe conjecture, Ann. of Math. (2) 129 (1989), no. 2, pp. 237-251. 7. -, Sur une conjecture de Howe. I, Compositio Math. 56 (1985), no. 1, pp. 87-110. 8. S. DeBacker and P. J. Sally, Jr., Germs, characters, and the Fourier transforms of nilpotent orbits, The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998), Proc. Sympos. Pure Math., 68,Amer. Math. SOC.,Providence, RI, 2000, pp. 191-221. 9. S. DeBacker, Homogenefty results for invariant distributions of a reductive p-adic group, Ann. Sci. Ecole Norm. Sup., 35 (2002), no. 3, pp. 391-422. 10. -, Parametrizing nilpotent orbits via Bruhat-Tits theory, Ann. of Math., 156 (2002), no. 1, pp. 295-332. 11. ~, Some applications of Bruhat-Tits theory t o harmonic analysis o n a reductive p-adic group, Mich. Math. J., 50 (2002), no. 2, pp. 241-261.

Lectures on Harmonic Analysis for Reductive p-adic Groups

93

12. R. Kottwitz, Harmonic analysis o n semisimple p-adic Lie algebras, Proceedings of the International Congress of Mathematicians, Vol. I1 (Berlin, 1998). Doc. Math. 1998, Extra Vol. 11, pp. 553-562. 13. Harish-Chandra, Admissible invariant distributions o n reductive p-adic groups, Preface and notes by Stephen DeBacker and Paul J. Sally, Jr., University Lecture Series, 16,American Mathematical Society, Providence, RI, 1999. 14. -, T h e characters of reductive p-adic groups, Contributions to algebra (collection of papers dedicated to Ellis Kolchin), Academic Press, New York, 1977, pp. 175-182. 15. -, A submersion principle and its applications, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), no. 2, pp. 95-102. 16. R. Howe, T h e Fourier transform and germs of characters (case of GL over a p-adic field), Math. Ann. 208 (1974), pp. 305-322. Kirillov theory f o r compact p-adic groups, Pacific J. Math. 73 (1977), 17. -, no. 2, pp. 365-381. 18. -, S o m e qualitative results o n the representation theory of G1, over a p-adic field, Pacific J. Math. 73 (1977), no. 2, pp. 479-538. 19. ___ , T w o conjectures about reductive p-adic groups, Harmonic analysis on homogeneous spaces, Proceedings of Symposia in Pure Mathematics, vol. 26, American Mathematical Society, 1973, pp. 377-380. 20. R. Huntsinger, Vanishing of the leading t e r m in Harish-Chandra's local character expansion, Proc. Amer. Math. SOC.124 (1996), no. 7, pp. 2229-2234. 21. -, S o m e aspects of invariant harmonic analysis o n the Lie algebra of a reductive p-adic group, Ph.D. Thesis, The University of Chicago, 1997. 22. N. Jacobson, Basic Algebra, I, second edition, W.H. Freeman and Company, New York, 1985. 23. D. Kazhdan, Cuspidal geometry of p-adic groups, J. Analyse Math. 47 (1986), pp. 1-36. 24. A. A. Kirillov, Unitary representations of nilpotent Lie groups, Russ. Math. Surveys, 17 (1962), pp. 53-104. 25. B. Lemaire, Inte'grabilite' locale des caractkres-distributionsde G L N ( F ) OG F est un corps local non-archime'dien de caracte'ristique quelconque, Compositio Math. 100 (1996), no. 1, pp. 41-75. 26. C. Mceglin and J.-L. Waldspurger, Mod2les de Whittaker de'ge'ne're's pour des groupes p-adiques, Math. Z . 196 (1987), no. 3, pp. 427-452. 27. A. Moy and G. Prasad, Jacquet functors and unrefined minimal K-types, Comment. Math. Helvetici 71 (1996), pp. 98-121. Unrefined minimal K-types forp-adic groups, Inv. Math. 116 (1994), 28. -, pp. 393-408. 29. R. Ranga-Rao, Orbital integrals in reductive groups, Ann. of Math. (2) 96 (1972), pp. 505-510. 30. F. Rodier, Int6grabilit6 locale des caractbres du groupe GL(n,k) ozi k est un corps local de caracte'ristique positive, Duke Math. 3. 52 (1985), no. 3, pp. 771-792. 31. -, M o d d e de Whittaker et caractkres de repre'sentations. Non-

94

32. 33. 34.

35. 36. 37.

S. DeBacker

commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1974), Lecture Notes in Math., Vol. 466, Springer, Berlin, 1975. pp. 151-171. J. D. Rogawski, An application of the building t o orbital integrals, Compositio Math. 42 (1980/81), no. 3, pp. 417-423. G. Savin, Lectures o n representations of p-adic groups, this volume. P. Schneider and U. Stuhler, Representation theory and sheaves o n the B m h a t - T i t s building, Inst. Hautes tudes Sci. Publ. Math. No. 85 (1997), pp. 97-191. J.-L. Waldspurger, Homoge'nEitE de certaines distributions sur les groupes p-adiques, Inst. Hautes Etudes Sci. Publ. Math. No. 81 (1995), pp. 25-72. -, Quelques resultats de finitude concernant les distributions invariantes sur les algZbres de Lie p-adiques, preprint, 1993. -, Une formule des traces locale pour les algZbres de Lie p-adiques, J. Reine Angew. Math. 465 (1995), pp. 41-99.

On Classification of Some Classes of Irreducible Representations of Classical Groups

Marko TadiL Department of Mathematics University of Zagreb BijeniEka 90, 10000 Zagreb, Croatia E-mail: [email protected] r

Representation theory of reductive p-adic groups, besides its importance for harmonic analyses, is very important for Langlands program. It also gives us often better understanding of representation theory of reductive Lie groups. In these notes we review some parts of representation theory of reductive groups over local fields, in particular over p a d i c fields. We discuss classifications of some families of irreducible representations, which are important for harmonic analysis on these groups. We start with general principles of harmonic analyses on groups (which are given in terms of unitary representations). Then we explain algebraization of the problem of classification of irreducible unitary representations. Langlands classification reduces classification of irreducible representations to tempered representations, which come from square integrable representations by parabolic induction. We present existing classifications of such classes of representations for general linear and classical groups, and discuss connection of this with Langlands correspondences. Special attention is devoted t o classification (modulo cuspidal data) of irreducible square integrable representations of classical padic groups, which implies parameterization of non-unitary duals. This opens possibility t o work on the (very hard) problem of classification of irreducible unitary representation of these group.

Contents

1 Harmonic analysis and unitary duals 2 Non-discrete locally compact fields, classical groups, reductive groups

97 99

3 KO-finite vectors

103

4 Smooth representations

104 95

96

M. TadiC

5 Parabolically induced representations 6 Jacquet modules 7 Filtrations of Jacquet modules 8 Square integrable and tempered representations 9 Langlands classification 10 Geometric lemma and algebraic structures 11 Square integrable representations of padic general linear groups 12 Two simple examples of square integrable representations of classical padic groups 13 Invariants of square integrable representations of classical padic groups 14 Reduction to cuspidal lines 15 Parameters of D ( p ; a) 16 Integral case 17 Non-integral case 18 Local Langlands correspondences 19 Non-unitary duals of classical padic groups 20 Unitary duals of general linear groups over local fields 21 On the unitariaability problem for classical padic groups References

105 109 111 112 114 120 123 125 127 135 137 138 141 143

145 147 154 158

Introduction In these notes of the lectures given during the special period on representation theory of Lie groups in IMS, NUS, Singapore, we shall discuss the problem of classification of some important series of irreducible representations of general linear and classical groups, having in mind unitary representations. We shall discuss more padic groups, but a part of notes deals also with real groups. One of the main goals of the notes is to give an introduction to the classification modulo cuspidal data, of irreducible square integrable representations of classical padic groups. After that we shall describe unitary duals of general linear groups over local fields, and describe the proof of the classification theorem in the case of complex general linear groups. We shall finish the notes with a series of questions regarding unitary representations of classical padic groups. In these notes, the fields of real and complex numbers are denoted by R and C, and the ring of rational integers is denoted by iz (as usual). Further

z+= {k E Z ; k 2 0) N = {t E z; t 2 1).

Some Classes of Irreducible Representations

97

We are thankful to the organizers of the special period for providing a very stimulating atmosphere in which we had opportunity to present the lectures.

1. Harmonic analysis and unitary duals 1.1. One can interpret classical harmonic analysis in terms of unitary representations of Rn and (R/Z)n. This point of view opens a possibility of generalizing classical harmonic analysis, and building such a type of theory for a general locally compact group G (in general, neither compact, nor commutative). We shall briefly describe the main problems of harmonic analysis on such a group G. First, we shall introduce a few notions which we shall need for this description. 1.2. A representation (r,V) (or simply r or V) of a group G is a group homomorphism r from the group G to the group of all invertible linear operators on a complex vector space V (there is no requirement on continuity in this definition). A representation r on a non-zero vector space V is called irreducible (or algebraically irreducible) if (0) and V are the only vector subspaces of V which are invariant for all r ( g ) , g E G. A representation (r,H ) is called unitary if H is a Hilbert space and: (1) the mapping (g,v)H r ( g ) u ,

GxH

-+

H

is continuous; (2) each operator .rr(g),g E G is unitary.

If we omit the second requirement in the above definition, then the representation defined in this way will be called continuous (one can consider much more general continuous representations, but we shall not need them in these notes). A unitary (or only continuous) representation ( r , G ) is called irreducible (or topologically irreducible) if (0) and H are the only closed subspaces of H which are invariant for all r ( g ) , g E G. 1.3. Now we can describe the main goals of harmonic analysis on a locally compact group G (which satisfies some technical requirements, which we shall not discuss here, but which are satisfied for the groups that we shall consider in these notes, i.e. for general linear and classical groups over local fields).

M. TadiC

98

The first problem is to

(1) understand in a convenient way (possibly classify) the set of all the equivalence classes of irreducible unitary representations of G. This set is called the unitary dual of G, and it is denoted by G. The second problem is to

(2) interpret other important unitary representations of G in terms of G. Such important unitary representations are usually given on functional spaces. The most important examples of such representations include representations of G on spaces of square integrable functions (with respect to an invariant measure, assuming that it exists) on a space X where G acts transitively. Then X % H\G for some closed group H of G and G acts by right translations on the space L2(H\G) of the square integrable functions on H\G (in this case H\G carries an invariant measure for right translations of G). The first example of such representation would be when H is the trivial subgroup of G, i.e. the representation of G on the space L2(G)of the square integrable functions on G with respect to right invariant measure on G. This representation is very important. A significant portion of Harish-Chandra’s work is closely related to this representation in the case of semi simple real Lie groups (among others, he described the representation from G necessary for decomposing L2( G ) , and found Plancherel measure by which one decomposes L2(G)in terms of these irreducible representations). In this lectures we shall be more related to the problem (1) of harmonic analysis, although we shall be also related to the problem (2). Irreducible square integrable representations, which are one of the main topics of our notes, are part of both problems, (1) and (2). They are subrepresentations of L2(G)if the center of G is compact.

Remark: Some of the most important parts of the Langlands program can be considered as a kind of problems from harmonic analysis on groups in the above sense. For example. the origin of the Langlands program one can view as a kind of problem of harmonic analysis. The program started as a strategy for proving the Artin’s conjecture that Artin’s L-functions are

Some Classes of Irreducible Representations

99

entire. Roughly, Langlands proposed a strategy that irreducible representations of the absolute Galois group of a number field would parameterize irreducible subrepresentations of adelic general linear groups on the spaces of cuspidal automorphic forms (which are unitary representations on functional spaces), in a way that corresponding L-functions match (this can clearly be regarded as a kind of problem of type (2) of harmonic analysis on groups). Realization of this strategy would imply the Artin’s conjecture. The above philosophy has its local counterpart (with corresponding parameterizations). In the local case of the Langlands program, we are more related to the problem of type (1) of harmonic analysis on groups. One can extend the above considerations to other reductive groups, and one can consider also different fields. A question may be, can one do the above mentioned parameterizations in a naturally compatible way. Such a question is related to the functoriality problem. Above we gave only a very rough comments regarding the Langlands program. Details regarding this program can be found in [19] or [36].

2. Non-discrete locally compact fields, classical groups, reductive groups

2.1. Let F be a non-discrete locally compact field. Such field will be called local field. If F is connected, then the field is called archimedean. Otherwise, it is called non-archimedean. Non-archimedean fields are totally disconnected. They contain a basis of neighborhoods of 0 consisting of open compact subrings. If a local field is archimedean, then it is isomorphic to R or C. Let p be a prime integer. Ideals p k Z , k E Z+, define a basis of neighborhoods of 0 in Z. The completion of 2 with respect to this topology (more precisely, uniform structure defined by this topology) is denoted by Z,. The field of fractions of Z,is denoted by Q p . This is the field of padic numbers. We can introduce Q p also as a completion of Q with respect to the absolute value

Any finite extension F of Q, is in a natural way a topological space, and with this topology, F is a local non-archimedean field of characteristic 0. One gets each non-archimedean field of characteristic 0 in this way. Let F,[[X]] be the ring of all formal power series CEO u,Xn over a finite field F, (with q elements), and let F,((X)) be the field of all Laurent

M. TadiC

100

power series CE-, a,Xn over IF, for which there exists no E Z such that a, = 0 for all n 5 no. Then the powers of the ideal X F , [ [ X ] ]in IF,[[X]] define a basis of neighborhoods of 0 in P,((X)), and therefore a topology on F,((X)).In this way F,( ( X ) )becomes a local non-archimedean field of positive characteristic. One gets each local non-archimedean field of positive characteristic in this way. Topology on a local field can be always defined using an absolute value. Moreover, there exists a unique absolute value 1 JFon a local non-discrete field F such that

for any a E F X and for any continuous, compactly supported function f on G, where dx denotes an invariant (for translations) measure on F . We shall always fix such an absolute value on F . Let us note that for @, this absolute value is a square of the standard one.

2.2. We shall recall now of a definition of the classical groups. A classical group over a local field F is the group of isomorphisms of either symplectic, or orthogonal or unitary space over F (of finite dimension). For the study of representations of classical groups, it is important to understand the representation theory of general linear groups GL(n,F)’s,i.e. of the groups of all isomorphisms of finite dimensional vector spaces over F (soon it will become clear why this is important). In the study of classical groups, we shall use very convenient language of structure theory of reductive groups without going into this theory. In general, we shall try to keep the technicalities as low as possible.

For the simplicity, we shall consider in these lectures two series of classical groups (these series consist of split and connected groups). The reason for this is only t o simplify the notation. 2.3. The first is the series of symplectic groups: Denote 0 0 . . . 0100 ... 10

J, =

: 01 .. . 00

1 0 . . . 00-

E

GL(n,F).

Some Classes of Irreducible Representations

Then the symplectic group is

S

E

GL(2n, F ) ; tS

[

-Jn O

101

"1 s = [ "I). -Jn O 0

Here t S denotes the transposed matrix of S. 2.4. The second series consists of split odd-orthogonal groups:

Denote by Inthe identity matrix in GL(n, F ) . Let SO(2n

+ 1,F )

{ S E SL(2n + 1,F ) ; ' S S = 1zn+l}.

Here S L ( n , F ) = {g E GL(n,F);det(g) = 1) and ' S denotes the transposed matrix of S with respect to the other diagonal. We could work also with O(2n 1,F ) instead of SO(2n 1,F ) .

+

+

In these notes, we shall always deal with matrix forms of classical groups. 2.5. The above groups are connected, split, semi simple algebraic groups over F . They are topological groups in a natural way. In the case when F is a non-archimedean field, these groups are totally disconnected. Then one can write a basis of neighborhoods of identity which consists of open (and closed) compact subgroups. If F is archimedean, then symplectic and odd-orthogonal groups are connected semi simple Lie groups. If G is GL(n,F ) , or Sp(2n,F ) or S0(2n+1, F ) , then we shall denote by Po the subgroup of all the upper triangular matrices in G. Then Po is called standard minimal parabolic subgroup of G. Any subgroup of G containing Pa, is called standard parabolic subgroup of G. There are finitely many of them, and we shall describe them precisely. Any subgroup conjugate to a standard parabolic subgroup is called parabolic subgroup. Let Q

= (n1,...,n k )

be an ordered partitions of n into positive integers. Consider matrices of GL(n,F ) as block matrices with blocks of sizes ni x nj. Let P,"" (resp. M:"), be the upper block-triangular matrices (resp. block-diagonal matrices) in GL(n,F ) . Denote by N z L the (block) matrices in P,"" which have identity matrices on the block-diagonal. Now

is one-to-one mapping of the set of all ordered partitions of n onto the set of all standard parabolic subgroups of GL(n,F ) . We have Levi decomposition

M. TadiC

102

P,"" = M,""N,"". This means that P,"" is a semi direct product of M,"" and N,"", where N,"" is a normal subgroup in P z L , i.e.

P,""

= M,GL K

N,GL.

This decomposition of P,"" is called the standard Levi decomposition of P:", where M,GL is called the standard Levi factor of P,"" and N:" is called the unipotent radical of P,"".

+

2.6. Standard parabolic subgroups of Sp(2n,F ) and SO(2n 1,F ) are parameterized by ordered partitions a: = (711,. . . ,nk) of integers m, where 0 5 m 5 n. If we consider the group G = Sp(2n,F ) set a' = (721, ...,n k , 2n - 2m, n k , ..., n l ) , while in the case of the group G = SO(2n

a' = (n1,..., n k , 2n

+ 1,F ) set

+ 1 - 2m,nk,...,

n1).

Then a: H Pa = P$" n G

gives a parameterization of standard parabolic subgroups in G. In similar way as in the case of general linear groups, one defines standard Levi decompositions in this case, using standard Levi decompositions of P2". In the sequel, we shall denote by G one of the groups GL(n,F ) , Sp(2n,F ) or SO(2n 1,F ) . We shall denote by A0 the subgroup of all diagonal matrices in G. This is a maximal split (over F ) torus in G (it is also a maximal torus in G).

+

2.7. We shall denote by KOa maximal compact subgroup of G. If F is non-archimedean, let OF =

PF

{x E F ; l x l F 5 I},

= {x E

F ; 1 x 1 ~< 1).

In the non-archimedean case one can take

KO= GL(n,O F )n G. For F = W (resp. F = C ) ,one can define KOin a similar way as above, taking the group O ( n ) (resp. U ( n ) ) of orthogonal matrices in G L ( n , R ) (resp. unitary matrices in GL(n,C)) instead of GL(n,O F ) . It is important to note that KO is an open subgroup if F is nonarchimedean field, which is not the case (in general) in the archimedean case.

S o m e Classes of Irreducible Representations

103

3. KO-finite vectors

3.1. Let ( T ,H) E G. For T E ko denote by m(r : T ) the multiplicity of r in T . The basic property of KO is that it is a large subgroup of G (this was proved by Harish-Chandra in the archimedean case, and by J. Bernstein in the non-archimedean case). It means the that the function T H m(T

:T)

is a bounded function on G , for any fixed I- E KO.This fact has a number of important consequences. Among others, it enables algebraization of the problem of determining of the unitary dual G of G.

3.2. Let

( T , H)

E G. Denote by

H" the set of all vectors w E H such that

dim@spanc T(KO)V < 00. Then H" is a dense KO-invariant vector subspace of H". Suppose that F is non-archimedean. Since for each g E G the group sKog-l n KO has finite index in KO,H" is G-invariant. It follows easily that the following property holds for H": For any u E H" there exists an open subgroup K of G such that T ( k ) w = w for any k E K . This follows from the fact that each continuous representation of KO is trivial on an open subgroup (since open subgroups in KO form a basis of neighborhoods of identity, and GL(n,cC) does not contain small (nontrivial) subgroups).

3.3. In the archimedean case, gKog-l n KO is not (in general) of finite index in K O .Because of this, H" is not (in general) G-invariant. But then one can prove that it is invariant for the natural action of Lie algebra g of G. Moreover, the action of g and KO satisfies a natural condition. Such a structure is called (8, KO)-module. 3.4. At this point usually archimedean and non-archimedean theory continue to develop separately. In the sequel, we shall more discuss the nonarchimedean theory, but a number of topics hold for both theories (these will be examples of Lefschetz principle). We shall usually comment the results which hold in both theories.

M. TadiC

104

4. Smooth representations

We shall assume in the sequel that F is a local non-archimedean field (if it is not otherwise specified).

4.1.A representation ing condition:

(7r, V

)of G is called smooth if it satisfies the follow-

For any u E V there exists an open subgroup K of G such that 7r(k)v = u for any k E K . Denote by

G the set of all equivalence classes of irreducible smooth representations of G. This set is called non-unitary dual of G, or admissible dual of G. 4.2. The mapping (7r,H

)w

H”);

(Too,

G4G

is injective (here 7r” denotes the restriction of 7r to H”). Therefore, the unitary dual can be identified with a subset of G. We shall assume this identification in further. It can be shown that in this way the unitary dual is identified with the subset of all ( T , V )E G such that on V there exists an inner product which is invariant for the action of G. The problem of classification of G has appeared much more manageable then the problem of classification of G.

4.3.The problem of classification of unitary dual of G now breaks into two parts: problem of classification of G , which is called the problem of nonunitary dual; problem of determining the subset G of G (in other words, the problem of identifying unitarizable classes in G), which is called the unitarizability problem. We shall discuss both problems in these lectures.

Some Classes of Irreducible Representations

105

4.4. Regarding the problem of non-unitary duals, let us note that there is Langlands classification of non-unitary duals, which reduces the problem of classification of non-unitary duals to the problem of classification of a special kind of irreducible representations of Levi subgroups, namely to the problem of classification of tempered representations, which will be introduced later. In the moment, let us just note that these tempered representations are unitarizable. The problem of classifying of irreducible tempered representations is very far from being easy. Before we describe Langlands classification, we shall recall of a more simple (and less precise) reduction of the non-unitary duals. We shall need to have a tool by which we shall be able to produce new representations. This tool is provided by parabolic induction, a construction which generalizes in a natural way induction studied already by Schur and Frobenius in the case of finite groups. Further, we shall need a tool for analyzing induced representations. Jacquet modules will be of great help for this.

5 . Parabolically induced representations

5.1. Smooth representations of G and intertwinings form an Abelian category, which will be denoted by Alg(G). Let ( T , V ) be a representation of G. Denote

V” = {Y E V ;there exists an open subgroup K such that 7r(k)v = Y, k E K}. The space V” is a G-subrepresentation of G, and it is called the smooth part of V . For a compact subgroup K of G let

V K = (Y E V ;~ ( l c ) w= Y for any k E K } . This vector space is called the space of K-invariants of V . Further, (T,

V )H V K

is an exact functor on the category Alg(G). 5.2. If ( T , V )is a smooth representation of G, then there is a natural representation d on the space of all linear forms V’ on V defined by (r’(g)v’)(v)= d(7r(gP1)v). The smooth part of this representation is called the contragredient of (7r, V). This representation is denoted by

(*,

V)

M. TadiC

106

(recall (ii(g)V)(w) = V(n(g-l)w)). Then the mapping

(w,V)

H=: q w ) ,

is called canonical bilinear form. This form is G-invariant. A function g ++

is called a matrix coefficient of n. Further (r,V )

(El V )

extends to a contravariant functor in a natural way. This functor is exact. For a representation ( n I V ) of G, the representation on the complex conjugate vector space 8 of V will be denoted by (ii 8). ,

A smooth representation

( T , V)

will be called Hermitian if -

( T , V)

= (%,V ) .

5.3. A smooth representation (r,V ) of G is called admissible if dim@VK

< 03

for any open compact subgroup K of G. For a smooth representation (n, V ) of G we have always a natural in-

tertwining of V into V . If the representation is admissible] then this is an isomorphism. The converse also holds, i.e. if V and V are isomorphic, then (T] V ) is admissible. It is easy to show that each unitarizable admissible representation of G is Hermitian. 5.4. We shall fix the group G of rational points of a connected reductive

group defined over a local non-archimedean field F . One of the main examples for us are general linear groups and classical groups. We shall fm a maximal split torus AP)in G and a minimal parabolic subgroup P0 of G which contains A @ .Standard parabolic subgroups of G are subgroups of G which contain P0. For a standard parabolic subgroup P of G, a Levi decomposition of P into semi direct product of a reductive subgroup M and a

Some

Classes of Irreducible Representations

107

normal unipotent subgroup N will be called standard if A0 C M. For standard parabolic subgroups we shall always assume that Levi decompositions are standard. Parabolic subgroups and their Levi decompositions one gets from standard parabolic subgroups and their standard decompositions by conjugation with elements of G. We shall fix a maximal compact subgroup KO of G for which Iwasawa decomposition

G = Pa KO holds (such a maximal compact subgroup always exists). 5.5. Let for a moment 0 be a locally compact group. Then there always exists a positive measure which is invariant for right translations. Such a measure will be denoted by d g . Right invariance means that

for any continuous compactly supported function f on Q and any x E 0. This measure is unique up to a multiplication by a constant, and it is called a right Haar measure on B. A right Haar measure does not need to be left invariant (if it is, then the group is called unimodular; reductive groups are unimodular), but there exists a character A, of 6 (which is called the modular function or modular character of G), such that holds

holds for any f and x as above. 5.6. Let us return back to the case of a connected reductive group G over a non-archimedean field F . Fix a parabolic subgroup P of G with a Levi decomposition P = M N (more preciseIy, the group of rational points). Let (a,U ) be a smooth representation of M . Denote by Ir&(a) the space of all functions f G t U which satisfy f(72.V)

= AP(m)1’24m)f ( 9 )

for each m E M , n E N , g E G. Then G acts on In&(a) by right translations (R,f)(rc)= f ( x g ) , x,g E G. The smooth part of the representation Indg(a) is denoted by Indg (D )

M. TadzC

108

and called a parabolically induced representation of G from P by a. Parabolic induction becomes in an obvious way a functor from Alg(M ) into Alg(G). The functor of parabolic induction is exact. 5.7. If a is unitarizable, then Indg(a) is also unitarizable. The inner product

( , ) on Indg(a) is given by

Further, Indg (a)-Z Indg (8) The canonical bilinear form is given by the same formula as the above inner product:

5.8. Suppose that P = M N is a standard parabolic subgroup of G and P' =

M'N' another standard parabolic subgroup of G (the above decompositions are considered to be standard Levi decompositions). Let

P 2 PI. Then Ind$(a) E Indg,(Ind&L, (a)). This fact is called induction by stages (which gives the same result as the original, direct parabolic induction). It is easy t o prove it (one writes an explicit isomorphism). 5.9. Iwasawa decomposition implies that Indg(a) is an admissible representation if a is admissible. It is less obvious to prove that if (T is a representation of finite length, then Indg(a) is also a representation of finite length (of G).

5.10. Suppose that we have a parabolic subgroup P with Levi decompositions P = M N and P = MINI, which do not need to be the standard one (in the case that really interests us, a t least one Levi decomposition should not be the standard one). Suppose

M=M'.

Some Classes of Irreducible Representations

Let

CT

109

be a smooth finite length representation of M. Then Indz(a) and Indg, (a) have the same Jordan-Holder series.

This is an important fact, called induction from associate parabolic

subgroups. It is not quite simple to prove it. It relies on the theory of characters. Since we shall not introduce characters in these notes, we shall not comment the proof here. 6. Jacquet modules

In this section we shall introduce a functor which is left adjoint to the functor of parabolic induction.

6.1. Suppose that ( T , V ) is a smooth representation of G and let P = M N be a parabolic subgroup of G (actually, it is enough to assume that ( T , V ) is a smooth representation of P only). Let

V ( N )= spanc { ~ ( n )-ww;n E N , 21 E V } . Since N is normal in P, V ( N )is P-invariant. In particular, it is M-invariant. We have a natural quotient action of M on

r$(V) = V / V ( N ) . We shall consider the action of M on r $ ( V ) which is the quotient action of the action of M (through T ) on V , twisted with A;'". This action will be denoted by

rE ( T ) . The representation (.$(T)l.$(V))

is called the Jacquet module of ( T , V )with respect to P = M N . One defines in a natural way Jacquet functor from A1g(G) into A l g ( M ) . Jacquet functor is exact.

6.2. If P = M N and P I = MINI are standard parabolic subgroup, with standard Levi decompositions, such that

P2

PI,

then

r & ( r g ' ( T ) )E r g ( T ) .

110

M. TadiC

This fact will be called transitivity of Jacquet modules.

6.3. The fact that Jacquet functor is left adjoint to the functor of parabolic induction means that we have a natural isomorphism HomG ( T , Indg(a)) 2 HomM ( r $ ( r ) ,a).

The above isomorphism is called Frobenius reciprocity. One constructs this isomorphism using evaluation of f E Homc (T,IndZ(a)) at 1. 6.4. A smooth irreducible representation ( T , V )of G is called cuspidal (or supercuspidal) if all the Jacquet modules for proper parabolic subgroups are trivial modules. It is natural to distinguish these representations, as will become clear very soon. Actually, in the definition of cuspidal representations, it is enough to require triviality Jacquet modules only of proper standard parabolic subgroups. Cuspidal representations are very special representations, as we shall see later. 6.5. One easily sees that if ( T , V ) is a finitely generated smooth representation of G, then r $ ( T ) is finitely generated representation of M . From this

follows that it has an irreducible quotient. Let (n, V) be an irreducible smooth representation of G. Among all the parabolic subgroups P = M N which satisfy T $ ( T ) # {0}, chose a minimal one. Then by the above observation, r Z ( T ) has an irreducible quotient, say a. Minimality of P and transitivity of Jacquet modules (together with exactness) imply that a is cuspidal. Now Frobenius reciprocity implies that n embeds into Indg(a).In this way we have obtained a simple but important

Theorem: An irreducible smooth representation T of G i s cuspidal, o r there exists a proper parabolic subgroup P = M N of G and a n irreducible cuspidal representation a of M such that n is isomorphic t o a subrepresentation of In@ (a). In this way the problem of classification of non-unitary dual C? breaks into two problems. One problem is to classify cuspidal representations of Levi factors, and the other one is t o classify irreducible subrepresentations of representations parabolically induced by cuspidal representations. From the above theorem it is not clear at all how to classify irreducible subrepresentations of representations parabolically induced by cuspidal representations. Langlands classification will provide a strategy for it. There is

S o m e Classes of Irreducible Representations

111

another way to describe irreducible cuspidal representations. They can be characterized as those representations which never show up as subquotients of representations parabolically induced from proper parabolic subgroups.

6.6. Remarks: (i) Let G be GL(2,F ) and 7~ be an irreducible representation of G which is not cuspidal. Then Schur lemma and the above theorem imply that 7r is isomorphic to a subrepresentation of Ind$&), for a character x of the Levi factor M0 of 4 (note that M0 is commutative). (ii) There is an archimedean version of the above theorem (see [17]).

6.7.One important property of cuspidal representations is that their matrix coefficients are functions which are compactly supported modulo center. W. Casselman has shown that this property characterizes cuspidal representations. H. Jacquet has proved that each cuspidal representation is admissible (a nice argument for this can be found in [50]).Now 5.9 and Theorem 6.5 imply that each irreducible smooth representation is admissible. This is the reason that G is also called admissible dual. 6.8. Let us note that Jacquet functor carries admissible representations to admissible ones (this is not quite easy to prove). Further, it carries smooth representations of finite length of G to smooth representations of finite length of M . For proofs one can consult [IS]. 7. Filtrations of Jacquet modules

Jacquet modules are very important in analysis of admissible representations, in particular of the induced ones. In general, it is not easy to determine their structure. There is geometric lemma, obtained independently by J. Bernstein and A. V. Zelevinsky, and by W. Casselman, which describes filtrations of Jacquet modules of Ind$(a) in terms of representations parabolically induced by Jacquet modules of a. We shall illustrate this lemma on the example of G = GL(2,F ) . Later we shall describe how one can realize the geometric lemma as a part of an algebraic structure in the case of general linear and classical groups. In this section we assume that

G = GL(2,F ) . 7.1. For G = GL(2,F ) we have P0 = PGL . Irreducible smooth representa(191) tions of M g f ; , are one dimensional, i.e. characters. Since M g f ; , is naturally

M. TadiC

112

isomorphic to F X x can be written as

F X, each irreducible smooth representations of x 1 8 xz

where x1 and x2 are characters of We shall consider

1

F X.

IndgO(X18 x2). Denote

-

It is not hard to show that the following obvious sequence 0 -+

-

{f E IndgO(xl8 XZ);supp(f) C P0wop0) restriction

IndZO(xl8 XZ)

(flP0; f E IndZO(xlc3 x2)) 0. (1) is exact. Further, considering the mapping f H f ( l ) ,one gets easily that +

G0((fJP0;f E I n d g (Xl 8 x2))) Ei x1 8 xz. It requires a little bit more efforts to show that G T M @ ( {E ~

IndgO(xi8 xz);supp(f) C_ p0wop0))

x2

8 XI.

7.2. Applying Jacquet functor to the exact sequence (1) (recall that the Jacquet functor is exact), we get the following exact sequence 0 4 x2 8 XI

---$

TEO(IndZO(xl 8 XZ))

-+

XI 8 xz

-+

0.

As a consequence of this exact sequence, we can conclude that Indgo(XI @ X Z ) has at most length 2. 8. Square integrable and tempered representations

8.1. Let ( T , V ) be an irreducible smooth representation of G. Then Schur lemma implies that the center Z(G) of G acts by scalars. Corresponding character will be denoted by WT

and called the central character of

T.

113

Some Classes of Irreducible Representations

A smooth representation does not need to be irreducible, but the center can act by scalars. Then we shall say that the representation has a central character.

8.2. An admissible representation (7r,V)of G will be called square integrable (more precisely, square integrable modulo center) if it has central character, if the central character is unitary, and if absolute values of all the matrix coefficients g

H

I < 7r(g)v,V > 1,

21

E

v,v E v,

are square integrable functions modulo center (i.e. square integrable functions on G/Z(G)). In this notes we shall consider only irreducible square integrable represent ations. An admissible representation (7r,V) of G will be called essentially square integrable (or essentially square integrable modulo center) if there exists a character x of G such that x7r is square integrable. 8.3. Each (irreducible) square integrable representation ( 7 r , V ) of G is unitarizable. To see this, take Go E different from 0. Now for u,w E V set

v

One sees directly that this is a G-invariant inner product on V (if irreducible, one proceeds similarly; see [73]).

7r

is not

8.4. Irreducible square integrable representations are very important. First,

they are (very distinguished) elements of the unitary dual. Then, via Langlands classification, they are crucial in the parameterization of non-unitary duals. Further, using matrix coefficients one gets that they are (irreducible) subrepresentations of L2(G) if G has compact center (in the non-compact case, we have similar situation when one fixes central character). Therefore, they are very important for understanding decomposition of L2(G). 8.5. An irreducible smooth (which implies admissible) representation ( 7 r , V ) of G is called tempered if there exists a parabolic subgroup P = M N of G (not necessarily proper) and an irreducible square integrable representation 6 of M such that 7r is isomorphic to a subrepresentation of

Indg (6).

M. Tadic‘

114

An irreducible smooth representation ( T , V )of G is called essentially tempered if there exists a character x of G such that XT is tempered. One usually defines tempered representations without irreducibility requirement, but since we shall work only with irreducible tempered representations in these notes, the above definition is not restrictive for us. 8.6. We shall now introduce notation for general linear groups. The character

will be denoted by

v. If r is an (irreducible) essentially tempered representation of GL(n,F ) , then one easily sees there exists a unique e(T) E

R

and a unique tempered representation rU such that = ye(T)TU.

This requirement uniquely defines e(r).

8.7. There are a very useful criteria of Casselman for checking if an irreducible admissible representation is square irreducible or tempered. We shall explain this criterion on the simplest case, on G = GL(2,F ) . Let ( T , V ) be an irreducible essentially square integrable (resp. essentially tempered) representation of GL(2,F ) . Then for any irreducible subquotient x = x1 @ x 2 of rg:(2’F)(n) ( X I and x2 are characters of F X )we have

4 x 1 )> 4x2)

(resp.

4x1) 2 4x2)).

Moreover, the converse also holds for an irreducible admissible representation ( T , V ) . 9. Langlands classification

9.1. Langlands classification parameterizes representations of 6, by (irreducible) essentially tempered representations of Levi factors of standard

S o m e Classes of Irreducible Representations

115

parabolic subgroups. These essentially tempered representations need to satisfy certain positiveness condition (which will be discussed later). Langlands classification claims the following: For an irreducible essentially tempered representation T of Levi factor A4 of standard parabolic subgroup P of GI which satisfy the above mentioned positiveness condition, the representation Ind$(T) has a unique irreducible quotient. This irreducible quotient is called the Langlands quotient (Ind$ ( T ) is called a standard module of G). Each 7r E G is isomorphic to some Langlands quotient, and moreover 7r determines uniquely the standard parabolic subgroup P and essentially tempered representation T . We shall now explain the positiveness condition for the groups that we consider in these notes. We shall start with general linear groups.

9.2. Each Levi factor of a parabolic subgroup of a general linear group, is a direct product of general linear groups. Because of this, each irreducible essentially tempered representation of a Levi factor of a general linear group is a tensor product of such representations of general linear groups. Therefore, the essentially tempered representations of a Levi factors of a general linear groups are of the form 71

'8 72 '8 . ' ' '8 Tk,

where 71, 7 2 , . . . ,'rk are essentially tempered representations of general ]inear groups. The positiveness condition here is simply e(T1)

> e(7-2) > . . . > e ( n ) .

9.3. We shall see how one gets the Langlands parameterization in the case of the simplest possible example, in the case of G = GL(2,F ) . Let T E G. If 7r is essentially tempered, then it is its own Langlands parameter. Suppose therefore that 7r is not essentially tempered. Then, in particular, it is not cuspidal. Therefore

ii

is a subquotient of

Indga (XI 8 x 2 )

by Theorem 6.5, for some characters x 1 and x 2 of F X (see also Remarks 6.6, (i)). Since 7r is not essentially tempered, ii is also not essentially tempered. Now by 8.7 4x1)

Therefore by 7.2

# 4x2).

M. Tadic'

116

Without lost of generality we can assume

4x1) > 4 x 2 1 7 since IndgO(xl @I x2) and IndgO(x2€3 XI) have the same Jordan-Holder series (one sees this using induction from associate parabolic subgroups; see 5.10). Since i f is not essentially square integrable (recall that 7r it is not essentially tempered), from criterion for essentially square integrability follows that x 2 €3 x1 must be a subquotient of rga(?). Now from (2) we see that there exits a non-trivial homomorphism

rgO(5)

-+

-

x2

63 x1.

Frobenius reciprocity implies 77

Indg(X2 63 X l ) .

Passing to contragredients we get an epimorphism IndZO(x;' 8 x;')

---t

71

Note that e ( x z l ) = -e(xz)

> -e(xl)

= ~(xT').

This implies that we have shown the existence of Langlands parameters for irreducible representations of GL(2,F ) . Their uniqueness follow from the filtration of Jacquet modules (see 7.2). Thus, we have "proved" Langlands classification for GL(2,F ) . This case is too simple to illustrate the proof of the Langlands classification in general, but one can get a t least some idea from this simplest case how proof goes in general. In any case, we see the importance of the Geometric lemma. 9.4. Since we have defined tempered representations by square integrable representations, it is natural to try to express Langlands classification in terms of square integrable representations, if this is possible. In the study the representation theory of general linear groups, it is convenient to use notation that was used by Bernstein and Zelevinsky in their work on the representation theory of general linear groups. We shall now recall of ( a very small part of) it. For smooth representations 7r1 and 7rz of G L ( n l ,F ) and GL(n2,F ) denote 7r1

x

7rz

GL(n1f n 2 ,PI = IndpyL (Tl€3.2). ni.nz)

117

S o m e Classes of Irreducible Representations

Then 7r1 x

(“2

x

“3)

s (“1

x

7T2)

x

(3)

7T3.

This follows from induction by stages. Further, for smooth representation and “2 of finite length,

“1

“1

x

“2

and

7r2

x

7 ~ 1have

the same Jordan-Holder series.

(4)

This follows from induction from associate parabolic subgroups.

Remark: In the case of an archimedean field F , using parabolic induction we define multiplication x between (g, KO)-modules of general linear groups (over F ) in the same way as above. Then (3) and (4) hold also in this case. 9.5. A principal result regarding tempered induction for general linear groups is that this induction is irreducible (this fact holds for all the local fields). This fact has been proved independently at several places, but it seems that the first proof in this setting belongs to H. Jacquet, who proved it is for all the local fields (see [29]). This principal result claims the following:

If “1, “2, . . . ,“k are (unitarizable) irreducible square integrable representations of general linear groups, then irreducible.

“1

x

“2

x

- - .x

%

is

Either from general facts regarding tempered representations, or from A. V. Zelevinsky paper [76], follows that the tempered representation “1 x 7r2 X . . . X “rk determines irreducible square integrable representations “1 ,“2, . . . ,nk up to a permutation. 9.6. Using 9.4 and the fact that for a character

x of F X we have

[(xo d e t ) ~x~ [(x ) o det).irz] ”= (xo det)(7rl x “2) (which one proves directly), we can reformulate the Langlands classification for general linear groups in the following way. Denote by D the set of all the irreducible essentially square integrable representations of GL(n,F)’s for all n 2 1. Let M ( D ) be the set of all finite multisets in D. These are functions from D into Z+with finite supports. We shall write them similar as sets, but repetitions of elements will be allowed. We shall write them as (61,62,.. . ,6k),

where 6i E D.

M. TadiC

118

Take any d = (1,.. . , k} such that

(S1,62,,

. .,Sk)

E

M ( D ) . Take a permutation p of

> e(dp(2))2 . . . 2 4 b p ( k ) ) .

e ( b p ( 1 ) )-

Now the representation Sp(1)

x

bp(2)

x

.. . x

bp(k)r

which will be denoted by

has a unique irreducible quotient (the representation X ( d ) is determined by d E M ( d ) up t o an isomorphism.). This quotient will be denoted by

L(d). In this way one gets bijection between M ( D ) and the set of all the irreducible smooth representations of all general linear groups over F . This is just a reformulation of the Langlands classification for general linear groups. The Langlands classification has a number of natural properties. Let us mention three:

L ( b l , b 2 , .. . , b k ) -

L(&,&, . . . ,&),

x L ( 6 1 , 6 2 , .. . , b k ) 2 L(X61,x s 2 , . . ., X 6 k )

(5)

(7)

( x is a multiplicative character of the field, and further, for a representation of G L ( n ,F ) , X T denotes the representation ( x o det) T ) .

7r

Remark: The Langlands classification for general linear groups holds also if the field is archimedean F . In this case the non-unitary dual GL(n,F ) of G L ( n , F )is the set of all the equivalence classes of irreducible (g,Ko)modules of GL(n,F ) . The irreducible representations (i.e. non-unitary duals) are classified by M ( D ) , where D is the set of all the equivalence classes of irreducible essentially square integrable (8, Ko)-modules of all GL(n,F ) ' s , n 2 1 (if F = C, then D = (C")-, while for F = R we have

D

c ( R x ) - u GL(2,R)-).

9.7.Now we shall describe the Langlands classification for symplectic and odd-orthogonal groups.

S o m e Classes of I n e d u c i b l e R e p r e s e n t a t i o n s

119

It is convenient to introduce for classical groups the following notation, which simplifies notation when one works with parabolically induced representations. Let 7r be a smooth representation of GL(n,F ) and let u be a smooth representation of Sp(2m,F ) (resp. SO(2m 1,F)).Denote

+

7r

x u = Ind S p ( Z ( n + m ) , F ).( '(n)

@ ).

From induction by stages follows 7r1

x

(7r2

x

0)

E (7rl x 7r2) x

Further, for smooth representations 7r

x

(T

and ii x

(T

7r

and

(T.

of finite length,

have the same Jordan-Holder series.

(8)

This follows from induction from associate parabolic subgroups. 9.8. Regarding the Langlands classification for symplectic and oddorthogonal groups, one can first describe it in terms of essentially tempered representations, and after that pass to a description which include only essentially square integrable representations of general linear groups (and tempered representations of symplectic or odd-orthogonal groups), similarly as we did in the case of Langlands classification for general linear groups. Instead of this, we shall skip over the first description and go directly t o the second description. Set

D+ = (6 E D ; e ( 6 )> 0 ) . Denote by T the set of all equivalence classes of tempered representations of all Sp(2m,F ) (resp. SO(2m 1,F ) ) , for all m 2 0. Take

+

((61,62,...,6k),7)

E

M ( D + )x T .

After a renumeration, we can assume

61 2 62 2

" '

26k

Then the representation

61 x

62

x

...x

6k

x 7

has a unique irreducible quotient, which will be denoted by L(61762,. . . , 6 k ;7)

M. TadiC

120

Now the mapping

((61, Jz, . . ., Jk), 7 )

-

L(J1,Jz,

. . .,6kk;7 )

defines a bijection from the set M ( D + ) x T onto the set of all the equivalence classes of irreducible smooth representations of all Sp(2m,F ) (resp. SO(2m 1,F ) ) , m 2 0. This is the Langlands classification for (these) classical groups.

+

Remark: One can define >a also for the case of archimedean fields. The Langlands classification holds here in the same form. 10. Geometric lemma and algebraic structures Geometric lemma, which is a technical result describing filtrations of Jacquet modules of induced representations in terms of representations induced by Jacquet modules of inducing representations, is very important for number of purposes. For general linear and classical groups we can “incorporate” it into an algebraic structures on the representations of these groups.

10.1. Let for a moment G be a reductive group over a non-archimedean field

F . The Grothendieck group of the category of all smooth G-representations of finite length will be denoted by

R(G). This is just a free Z-module over basis G (it is isomorphic to the group of virtual characters of G). For a finite length representation T , let s . s . ( ~ )=

C m(7 :

T)T.

T€C

This is called semi simplification of T . We consider it as an element of R(G). There is a natural order on R ( G ) . We have

R(G1 x Gz)

R(G1)@ R(G2).

(9)

Further, r$ factors in a natural way to a homomorphism from R(G) into R ( M ) , which is denoted again by r z . This is a homomorphism of ordered groups (i.e. it respects also orders). We have analogous definition for the parabolic induction: Indg : R ( M ) -+ R ( G ) ,which is again a morphism of ordered groups.

S o m e Classes of Irreducible R e p r e s e n t a t i o n s

121

10.2. Set

R = & E Z + R ( G L ( ~F,) ) . Then one lifts x to a multiplication on R in a natural way. The mapping x : R x R + R factors in a natural way through a mapping

m : R @R Let

T

+ R.

E GL(n,F)-, We will consider

using the isomorphism (9). Define n

k=O

We can (and will) consider

m*(*)E R @R , since each Rk @ Rn-k

L)

R @ R. We can lift m* to an additive mapping rn*:R

.--)

R @R .

This mapping is called comultiplication. With the multiplication and comultiplication, R is a commutative Hopf algebra (over Z). This algebra was constructed by A. V. Zelevinsky. The most important part of this Hopf algebra structure is the formula m*(7r1x ~

2= ) m*(~1 x )m*(r2),

which explains how to get composition factors of Jacquet modules (for maximal parabolic subgroups) of induced representations, by induction from J acquet modules of inducing representations.

10.3. Suppose (only) in this paragraph that F is an archimedean field. One defines R ( G ) in the same way as in the non-archimedean case, considering the category of (8, K)-modules of finite length (G is a connected reductive (it is isomorphic to group over F ) . This is a free Zmodule over basis the group of virtual characters of G). Now for general linear groups over F one defines R in the same way as in the non-archimedean case. By the same formula as in the non-archimedean case, one defines multiplication x on R (using parabolic induction). In this way R becomes a commutative

M. Tadic‘

122

ring with identity. In the archimedean case, there is no comodule structure on R as in the non-archimedean case.

Remark The Kazhdan-Patterson lifting for G L ( n ,C ) has a very nice and natural description in terms of this algebra (see [64]). 10.4. Assume in this section that F is any local field (archimedean or nonarchimedean). For a = (61,. . . ,bk) E M ( D ) consider S . S ( X ( a ) ) = bl X

. .. X

bk E

R.

Then a simple fact regarding Langlands classification implies that s.s(A(a)),

aE

M(D),

form a Z-basis of R (this simple fact is usually expressed in the following form: characters of standard modules form a Z-basis of R ( G ) ) .In other words, for any local field F holds the following

Proposition: The ring R is a polynomial Z-algebra over

D.

10.5. We shall denote

R ( S ) = & E Z + R ( S PF( )~)~ , if we consider symplectic groups, and

+

R ( S ) = c B ~ , z + R ( S O ( ~1,~F ) ) if we consider odd-orthogonal groups. In this setting, one again lifts x in a natural way to a multiplication R x R ( S )+ R ( S ) .This multiplication factors through a mapping p : R 8 R ( S )4 R ( S ) .

For

7r

E

Sp(2n,F ) - (resp. 7r E SO(2n

+ 1,F)-) set

n

k=O

Consider s.s of

as an element using (9), and consider further

Some Classes of Irreducible Representations

123

Lift p* to an additive mapping p * : R(S) + R @R(S)

,

which will be called comultiplication on R(S). With the above multiplication and complication, R ( S )is a module and a comodule over R. It is not a Hopf module over R, but is also far from this structure as we shall explain now. Define

M * = ( m @ l ) o ( ~ @ m * ) o s o mR*+: R @ R , where 1 denotes the identity mapping, the contragredient mapping and s the transposition mapping C xi @ yi H C yi @ xi. Then N

p*(7r >a

0)= M*(7r) M

p*(o)

(R @ R(S) is a R @ R-module in an obvious way). We say that R(S) is an M*-Hopf module over R. This is again a (combinatorial) formula from which we can again in a simple way get compositions factors of Jacquet modules of parabolically induced representations for classical group. 11. Square integrable representations of p-adic general linear groups 11.1. Denote by

C the set of all equivalence classes of irreducible cuspidal representations of all G L ( n , F ) ,n 2 1. A segment in C is the set of form

A = [PI &I

= { P , v p , . . ., ~ I c P ) ,

where p E C, k E Z+. Denote the set of all such segments by S.

For a segment A = [p, vkp] = { p , v p , . . . , v k p } E S, the representation ukp x

2-1p

x

.. . x

up x p

contains a unique irreducible subrepresentation, which will be denoted by

fi(A).

M. T a d 2

124

Then 6(A) is essentially square integrable representation, and in this way one gets a bijection from S onto D (which is the set of all the irreducible essentially square integrable representations of general linear groups GL(n,F ) , n 2 1).This is one of the consequences of Bernstein and Zelevinsky theory, which is based on Gelfand-Kazhdan theory of derivatives. One can obtain these results also by different methods. In applications of square integrable representations of general linear groups, it is important to know what are Jacquet modules of these representations. This tells the following simple formula

c k

."I>>

m*(6([Pl

=

6([Vi+'P,

&I>

@

S ( [ P , YZPl)

(10)

i=-l

(see 1761).

11.2. As we have seen, each segment of S determines uniquely essentially square integrable representation. Let us explain how t o "read" corresponding segment from an essentially square integrable representation

6

= 6 ( A ) ,A E S .

For this, we shall introduce two natural invariants of 6. There exists exactly two (inequivalent) p 1 , p 2 E C such that p2 x 6 reduce. We can, after a possible renumeration, assume e(Pd

p1

x 6 and

Ie(p2).

Representations p1 and p2 will be called cuspidal reducibilities of 6 the lower one and p 2 the upper one). Then

(p1

Thus cuspidal reducibilities of 6 determine completely the segment in C corresponding to 6.

11.3. Let

61 E

D have cuspidal reducibilities

p1,

and p2. Take 6 2 =-6 ( A 2 )E

D. Then 61

x 62

reduces if and only if

(1) card((LJ1,P 2 ) n A2) = 1; ( 2 ) neither p i nor p 2 is a cuspidal reducibility of d(A2).

Some Classes of Irreducible Representations

125

12. Two simple examples of square integrable representations of classical p-adic groups In the rest of these notes, we shall fix one of the series of classical groups, symplectic or odd-orthogonal, and denote by S, either Sp(2n,F ) or SO(% I, F ) . Before we proceed further with description of general square integrable representations, we shall give two examples of square integrable representations of classical groups. The trivial one-dimensional representation of a group G will be denoted by I G .

+

12.1. Example:In this example we shall describe Steinberg representations

for symplectic groups. Steinberg representation can be constructed for any reductive group. Here we consider the series S, = Sp(2n,F ) . An easy computation of modular character of Pa in S1 = S p ( 2 , F ) = SL(2,F ) implies that

Is,

-

v-l

1 F X

x ls,,

since v-l1FX x Iso contains constant functions. The length of the Jacquet module of v-l 1 F x x IS, for the (standard) minimal parabolic subgroup is 2 (and irreducible subquotients are not isomorphic). This implies that v-' 1 F x x IS is a, length two representation. Further, Frobenius reciprocity implies that v-' I F x x IS, is not completely reducible (i.e. it is not a sum of irreducible subrepresentations). Now passing to contragredients we see that

v

1 F X

x Is,

contains a unique irreducible subrepresentation. This representation will be denoted by Sts, , and called Steinberg representation of S1.We can see (from the algebraic structure of R ( S ) over R ) that the Jacquet module for the minimal parabolic subgroup is v 1 ~ @x Is,. Now Casselman's square integrability criterion in this situation implies that Sts, is square integrable. Define Sts, t o be IS,. Now both v2

~

F

x Sts, and S([v IFX , v 2 1 ~ ~x1Sts, )

X

embed into v2 1FX x u

1Fx

x Sts,.

M. TadiC

126

Analyzing Jacquet modules, we would see that these two subrepresentations have exactly one irreducible subquotient in common, and that this subquotient is square integrable. We shall denote it by Sts,. It is a unique irreducible subrepresentation of u2 1FX x u ~ F Xx Stso. Continuing recursively in the above way, we define the Steinberg representation StSP, for any S,. It is a unique irreducible subrepresentation of un

1p.x

x un-l 1FX x

.. . x

u2 1p.x x u 1FX >a Sts,

(this defines StSp,). It is again easy to write what are the Jacquet modules of these representations: n

p*(Sts,)

= ~ s ( [ u ~ + + l l F x , u ~ l@ FS xt]S)k . k=O

+

12.2. Example: Let we now consider the series S, = SO(2n 1,F ) . An easy computation of modular character of Po in S1 = S O ( 3 , F ) implies that ISl Lf u-l12 1px

x Iso

(since u-l12 lFx x IS, contains constant functions). Consider the representation U1I2

1FX x u-l12 1FX x

Is,.

Here 6([u-1/2 1FX,u112 IFXI)x IS, and u 1 / 21Fx >a IS,

are subrepresentations. Looking a t Jacquet modules, one sees that S ( [ u - 1 / 2 l p x , u 1 / 2lpx])x

Is,

(11)

reduces. This follows from S.S.

>+

( b ( [ v - 1 / 21FX,Y112 1FX])x ISo

g S.S. and

S.S.

(

Y1/'

1 p X

(u112 1FX x

1

>a ISl

u-l12 1FX >a ls,)

Some Classes of Irreducible Representations

127

what one checks using the structure of R ( S ) . The representation (11) is unitarizable, so it is completely reducible. considering the Jacquet module of this representation for P(2,, and applying Frobenius reciprocity, we get that the representation 6 ( [ ~ - l /l~ ~u 1 / 2~I F X, ] )x Iso reduces into a sum of two inequivalent irreducible subrepresentations, say TI and T2. Thus

6([u-1’2 IFx, Y

~ IFx]) / ~x

Iso = Ti @ T2.

Now 6([v-1/2

l F X ,

u 3 / 21FX I) x Iso

L-)

u3’2 1FX x 6([u-1/2 l p x , u1/2 l F X ] ) x Iso = U3l2 1 F x X

(Ti @T2)

Eu

~ 1Fx / x~ Ti @ u ~ 1 F/x x~ T2.

Now the multiplicity of u3l2 l F x @ Ti in corresponding Jacquet module of u3l2 1 ~ xTi x is one. This implies that u3I2 1~~ xTi has a unique irreducible subrepresentation. These two irreducible subrepresentations (for i = 1,2) are square integrable. One can show that u 3 / 2 lFXx Ti are subquotients of corresponding Jacquet module of 6 ( [ ~ - l /I~F x ,v 3 / 2l F x ] ) x 1s0. From this we see that 6([v-1/2 l F X , U 3 I 2

l p ] ) x IS,

has exactly two irreducible subrepresentations. They are inequivalent and square integrable. Next question is how to distinguish these two irreducible square integrable subrepresentations. One can show that they have Jacquet modules of different length. This is one possible way to distinguish them.

13. Invariants of square integrable representations of classical p-adic groups

13.1. Let 7r be an irreducible square integrable representation of a classical group S,. C. Mceglin has attached to it a triple (Jord(.rr),7rcu,p,

Ex).

Each of these three parameters was considered earlier (at least in some form), but C. Mceglin was the first who considered them in this form.

M. Tad2

128

We shall describe each of these parameters. Our goal will be to explain their meaning from the point of harmonic analysis (they have a clear meaning from the point of view of Langlands program, what will be discussed later).

13.2.Jordan block of n:This is probably the most important of the three parameters (this parameter should determine the L-packet in which the representation lies). As we shall see later, the definition of J o r d ( n ) is very natural from the point of harmonic analysis (and can be given completely in terms of harmonic analysis). For p E C and a E N denote

We shall start with the first definition of Jord(n-) (this is a little bit modified definition, to avoid L-functions). Jord(n-) is called the Jordan block of n- and it consists of all (p, a ) E C x N such that (1) p is selfdual (i.e. Z; "= p; then p is unitarizable) and (2) if d / ' p >a Iso is reducible (resp. irreducible), then a is even (resp. odd) and

is irreducible. Although the above definition is simple, the clear meaning and importance of Jordan blocks is not evident from it. This is the reason that we shall give another description of Jordan blocks, from which will be much more clear importance of Jordan blocks for the harmonic analysis.

13.3. As we have mentioned already above, there is a very natural way t o come to Jordan blocks from the point of view of harmonic analysis, and we shall explain it bellow. Besides, because of the importance of Jordan blocks, they deserve to be understood as good as possible. Before we start to explain it, let us note that the classification of irreducible square integrable representation of classical groups is done under a natural assumption, which will be explained in section 15.1. This assumption shows up in proofs, not in the expression of the parameterization of irreducible square integrable representations. We shall assume that it holds in further.

129

S o m e Classes of Irreducible Representations

Now we are going to explain the importance of J o r d ( ~ for ) harmonic analysis. Once we have an irreducible square integrable representation T of a classical group, having in mind classification of the non-unitary dual via the Langlands classification, the first question that arises is: Which irreducible tempered representations can be obtained from this

T.

In other words, we would like to understand how a representation of the form

61 x

62

x

... x

&I,

x

T

(12)

reduces, when 6i are (unitarizable) irreducible square integrable representations of general linear groups. If we would know the answer to this question, we would have a reduction of understanding of irreducible tempered representations of the classical groups (and in this way also of all the irreducible representations) to the problem of understanding of irreducible square integrable representations of the classical groups. Therefore, understanding of such a reduction would be of the first class importance. The theory of R-groups reduces this question to the question when

6xlr

(13)

reduces, for 6 an irreducible (unitarizable) square integrable representation of a general linear group.

Remark: For further discussion of Jordan blocks, one does not need to understand this reduction. But for the classification of the non-unitary duals, one needs it. Therefore, we shall explain the reduction that gives the theory of R-groups, without going deeper in this theory. Consider representation from (12). Denote by e the number of inequivalent 6i among 61,62, . . . ,6k such that 6a x

lr

reduces. Then 61 x 62 x . . . x 6 k x lr is a multiplicity one representation and it reduces into a direct sum of 2j

irreducible (tempered) representations.

M. T a d 2

130

If p is a permutation of {1,2,. . . , k} and e l l € 2 , . . . , Ek E {fl},then representations 61 X 62 x . . ’ X bk x 7r and bitl) X dG:2) X + ’. x b;;, >a 7r are equivalent, where bit.) denotes 6p(i) if E . = 1 and it denotes &(.) if ~i = -1. Let 6; x S$ x . . . x Sk, x 7r‘ be another representation] such that 6; are (unitarizable) irreducible square integrable representations of general linear groups and 7r’ is an irreducible square integrable representation of a classical group Sq,. Suppose that S1 x 62 x . . . x bk >a 7r and 6; x S$ x ... x Sh, XI 7r’ have an irreducible subquotient in common. Then 7r 2 7r’ (and therefore q = q ’ ) ] k = k’ and there exists a permutation p of {II2 , . . . k } and €1, E Z ] . . . ,E k E {fl}such that

s;

E S”

P(%)

for all i = 1 , 2 , . . . , k.

13.4. As we already have mentioned, to understand irreducible tempered representations] we need to understand when representations S x 7r (from (13) reduce. Having in mind the classification of irreducible square integrable representations of general linear groups] one needs to understand when %%a) x

77

reduces, for unitarizable p E C and for a E N. When we fix p , the reducibility of these representations can be described in a very nice way (a crucial role in this is played by Jord(7r)). Frobenius reciprocity implies irreducibility if p is not selfdual. Therefore, it remains to understand the reducibility for selfdual pis. The following two examples are very simple but nice examples, from which one can get an an idea what happens regarding these reducibilities in general.

13.5. Examples: Let S, = Sp(2n, F ) and

7r

= Iso.

(1) Suppose $I is a character of order two of F X = GL(1, F ) . Then

S(+, a ) x Iso is irreducible for all even a; S($I,a) x Is, is reducible for all odd a. We see that understanding of reducibility in this case is very simple. One needs only to know the parity of N for which we have reducibility. Unfortunately, this is not always the case for other square integrable representations.

131

S o m e Classes of Irreducible Representations

Now we shall give a simple example of a situation of a slightly different type.

+

(2) We shall consider now instead of the trivial representation GL(1,F ) (on one-dimensional space). Then

~

F

X

of

5(1Fx, a ) x l s o is irreducible for all even a ;

6 ( l F X a, ) x Iso is reducible for all odd a , except for a = 1. Now we shall explain what happens in general regarding such reducibility. Fix selfdual p E C. Then for exactly one parity in

N holds:

(1) b ( p , a ) x x is reducible for all a from that parity, with possibly finitely many exceptions; (2) 6(p,a) x x is irreducible for all a from the other parity. The parity of W for which ( 1 ) holds, will be called the parity of reducibility of p and x (note that for this parity we can have finitely many exceptions of reducibility), and the other parity will be called the parity of irreducibility of p and x (in this parity we have always irreducibility). Therefore, for understanding tempered representations we need to know which is the parity of reducibility for selfdual p E C, and what are exceptions (if there are exceptions, then clearly they determine the parity of reducibility). Therefore, it is very important to know these exceptions. This is just Jord( x):

New definition: Jord(7r) is the set of all exceptions ( p , a ) in (l),when p runs over all selfdual representations in C ( a E N). Suppose that x is an irreducible square integrable representation of S,. C. Mceglin has proved that

where dp is determined by the fact that p is a representation of GL(d,, F ) . The above inequality clearly implies that Jord(x) is finite. The above inequality is expected t o turn to be an equality.

13.6. Partial cuspidal support of x: In general, (conjugacy class of) an irreducible cuspidal representation T of a Levi factor M of a parabolic

M . Tadid

132

subgroup P in a reductive group G is called cuspidal support of 7 r , if 7r is a subquotient of Indg(7). For classical groups, Levi factor M is a direct product of general linear groups and a classical group. Therefore, 7

p1 63 p2 63 . . . 63 pz 63 cl,

where pi E C and CT is an irreducible cuspidal representation of a classical group. Now the definition of partial cuspidal support of 7r is

cusp = CT. We can define partial cuspidal support of 7r also in the following way: an irreducible cuspidal representation CT of a classical groups is called partial cuspidal support of 7r (and denoted by 7rcusp) if there exists a smooth representation 7r' of a general linear group, such that 7r

L)

7r'

x IY.

13.7. Partially defined function E , on Jord(7r):As we could see already from the simple Example 12.2, the construction of irreducible square integrable representations of classical groups involve reducibility of tempered induction, and thus R-groups (which for classical groups are sums of 2/22). This is roughly behind the fact that parameters of irreducible square integrable representations of classical groups will involve functions with values in {fl}. The definition of the domain of partially defined functions E , on Jord(7r) is quite technical. Because of this, we shall not give a complete definition of the partially defined functions (besides, from the general definition of partially defined functions, it is not quite easy to understand what are these functions). Rather, we shall try only to explain the main properties of these functions. One can understand pretty well the classification of irreducible square integrable representation of classical groups without knowing all the details of the definition of partially defined functions E , . Later, we shall give a constructive definition of these functions. Let X be a free Z/2Z-module with basis Jord(7r). We shall denote operation in the module X multiplicatively. Characters of this group are in a natural bijection with functions

Jord(7r)

--$

{fl}.

133

Some Classes of Irreducible Representations

We can think of of

E,

as a function on a subset of X, in our case on a subset

Jord(7r) u ( 2 1 2 2 ;

21,Q

E Jord(7r),z1 #

which can be extended to a character of X . Further, for 2 1 , E~ Jord(7r) we shall write also as

22),

~ ( 1 ~ 1 when x 2 ) ~it

is defined,

GI (21 ) E x ( 2 2 )

even if E , ( z ~and ) ~ ~ ( 2 are 2 ) not defined. The fact ~ ( 2 1 x 2= ) 1 (resp. ~ ( 2 1 x 2 = ) -1) will be written also as Gr(Z1) = %(Q)

even if

~ ~ ( 2 and 1 )

(resp.

E,(21)

#452))

E , ( z ~are ) not defined.

13.8. To give an idea of definition of E , , we shall describe one important case. The function E , is always defined on (p,a)(p’,a’) if p = p’, a # a‘ (and both ( p , a ) , ( p ’ , a’) are from Jord(7r)).

< a , and ( p , a ’ ) $ Jord(7r) for any a- < a’ < a.

Suppose ( p , a - ) ,( p , a ) are in Jord(n), a-

Then E , ( ( P , a - ) ( p , a ) ) = ~ , ( p ,a - ) ~ , ( p , a ) is defined and €,(PI

Q-)E7r(P1

a) = 1

if and only if there exists a smooth representation such that 7r

L-)

S([v(”-+1)/2p, v

7r’

( ” - q ] )x

of a classical group

T’.

13.9. In general, ~ , ( p a, ) is not always defined for ( p , a ) E Jord(n). It is always defined if a is even. The definition in this case is the following: Suppose ( p , a ’ ) E Jord(7r) with a’ even. Chose a minimal a such that ( p , a ) E Jord(7r). Then GT(p,a) = 1

if and only if there exists a smooth representation such that 7r

-

7r’

b ( [ v 1 / 2 p ,v ( 4 / 2 p ] )>a 7r’.

of a classical group (14)

M. TadiC

134

If a is odd, then ~ , ( p , a ) is not always defined. It is not defined if and only if (P, b) E

JOT~(T~,,~)

for some b E N. From the above condition for ~ , ( p , a )to be defined in the case of odd a , one can show that if ~ , ( p ,a ) is defined (for odd a ) , then

p

Tcusp

(15)

reduces (this is related to basic assumption under which we consider the classification of irreducible square integrable representations of classical groups; this assumption will be explained later). C. Mceglin has used normalized intertwining operators t o define E , ( P , a ) in this situation. In the case when ~ , ( p , a ) is defined for odd a , as we have mentioned already p x T,,,~ reduces. It reduces into a sum of two inequivalent irreducible subrepresentations. One can chose one of these subrepresentations and attach to it 1, and to the other attach -1. Then one needs t o extend in a natural way this choice to other tempered representations coming from inducing representations including 6 ( p , a ) as a factor. One can do this using intertwining operators, but one can also do it without them. Now we have almost complete definition of 6 , . 13.10. C. Mceglin has shown that for an irreducible square integrable representation 7r of a classical group, the triple

satisfies some technical conditions. The triples that satisfy these technical conditions she called admissible triples. We shall not give in the moment this technical definition. We shall give later a different and more explicit description of admissible triples. Let us just say that admissible triples are combinatorial objects modulo cuspidal data. It will become soon clear what we mean by cuspidal data. C. Mceglin has proved that the mapping attaching an admissible triple to an irreducible square integrable representation of a classical group, is an injective map from the set of all the equivalence classes of irreducible square integrable representations of classical groups (we fix a series of classical groups and a non-archimedean field F ) into the set of all the admissible triples. Jointly, we have proved that this mapping is surjective. This means that we have a bijection between irreducible square integrable representations of classical groups and admissible triples. Since admissible triples

135

S o m e Classes of Irreducible Representations

are combinatorial objects modulo cuspidal data (as we mentioned already above) , we have a classification of irreducible square integrable representations of classical groups modulo cuspidal data.

14. Reduction to cuspidal lines The classification of irreducible square integrable representations of classical groups modulo cuspidal data will be easier to understand if we pass to cuspidal lines. We shall explain it in this section. This reduction could be important also for some other purposes.

14.1. Fix an irreducible cuspidal representation a of a classical group S, and fix inequivalent selfdual irreducible cuspidal representations P I , . . . , p k of general linear groups. Denote by D(Pl,.. . I P k ; 0)

the set of all equivalence classes of irreducible square integrable subquotients of representations ualrl x

v=2r2

x

. . . x Pert

>a

a,

where a, E R,7i E { P I , . . . ,pk}. Then there is a natural bijection from D ( p 1 , . . . , p k ; c) into the Cartesian product

This bijection is given in the following way. Fix T E D ( p 1 , . . . , P k ; a) and 1 5 5 k. Then there exists an irreducible representation r, of some Sn, which is a subquotient of some vplp, x vpzp, x . . . v P ' 3 p, x1 0 , pzE R,and there exists an irreducible representation 0, of a general linear group which is a subquotient of r1 x 7-2 x . . . x rm3with r2E UF=l,tz3 {v"p,; Q E R}, such that T

v 0,

x1 T,

C. Jantzen has proved in 1321 that representations T I , . . . , rk are uniquely determined by T , and that they are square integrable. Further T I---+ k ( T I , . . . , T k ) defines a bijection from D(p1,.. . , P k ; 0 ) onto D(pz;0). Each irreducible square integrable representation of a classical group belongs to some D(p1,.. . , P k ; a). Further, if a o', then

n,=,

D ( P l , . . . i P k ; c )nD(P;,-,Ph')

= @ I

M. TadiC

136

and if

then

v(pl,..., p k ; r ) n ’ D ( p ’ ,,..., ,&;a)=V(& ,..., p ; , , ; ~ ) In this way we obtain a reduction of the classification of irreducible square integrable representations of classical groups, to the problem of classification of sets

( p E C is selfdual).

14.2. Consider the projection OF

OF/PF.

Lift it to the level of groups. In this way one gets a natural homomorphism from the maximal compact subgroup KO in S, to the group S, over the field O F / P F . The preimage in KO of the standard minimal parabolic subgroup in S, over the field OF/pF is denoted by

1 This open compact subgroup is called Iwahori subgroup of S,. The study of irreducible smooth representations with Iwahori fixed vectors has attracted a lot of attention. In this case, for building the representation theory, one does not need non-trivial cuspidal representations (i.e. other than characters). Also, corresponding group algebras for this setting have a nice geometric realization. Using this, one can obtain construction of irreducible representations by geometric methods, what was done by D. Kazhdan and G. Lusztig. For classical groups, to determine irreducible square integrable representations with Iwahori fixed vectors, it is equivalent t o determining of D(1FX

; SO)

v($;lSo),

where +IJ.! is a (unique) character of order 2, which is unramified (i.e. which is trivial on 0 ; ) . Irreducible square integrable representations with Iwahori fixed vectors are parameterized by the Cartesian product 2)(1FX ; Is,) x

V($;l s o ) .

S o m e Classes of Irreducible Representations

137

15. Parameters of D ( p ; a) 15.1. For selfdual p E C and an irreducible cuspidal representation a of S,, A, Silberger has proved there exists a unique Qp,o

20

such that wapj,p

xo

reduces. Now we shall say what is the basic assumption (for p and o),under which V ( p ,a) is classified: (BA) for p and o

ap,a

- Qp,ls, E

z.

This assumption is needed (essentially only) in proofs. F. Shahidi has proved that (BA) holds if a is generic. It is also known that (BA) holds in some other cases. In general, (BA) would follow from the truth of some general Arthur’s conjectures. F. Shahidi has proved that

15.2. We shall fix p and a as above, and assume that (BA) holds for p and a. Denote in the sequel (Y

=

Now since we assume that (BA) holds for p and a, (16) implies CY

Note that for

7r

= ( Y p , o f (1/2)2+.

E V ( p , a),

cusp

= 0.

Therefore, since 0 is fixed, for classification of V ( p ,0)it is enough to consider instead of triples (Jord(7r),7Tcuspr E 7 r )

pairs

(Jord(r) E7r) 7

M. TadiC

138

(which form with g admissible triples). Further, for classification of D(p, 0 ) it is convenient (and enough) to work with

Jord,(n) = { a E W; (p,a) E Jord(n)}. instead of Jord(n). Now pairs

(Jordp(n)E 7 T ) 7

will be parameters of representation in D ( p , o ) . Note that Jord,(n) is a finite subset of either 2N or 2N - 1 and ex is now regarded as a partially defined function on Jord,(n). Because of this, the parameters of D ( p , a ) are now simpler than before. There are two possibilities for a. The first is

a E Z+, which will be called integral case, and the second is Q

E ( ( 1 / 2 P + \ Z+),

which will be called non-integral case. 16. Integral case

We shall suppose in this section that

and describe parameters (Jord,,

z+,

Q

E

E)

of elements of D(p,u ) in this case.

16.1. In the integral case we have always

Jord,

C 2N - 1.

Here partially defined function is defined on elements of Jord, if and only if a = 0. If it is defined on Jord,, then the values on Jord, completely determine the partially defined function. If Q 2 1, then E is defined only on pairs from Jord, (and this partially defined function can be extended to a character of a free Z/2Z-module with basis Jord,). 16.2. In the integral case, Jord, will be called of alternated type if

card(Jord,) = a.

Some Classes of Irreducible Representations

139

Here always exists a unique partially defined function E such that Jord,, and (T form an admissible triple. If cr. = 0, then there is nothing t o define. If a 2 1, then E is not defined on elements on Jord,, but it is defined on pairs. It is completely defined by the following property:

E

For each a _ , a E Jord,, a-

< a , such that

[a-,a] n Jord, = { a - , a } we have 4a-)

# 4a)

(this is where the name alternated type comes from). Now we shall define the representation corresponding to alternated Jord, (and E ) . Write Jord, = { a l ,a2,. . . ,u a } . After a renumeration we can assume

Now the representation

has a unique irreducible subrepresentation, which will be denoted by n(Jordprc,E).

This representation is square integrable. An example of such representations are Steinberg representations for symplectic groups. 16.3. We shall describe now general parameters of elements of D ( p ; a ) . Take Jord, (and E ) of alternated type. Take any a _ , a E 2N - 1, a- < a, such that [ a _ ,a] n Jord, = 0.

Denote

J o r d r ) = Jord, u { a - , a } . Extend

E

in a way that

€ ( a _ )= € ( a ) .

M. Tadid

140

It is easy to see that there are exactly two such extensions. Denote them by €1,€ 2 . Now ( J o r d y ) ,(T,~ i ) i, = 1 , 2 , are (new) admissible triples (in this setting). These triples are no more of alternated type. We can continue this construction, but now starting from J o r d r ) ,(T,~ i . Continuing this process, we construct Jord,(2) , J o r d f ) , . . . (and corresponding partially defined functions). In this way, we shall get all the parameters of 'D(p; 0).

16.4. We shall describe now representations corresponding to these (new) parameters. Let (Jord,,E),a-,a and (Jordr),c,ci)be as in 16.3. To alternated (Jord,,e) we have already attached in 16.2 a square integrable represent ation *(Jord,,U,E).

Now the representation

S( [,-(.--Wp,v(a--1)/2 PI)

>a

~(JO?-dp,U,€)

contains exactly two irreducible subrepresentations. One shows this using the strategy that we have used in Example 12.2. These irreducible subrepresentations are square integrable, and their parameters are (Jord,(1), CT,€1) and (Jord;) , (T,€2). If Jord,, # 0, one determines from 13.8 which subrepresentation corresponds to which ~ i .

16.5. It remains to say which subrepresentation t o attach t o which ~i if Jord, = 0. C. Mmglin has used normalized intertwining operators for this attaching. One possibility would be to proceed in the following way. Suppose

Jord,, Then a

= 0.

= 0.

Write p >a

(T

= 7-1 @ 7 - 1

as a sum of (two inequivalent) irreducible representations. The representations

S( [.-'a-

- W p J", -1)/2 PI)

f J

and 6 ( [ v p ,v(a--1)/2 p ] ) x b ( [ v p ,v(a--1)/2 PI) >a Ti

141

Some Classes of Irreducible Representations

have exactly one irreducible subquotient in common (for each i = - l , l ) . Denote it by Ti. Now ,J([v("-+1)/2p, v(a-1)/2p])

Ti

contains a unique irreducible subrepresentation. Denote it by 7 r i . Then 7r1 and 7r-1 are two inequivalent irreducible subrepresentations of ,J( [ v - ( " - - l ) / 2 p, v(a-1)'2Pl) CT (here 7 r ( J O T d p , f l r E ) = .). One natural possibility to distinguish irreducible subrepresentations of

,J([ v - ( a -

-1)/2

p, v ( a - q ) M

c7

is to attach ~i t o 7 r E i ( a ) (for i = - 1 , l ) . Let us note that we have not checked that this choice is the same as the choice that C. Moeglin made using normalized intertwining operators. 16.6. One proceeds further from Jord;) t o J o r d p ) , J o r d p ) to Jord,(3) , . . . recursively in the same way as we did in passing from Jord, to Jord,(1)

(it is even less complicated here, since always Jord;) # 0, J o r d p ) and therefore we do not need to make choices as in 16.5).

# 0,. . .

17. Non-integral case

Now we shall assume that

a E ((1/2)2+\ Z+). 17.1. In this case.

Jord,

2N.

Partially defined functions are defined on elements of Jordp's (and values on Jord,'s completely determine partially defined functions).

17.2. In the non-integral case, Jord, will be called of alternated type if card(Jord,) = (I! f 1/2, i.e. if card(Jord,) = a - 1/2

or

card(Jord,) = a

+ 1/2.

In this case, there exists a unique partially defined function E on Jord, such that Jord,, CT and E form an admissible triple. This partially defined function E is defined (and uniquely determined) by the following conditions: For each a _ , a E Jord,, a- < a, such that [u- , u] n Jord, = { u - , u }

M. TadiC

142

we have

and c(min(Jord,)) =

1

-1

if card(Jord,) = a if card(Jord,)

+ 1/2;

=a -

1/2.

17.3. We shall now describe the representation corresponding to the above alternated Jord,. Let Jord, = { a l ,a 2 , . . . , aa+1/2}.After a renumeration we can assume

Consider first the case card(Jord,)

= CY

-

1/2

Then the representation

has a unique irreducible subrepresentation. This subrepresentation will be denoted by

n(Jord,,u,e). It is a square integrable representation which corresponds to (Jord,, Now assume card(Jord,)

=a

E).

+ 1/2.

Then the representation

has a unique irreducible subrepresentation. We again denote this subrepresentation by

n(JoTdp,urE). This is a square integrable representation which corresponds t o ( Jord,, E ) . We have described above how one attaches square integrable representations t o alternated parameters.

S o m e Classes of Irreducible Representations

143

17.4. Now we define general parameters of D(p;n) in the same way as in the integral case. We also attach square integrable representations in the same way. The only difference which occurs between integral and non-integral case is in passing from Jordp to Jordp) when Jordp = 0. In the non-integral case one uses (14) to determine which irreducible subrepresentation corresponds to which ~ i . 18. Local Langlands correspondences 18.1. Denote by WF the Weil group of F . This is a dense subgroup of the Galois group of the separable algebraic closure of F over F (the topology is not the induced one from the Galois group, but a slightly modified one). Let G be a split connected reductive group over F (as GL(n, F ) ,Sp(2n,F ) or SO(2n 1,F ) ) . By the Langlangs program, there should exist a natural partition of G into finite subsets, called L-packets, which are indexed by (conjugacy classes of) admissible homomorphism of WFx SL(2,C) into the complex dual group LGo of G (admissible here means that the homomorphisms are continuous, that they carry WF into semi simple elements and that they are algebraic on SL(2,C)). For the cases of the groups that we consider, complex dual groups are as follows:

+

L G L ( ~5'),' = G L ( n ,C), LSp(2n,F)' = SO(2n + 1,C), L S 0 ( 2 n+ 1,F)' = S p ( 2 n ,C) (a property of the complex dual group 'Go is that it has the root system dual to the root system of G). We shall now concentrate our attention regarding the above aspect of the Langlands program to irreducible square integrable representations. In this case square integrable L-packets should be parameterized by admissible homomorphisms whose image is not contained in any proper Levi factor. Further, elements of a square integrable L-packet, which is indexed by an admissible homomorphism $, should be parameterized by irreducible representations of the component group of $ (which is the quotient of the centralizer of the image of $ by the connected component). The correspondence that one would get in this way is called local Langlands correspondence for G.

144

M. Tadit

18.2. In the case of general linear groups, by the Langlands program there should be a bijection of irreducible square integrable representations of GL(n,F ) and n-dimensional irreducible representations of WF x SL(2,C) (which are admissible homomorphisms). Here component groups are trivial. In this bijection irreducible cuspidal representations of GL(n,F ) should correspond to irreducible representations of WF The work of Bernstein and Zelevinsky, which gave classification of irreducible square integrable representations of general linear groups modulo cuspidal representations, resulted in a reduction of establishing of local Langlands correspondence to the cuspidal case, i.e. to establishing a correspondence between (classes of) irreducible cuspidal representations of GL(n,F ) and (classes of) irreducible representations of W,. More precisely, suppose that

is such a correspondence for general linear groups between irreducible cuspidal representations of general linear groups and irreducible representations of W, (we consider all general linear groups over F together). From the representation theory of SL(2,C) one knows that for each a E W there exists a unique irreducible algebraic representation

of SL(2,C) on a-dimensional complex vector space (up to an equivalence). Then the formula for local Langlands correspondence on the set of (classes of) irreducible square integrable representations of general linear groups, which we shall denote also by 'p, would be CP(~(a P ), ) = P ( P ) 8 Ea.

Local Langlands conjecture for GL(n,F ) has been recently proved in full generality (by M. Harris and R. Taylor in [25], and by G. Henniart in

[261).

18.3. One may ask does the classification of irreducible square integrable representations of classical groups modulo cuspidal data give also a similar reduction. The natural candidate for the Langlands correspondence @ for classical groups is

Some Classes of Irreducible Representations

145

(p is the local Langlands correspondence for general linear groups, which we have considered before). But it remains a number of facts to prove even to see that it is a good candidate (for the beginning, it is not clear at all that @ ( T ) goes in the right group).

18.4. For classical groups, the centalizers of images of admissible homomorphism of W , x SL(2,cC) into the complex dual group, whose images are not contained in any proper Levi factor, are finite groups which are 2/22modules (these are component groups). Therefore, after choosing a basis, irreducible representations of the component group correspond to functions from the basis to {fl}. Now E~ should give a part of the irreducible representation (i.e. character) of the component group corresponding to 7 r . The rest should come from E ~ , , , ~(once the local Langlands correspondence is established for cuspidal representations of classical groups). Complete discussion regarding this reduction one can find in [41]. 19. Non-unitary duals of classical p-adic groups 19.1. The classification of irreducible square integrable representations of classical groups modulo cuspidal data implies also a classification of all the irreducible smooth representations of classical groups modulo cuspidal data (by cuspidal data we mean irreducible cuspidal representations of general linear and classical groups, and cuspidal reducibilities) . Suppose that a selfdual p E C and irreducible square integrable representation T of a classical group are given. For understanding tempered representations, one needs to know the parity of reducibility. If Jord,(~) # 0, then the parity which shows up in J o r d , ( ~ ) is the parity of reducibility of p and 7 r . If J o r d , ( ~ )= 0, then Jordp(7rcusp)= 0 and then the reducibility of p and 7rcuuSp(and also T ) is at 0 or 1/2. If the reducibility is at 0 (resp 1/2), then the parity of reducibility of p and T is odd (resp. even). 19.2. We can describe also the non-unitary dual by reduction to cuspidal lines. Let be an irreducible cuspidal representation of a classical group S, and let p 1 , . . . ,p k E C be unitarizable such that for z # j , sets { p i , pi} and { p j , & } have no equivalent representations (i.e. pi Pj). p j and pi

146

M. TadiC

Denote by

the set of all equivalence classes of irreducible subquotients of

where

Then by [32] there exists a bijection

similarly as in 14.1 (for complete definition of the bijection one should consult [32]). The classification of the non-unitary duals of classical groups reduces to the classification of the sets Z(pi;a) in a similar way as the classification of the irreducible square integrable representations in 14.1 reduces to cuspidal lines (we shall not write details here, but the reduction is analogous). 19.3. Fix a unitarizable p E C and fix an irreducible cuspidal representation of a classical group. If p is not selfdual, then the tempered induction in Z ( p ; o ) is always irreducible. Now irreducible tempered representations which show up as Langlands parameters of representations in Z(p;g) is easy to write down (using Remark 13.3 to know equivalences among them). Suppose now that p is selfdual. Let the reducibility of p and (T be a = ap,D2 0. Now the parity of reducibility of p and (T (and also each square integrable representation in Z ( p ;0 ) ) is odd (resp. even) if a E Z (resp. a @ Z). Further, one can describe easily irreducible tempered representations which show up as Langlands parameters of representations in Z(p; (T), since we know the parity of reducibility of p and (T (one needs also to use Remark 13.3). (T

S o m e Classes of Irreducible Representations

147

20. Unitary duals of general linear groups over local fields

20.1. Denote

D"

= (6 E

D ;e ( b ) = O } .

These are just all the square integrable classes in D. The following theorem describes the unitary duals of general linear groups over any local field (archimedean or non-archimedean).

Theorem: For a representation 6 E D" and m 2 1 denote U ( S ,m ) = L ( v ( " - 1 ) / 2 S ,

v(m-3)/2(5,.

. . ,v - ( m - 1 ) / 2 6 )

For 0 < a < 112 and S and m as above, denote r(u(S,m ) ,a ) = voIu(S,m ) x vPau(S,m).

Let B be the set of all possible u ( b , m ) and n(u(S,m),a)with S,m,a as above. T h e n (i) If rl,7 2 , . . . , r, E B , then the representation r1

x r2 x . . . x

T,

is a n irreducible unitarizable representation of a general linear group. (ii) Let r 1 , r 2 , . . . , r,, ri,ri,. . . ,r;, E B , T h e n

r l x r ~ x ~ ~ ~ x ~ , ~ r i f and only i f n = n' and if one can obtain the sequence ( 7 1 , 7 2 , . . . , r,) f r o m (T;, r;, . . . , T;) by a permutation. (iii) Each irreducible unitary representation of a general linear group is isomorphic to a representation r1

x rz x ... x r,,

forsomer1,r2,...,Tn E B. The above classification theorem is the same for all the local fields. The difference in the form of unitary duals comes from the difference of the sets D" for different fields. 20.2. This theorem is proved in [60] in the non-archimedean case. As we already mentioned, the theorem holds for archimedean fields in the same form (using the notion of (8,K ) modules), with the proof alocg the same

M. TadiC

148

strategy as in the non-archimedean case (see [71]). D. Vogan in [72] has made quite different approach to the classification of unitary duals of general linear groups over archimedean fields. Not to deal all the time with non-archimedean fields, we shall now describe the proof of the above theorem for F = @. Since the proofs in the archimedean and non-archimedean case are along the same strategy, one will be able t o get from this description quite good idea of the proof in the non-archimedean case.

20.3. In the sequel of this section, by a representation we shall mean corresponding (8, KO)-module. Denote by

Irr" = UZroGL(n,C)-. Consider algebra R for complex general linear groups (constructed in 10.3). We shall consider

Irr"

R.

Recall that R is a polynomial ring over D (Proposition 10.7). In particular, R is a factorial ring. Therefore, we can talk about prime elements in R. In the complex case we have

D = GL(1,C)-. Note also that I la: is the square of the usual absolute value in @. We shall now introduce several claims, whose proofs shall be discussed later:

(UO) (Ul) (U2) (U3) (U4)

x T E Irr". c,T E Irr" b E D" and n E N j u(b,n)E Irr". b E D", n E N and 0 < a < 1 / 2 ==+ ~ ( u ( b , n ) , E a )Irru. 6 E D and n E N ==+ u(b,n)is prime in R. a , b E M ( D ) + L ( a ) x L(b) contains L ( a b ) as a subquotient.

+

The addition of multisets (which shows up in (U4)) is defined in obvious way: (21,.

. . , 2") + (Yl,. . . ,YV)

Proposition: Claims (UO) -

= (21, . . . , x u ,Y1, . . . , YV).

(U4)imply Theorem 20.1

Proof: First observe that (Ul), (U2) and (UO) imply (i) of the theorem.

149

Some Classes of Ineducable Representations

Further, commutativity of R give implication +== in (ii). The implication + follows from (U3). It remains to prove (iii) (i.e. exhaustion). Suppose 7r E IT?. Then 7r

= L ( Y l , Y 2 , . . .,re)

for some y1,y2,. . . ,-ye E D (see Remark 9.6). Note that 7r is hermitian (since it is unitary). This, together with (5) and (6), implies that 7r

= L(VQ'bl,V-='&,

.. .,Va"k,V-=k6k,6k+l,.

. . , &),

for some ai > 0, 1 _< i 5 k and S j E D", 1 _< j 5 s. We shall use in the sequel the following simple fact

M ( D ) and L ( a l ) , L ( a z )., . . ,L(a,) E then L ( a 1 ) x L ( a 2 ) x . . . x L(a,) = L(a1 + a2 +. . . + a,). If

a1,a2,.

. . ,a,

E

(17)

This follows directly from (U4) and (UO) (by induction). To get an idea of proof of (iii), we shall now give a proof of it in the rigid case, i.e. when all ai E ( 1 / 2 ) 2 . Denote a(&,m ) = (v(,-1)&,

y(,--3)/26,.

. . , v-(m-l)/26).

Then obviously u(S,m) = L(a(6,m ) ) .

Using the fact that ( V a , S .a 1

v-**62)

+ a ( & , 2ai - 1) = a(&, 2a2 + 1)

and (17) (several times), from (Ul) we get that 7r

x u(62, 2a1 - 1) x ' . . x u(&,2 a k U(&,

2a1

-

1)

+ 1) x .. . x U ( b k , 2CYk + 1) x 6k+l x . . . x 6,.

By (U3), on the right hand side we have prime elements (from B ) . Since R is factorial, 7r must be a subproduct of the right hand side (up to a sign). So, 7r must be a product of elements from B . This proves (iii) in the rigid case. The proof of (iii) in the non-rigid case proceeds along a similar idea, but it is slightly technically more complicated in this case. 0 By the above proposition, to prove the theorem, it is enough to prove (UO) - (U4). Now we shall explain how the proofs of each of these claims go.

M. Tadid

150

20.4. ( U l ) : Considering the modular function of the standard minimal parabolic subgroup in G L ( m , C ) ,we get easily that for 6 E D = G L ( l , C ) ^ (i.e. a character of C " ) we have u(S,m ) = 6 o det : GL(m,C ) + C x .

Thus, u(6,m) is unitarizable. Therefore, ( U l ) holds. 20.5. (U2): The restriction of the representation r(u(6,m ) ,a ) (which we consider in (U2)) to SL(2m, C ) ,is a Stein's complementary series representation from [58] (if m > 1; if m = 1, then this is a well known complementary series representation of SL(2, C)). Therefore, it is unitarizable. From this one gets directly that r(u(6,m ) ,a ) is unitarizable as a representations of GL(2m, C ) (since it has unitarizable central character). One can get the unitarizability of representations r(u(6,m ) ,a ) by standard construction of complementary series representations (which are unitarizable). For this, see 20.11 bellow. 20.6. (U3): We shall illustrate the proof of (U3) on the example of u(6,2). Note first that R is a graded ring (by definition). The degree of ~ ( 6 ~ is two. The representation theory of SL(2,C ) implies that

4 4 2 ) = x1 x

x 2-x 3

x

x 4

(18)

for some Xi E D ,i = 1,2,3,4, where all Xi are different. Suppose that u(S,2) is not prime. Since it is primitive (the greatest common divisor of

coefficients is l), it must be a product of homogeneous elements of degree one. Write fz = c1 (4 XI

+ c p x 2 + cpx3+ c y x 4 ,

2 =

fl

and

f2

1,2.

Since X1 x X2 shows up in u(6,2) (see (18)), it follows that cI1) # 0 and cp) # 0 (after possible changing indexes of f1 and f 2 ) . Since X3 x X4 shows up in u(6,2), it follows that cy) # 0 and cf) # 0 (after possible changing indexes of X3 and X4). These observations imply that the total degree of u(6,2) in variables XI and X4 is 2. This obviously contradicts to the expression (18). This contradiction completes the proof that u(b,2) is prime (in R). The proof of (U3) in the general case follows the same strategy and uses only very basic facts about composition series of standard modules, i.e. principal series (which are standard facts of Langlands classification).

S o m e classes of Irreducible Representations

151

20.7. (U4): This claim follows from basic properties of composition series of principal series (which are standard facts of Langlands classification). We shall explain now how it follows. We shall consider s.s.(X(a)) E R for a E M ( D ) . For simplicity, we shall write s.s.(A(a)) as an element of R simply as A(a) E R. Let a1,a2 E M ( D ) . There exists a partial order 5 on M ( D ) (which is quite explicit and which is simple to describe), such that we have

L(a,) = X(a,) +

c

rn$!)A(b(t)),

2

= 1,2,

b ( % 1, these are called discrete series representations, as they occur discretely in the decomposition of the representation L2(G).The quotient of VA,,by the sum of these two submodules is an irreducible module of dimension k - 1. For k = 1 the two submodules are called the limits of discrete series, and their sum is all of

+

+

I.

V0,l.

All this can be found with many more details and proofs in Vogan’s book [28], Chapter 1. Actually, Chapters 0 and 1 of that book contain a lot of material from this section (plus more) and comprise a good introductory reading. Other books where a lot about (g,K)-modules can be found are [15] and [33].

2. Clifford algebras, spinors and Dirac operators 2.1. Clifford algebras From now on we will adopt the convention t o denote real Lie algebras with the subscript 0 and t o refer to complexified Lie algebras with the same letter but no subscript. For example, go = to @ po will denote the Cartan decomposition of the real Lie algebra go, while g = e @ p will be the complexified Cartan decomposition. We saw in 1.4 that Z(g) has an important role: it reduces representations, and defines a parameter (infinitesimal character) for the irreducible representations. We also defined the “smallest” non-constant element of Z(g), the Casimir element R. Let us imagine for a moment that Z ( g ) contains a smaller (degree one) element T , such that T 2 = R. Then T would (in principle) have more eigenvalues than R,as two opposite eigenvalues for T would square to the same eigenvalue for 0. As a consequence, we would get a better reduction of representations, and infinitesimal character would be a finer invariant. Of course, such a T does not exist; degree one elements of U ( g ) are the

Dirac Operators in Representation Theory

175

elements of g and g has no center being semisimple. The idea is then to twist U ( g ) by a finite dimensional algebra, the Clifford algebra C(p). The algebra U ( g )@I C(p) will contain an element D (the Dirac element) whose square is close to R @ 1. One can define the Clifford algebra C(p) as an associative algebra with unit, generated by an orthonormal basis Zi of p (with respect t o the Killing form B), subject to the relations

zzzj

= -zjzz

(2

z i2 = -1.

#j);

(There are variants obtained by replacing the -1 in the second relation by 1 or by 1/2.) This definition involves choosing a basis, so let us give a nicer one:

C(P) = T(P)/I, where T(p)is the tensor algebra of the vector space p and I is the two-sided ideal of T(p) generated by elements of the form

x @ Y + Y @ X + 2B(X,Y). This definition resembles the definition of U ( g ) ;the similarity extends to an analog of the Poincark-Birkhoff-Witt theorem. Namely, C(p) inherits a filtration by degree from T(p).The associated graded algebra is the exterior algebra A(p). One can obtain a basis for C(p) by taking an (orthonormal) ordered basis Zi for p and forming monomials over it to obtain

za

il 0 f o r any a E A(u)), then C s ( Z )is irreducible and nonzero. (v) If 2 is unitary, then C,(Z)is unitary.

+

+

Remark: There are two dualities:

where W h is the hermitian dual of W . 4.4. A,(X)-modules

Now we consider Z to be a one-dimensional representation. A : called admissible if it satisfies the following conditions: (i) A is the differential of a unitary character of L (ii) if a E A(u), then ( a , X l t ) 1 0 . Given q and an admissible A, define p(q,A) = representation of

K of highest weight XIt

[ -+

+ 2p(u n p).

Q1 is

(1)

J . 3 . Huang and P. PandiiiC

186

The following theorem is due to Vogan and Zuckerman.

Theorem: ( [ 3 2 ] ,[29])Suppose q is a 8-stable parabolic subalgebra of g and A: I + CC is admissible as defined above. Then there is a unique unitary (9,K)-module A, (A) with the following properties: (i) The restriction ofA,(X) to t contains p(q,X) as defined in (4.1); (ii) Aq(X) has infinitesimal character X p; (iii) If the representation o f t of the highest weight S occurs in Aq(X), then

+

S=p(q,X)+

c

npP

P E A (UmJ)

with np non-negative integers. I n particular, p(q, A) is the lowest K-type of A, (A). We note that the unitarity of Aq(X) in the above theorem was proved in [29].In the context of definition of 8-stable parabolic subalgebras, if we take X to be a regular element, then we obtain a minimal &stable subalgebra b = t, n. We call such a subalgebra b a &stable Bore1 subalgebra. The corresponding representation Ab (A) is called a fundamental series representation. It is the (9,K)-module of a tempered representation of G. If G has a compact Cartan subgroup, then Ab(X) is the (9,K)-module of a discrete series representation of G. Moreover, all (g, K)-modules of discrete series representations of G are of this form; this will be important in Section 6. For the proof of the main result of [12] (Section 5) it is only needed that Ab(X) has infinitesimal character X p and the lowest K-type p(b, A) = X 2pn, where pn = p(n n p). These facts are contained in Theorem 4.4.

+

+

+

4.5. Salamanca-Riba’s classification of the unitary dual

with strongly regular infinitesimal characters

As before, let be the complexification of a fundamental Cartan subalgebra t,o of go. Given any weight A E t,*, fix a choice of positive roots A+(A, t,) for A so that

Set

187

Dirac Operators in Representation Theory

Definition: A weight A E

b* is said to be real if A E it;

+ a;,

and t o be strongly regular if it is real and

Salamanca-Riba [24] proved: Theorem: (Salamanca-Riba) Suppose that X i s an irreducible unitary (8, K)-module with strongly regular infinitesimal character A E h*. Then there exist a 6-stable parabolic subalgebra q = I u and a n admissible char-

+

acter X of L such that X is isomorphic to A4(X). 5. Vogan's conjecture and its proof

In this section we explain Vogan's conjecture on Dirac cohomology and a proof of this conjecture. The presentation mostly follows [12].

5.1. Vogan's conjecture Let T be a maximal torus in K , with Lie algebra to. Let b be the centralizer o f t in g; it is a &stable Cartan subalgebra of g containing t. Since 9 = t@p', we get an embedding o f t * into b*. Therefore any element o f t * determines a character of the center Z(g) of U ( g ) . Here we are using the standard identification Z(g) 2 S(Ij)wvia the Harish-Chandra homomorphism ( W is the Weyl group), by which the characters of Z(g) correspond t o the W-orbits in b*. We fix a positive root system A+(t,t) for t in t; let pc = p(A4(t, t)) be the corresponding half sum of the positive roots. For any finite dimensional irreducible representation (y, E7) of t, we denote its highest weight in t* by y again. Vogan (301 made a conjecture which was proved as the following theorem: Theorem: ([12]) Let X be an irreducible (8, K)-module, such that the Dirac cohomology of X is non-zero. Let y be a K-type contained in the Dirac cohomology. Then the infinitesimal character of X is given by y pc.

+

In view of the remarks in 2.5, in case X is unitarizable, we get the following consequence:

J . 3 . Huang and P. PandiiC

188

Corollary: Let X be an irreducible unitarizable (g, K)-module, such that Ker D # 0. Let y be a K-type contained in Ker D. Then the infinitesimal character of X is given b y y p c .

+

In fact, Vogan first conjectured the above corollary and then he saw that the above theorem should be the right generalization t o non-unitary represent at ions.

5.2. An algebraic reduction of the conjecture

Vogan further reduced the claim of his conjecture t o an entirely algebraic statement in the algebra U ( g ) 8 C ( p ) . Let us first recall that in 2.2 we ) Z(~A of f! inside U ( g ) 8 C(p). U ( ~ Aand described a diagonal copy denote the corresponding universal enveloping algebra and its center. It is easy to see that they are also embedded into U ( g ) 8 C(p); namely, if u E U(t) is a PBW monomial, then its image in U ( g )8 C(p) is the sum of u 8 1 and terms of the form w @ a, with w having smaller degree than u. We can now state Vogan's algebraic conjecture that implies the theorem in 5.1. Theorem: Let z E Z(g). Then there is a unique ((2) in the center Z ( ~ of U(tA), and there are K-invariant elements a , b E U ( g )@C(p),such that

z 8 1 = C(z)

+ D a + bD.

To see that this theorem implies the theorem in 5.1, let Z E (X 8 S)(y) be non-zero, such that DZ = 0 and Z @ Im D. Note that both z 8 1 and ( ( z ) act as scalars on Z. The first of these scalars is the infinitesimal character A of X applied to z , and the second is the &infinitesimal character of y applied to C ( z ) , that is, (y p,)(C(z)). On the other hand, since ( z 8 1 - ((2)). = DaZ, and 2 $ I m D , it follows that ( z 8 1 - ( ( z ) ) Z = 0. Thus the above two scalars are the same, i.e.7 = (7 p c ) ( C ( . ) ) . In 5.6 we will show that under identifications Z(g) 2 S(b)w2 P(b*)w and Z ( ~ A ) Z(t) 2 S ( t ) w K 2 P(t*)wK the homomorphism C corresponds to the restriction of polynomials on Q* to t*. Here the already mentioned inclusion oft' into b' is given by extending functionals from t to b, letting them act by 0 on a = p f . It follows that A = y p c , as claimed.

+

+

+

189

D i m c Operators in Representation Theory

5.3. A differential complex induced by Dirac operator Let us first note that the Clifford algebra C ( p ) has a natural Z2-gradation into even and odd parts:

W )= C"P)

@ Cl(P).

This gradation induces a Z2-gradation on U ( g )@ C ( p )in an obvious way. We define a map d from U ( g )@ C(p) into itself, as d = do @ d l , where

do : U ( g )€4 C"P)

--f

U ( 8 )€4 C1(P)

is given by

do(.) = Da - aD,

(2)

and

is given by

In other words, if E , denotes the sign of a , that is, 1 for even a and -1 for odd a , then d ( a ) = Da - E,UD(for homogeneous a, i.e., those a which have sign). We will use the formula for D2 from Lemma 2.2, namely

D2=-flg@1+fle,

+c

to prove that our d induces a differential on the K-invariants in U ( g ) @ C ( p ) .

Proposition: Let d be the map defined in (2) and (3). T h e n (i) d is K-equivariant, hence induces a map from ( U ( g )@ C ( P ) into )~ itself. (ii) d2 = 0 o n ( U ( g )@ C ( P ) ) ~ .

Proof: (i) is trivial, since D is K-invariant. Let a E ( U ( g )@ C ( P ) be ) ~even or odd. Then ci2(a)= ~ ( D U - E , U D= ) D~U-ED,DUD-E,(DUD-E,DUD~)

=D ~ U - U D ~ ,

since obviously E,D = E D , = - E , . From the formula for D2 (Lemma 2.2), we see that a will commute with D2 if and only if it commutes with & ! ., If a is K-invariant, then this clearly holds, as a then commutes with all of UP,). 0

J.-S. Huang and P. PandEiC

190

Thus we see that d is a differential on ( U ( g )@ C ( P ) ) of ~ ,degree 1 with respect to the above defined &-gradation. Note that we do not have a Zgradation on ( U ( g )@ C ( P ) )so ~ that d is of degree 1, i.e., this is not a complex in the usual sense. 5 . 4 . Determination of cohomology of the complex

We want to calculate the cohomology of d. Before we state the result, let us note the following:

Proposition: Z(ta) i s in the kernel of d

Proof: Since D is K-invariant, it commutes with t ~and , thus with U(ta) and in particular with Z(ta). Since z ( t ~c)( U ( g )@ C o ( p ) ) Kthe , claim follows. 0 We now state the following theorem which implies Theorem 5.2.

Theorem: Let d be the diflerential on ( U ( g )8 C ( P ) )constructed ~ above. T h e n Kerd = Z(ta)@Imd.In particular, the cohomology of d is isomorphic to z ( t A ) . The proof uses the standard method of filtering the algebra (the filtration comes from the usual filtration on U ( g ) ) ,and then passing t o the graded algebra. This graded algebra is of course S ( g ) @J C ( p ) .The analogue of our theorem in the graded setting is easy; the complex we get is closely related to the standard Koszul complex associated to the vector space p. Namely, the operator d induced by d on S(g) @ C(p) is given by supercommuting with

and one easily calculates that

c k

d(u @ zi,. . . Zi,)= -2

uzz3@J

zi,. . . Zij . . .z2,.

j=1

Upon identifying C(p) and A(p) as vector spaces and writing

(s(P) 8 A(P)L

~ ( 88 ) C(P)= ~ ( t8)

we see that

d = (-2)id@d,,

Dirac Operators in Representation Theory

191

where d, is the Koszul differential for the vector space p. In particular, 2is a differential with cohomolgy S(e)8 @, which embeds into S(g) @I C(p) by embedding C into S ( p ) 8 A(p) as the constants. Passing t o K-invariants, we see that K e r 2 = S(t)K@ 1 $ I m d on (S(g) 8 C ( P ) ) This ~ . is the graded version of our theorem.

One can now go back to the original setting by an easy induction on the degree of the filtration. The main point is that one can reconstruct an element of Z(t,) from its top term. We refer the reader to our paper [I21 for the details of the above proof. Let us note a consequence, which immediately proves Vogan's conjecture, just put b = a.

Corollary: Let z E Z(g). Then there i s a unique < ( z ) E Z(ta), and there is an a E ( U ( g )@I C ' ( P ) ) such ~ , that z @ 1 = ( ( 2 ) t Da

+ aD.

Proof: This follows a t once from Theorem 5.4, if we just notice that z 8 1 commutes with D (indeed, it is in the center of U ( g )8 C(p)), and being even, it is thus in Ker d. Hence, it is of the form I(wp,a)I.

Hecht-Schmid [9] proved this is also a sufficient condition. Our assumption on the regularity of X with respect to the noncompact roots amounts to the condition that for all a E A+(p)

(A+ P,Q)

> l(P,Q)l.

Therefore] our condition is weaker than that assumed by Langlands and Hotta-Parthasarathy. 10.3. A final remark Bore1 and Wallach [2] proved that for any finite-dimensional representation of G,

~ * (F r) =] @ m ( r , T ) H * ( K, ~ , x, B F ) . ,€E

Dirac Operators in Representation Theory

217

We still assume t h a t rank G = rank K. If the highest weight of F is regular, is uniquely then it follows from a similar argument as in 10.2. t h a t determined as a discrete series Ab(X), and therefore,

x,

d i m H * ( I ‘ , F ) = vol(r\G)d,dimH*(g,K,X,

@ F).

Acknowledgements The research of the first author was partially supported by RGC-CERG grants of Hong Kong SAR and National Nature Science Foundation of China. The research of the second author was partially supported by a grant from the Ministry of Science and Technology of Republic of Croatia. P a r t of these notes was written during authors’ visit t o CNRS, University of Paris VII and IMS at the National University of Singapore. T h e authors thank these institutions for their generous support and hospitality.

References 1 . M. Atiyah and W. Schmid, “A geometric construction of the discrete series for semisimple Lie groups”, Invent. Math., 42 (1977), 1-62, 2. A. Bore1 and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Second edition, Mathematical Surveys and Monographs 67, American Mathematical Society, Providence, RI, 2000. 3. R. Cahn, P. Gilkey and J. Wolf, “Heat equation, proportionality principle, and volume of fundamental domains”, 43-54, Differential Geometry and

4.

5. 6. 7.

Relativity, Mathematical Phys. and Appl. Math., Vol. 3, Reidel, Dordrecht, 1976. H. Cartan, “La transgression dans un groupe de Lie et dans un espace fibre principal”, Colloque de Topologie alge‘brique, C.B.R.M. Bruxelles, (1950), 57-71. W. Casselman and D. MiliEiC, “Asymptotic behavior of matrix coefficients of admissible representations”, Duke Math. Jour. 49 (1982), 869-930. W. Casselman and M. S. Osborne, “The n-cohomology of representations with an infinitesimal character”, Compositio Math., 31 (1975), 219-227. C. Chevalley, The algebraic theory of spinors, Columbia University Press,

1954. 8. B. Gross, B. Kostant, P. Ramond and S. Sternberg, “The Weyl character

formula, the half-spin representations, and equal rank subgroups”, Proc. Nut. Acad. Sci. U.S.A., 95 (1998), 8441-8442. 9. H. Hecht and W. Schmid, “On integrable representations of a semisimple Lie group”, Math. Ann., 220 (1976), 147-149. 10. R. Hotta, “On a realization of the discrete series for semisimple Lie groups”, Jour. Math. SOC. Japan, 23 (1971), 384-407.

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J.-S. Huang and P. PandiiC

11. R. Hotta and R. Parthasarathy, “A geometric meaning of the multiplicities of integrable discrete classes in L2(r\G)”, Osaka Jour. Math., 10 (1973), 211-234. 12. J.-S. Huang and P. Pandiid, “Dirac cohomology, unitary representations and a proof of a conjecture of Vogan”, J . Amer. Math. SOC.,15 (2002), 185-202. 13. , J.-S. Huang, P. PandiiE and D. Renard, “Dirac operators and ncohomology” , in preparation. 14. V. Kac, “Lie superalgebras”, Adv. in Math., 26 (1977), 8-96. 15. A. W. Knapp, Representation theory of semisimple groups: a n overview based on examples, Princeton University Press, 1986. 16. A. W. Knapp and D. A. Vogan, Jr., Cohomological induction and unitary representations, Princeton University Press, 1995. 17. B. Kostant, “Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, the pdecomposition C(g) = End V, @ C ( P ) ,and the g-module structure of Ag”, Adv. in Math., 125 (1997), 275-350. 18. B. Kostant, “A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups”, Duke Math. Jour., 100 (1999), 447-501. 19. B. Kostant, “Dirac cohomology for the cubic Dirac operator”, Studies in memory of I. Schur, 69-93, Progress in Math. vol. 210, 2003. 20. B. Kostant, “A generalization of the Bott-Borel-Weil theorem and Euler number multiplets of representations”, Lett. Math. Phys., 52 (2000), 61-78. 21. R. Langlands, “The dimension of spaces of automorphic forms”, Amer. Jour. Math., 85 (1963), 99-125. 22. J . 3 . Li and J. Schwermer, “Automorphic representations and cohomology of arithmetic groups” 102-137, Challenges f o r the 21st century, Proceedings of International Conference on Fundamental Sciences: Mathematics and Theoretical Physics (Singapore, 2000) World Scientific Publishing. 23. R. Parthasarathy, “Dirac operator and the discrete series”, Ann. of Math., 96 (1972), 1-30. 24. S . A. Salamanca-Riba, “On the unitary dual of real reductive Lie groups and the A4(X) modules: the strongly regular case”, Duke Math. Jour., 96 (1998), 521-546. 25. W. Schmid, “Homogeneous complex manifolds and representations of semisimple Lie groups” Dissertation] UC Berkeley, 1967, 223-286, in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, Edited by P. Sally and D. Vogan, American Mathematical Society, 1989. 26. W. Schmid, “L2-cohomology and discrete series”, Ann. of Math., 103 (1976), 375-394. 27. P. C.Trombi and V. S. Varadarajan, “Asymptotic behaviour of eigenfunctions on a semisimple Lie group: the discrete spectrum”, Acta. Math., 129 (1972), 237-280. 28. D. A.Vogan, Jr., Representations of real reductive Lie groups, Birkhauser, Boston-Basel-Stuttgart, 1981. 29. D. A. Vogan, Jr., “Unitarizability of certain series of representations”, A n n .

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of Math., 120 (1984), 141-187. 30. D. A. Vogan, Jr., “Dirac operators and unitary representations”, 3 talks at MIT Lie groups seminar, Fall of 1997. 31. D. A. Vogan, Jr., “n-cohomology in representation theory”, a talk at “Functional Analysis V I P , Dubrovnik, Croatia, September 2001. 32. D. A. Vogan, Jr. and G. J. Zuckerman, “Unitary representations with nonzero cohomology”, Compositio Math., 53 (1984), 51-90. 33. N. R. Wallach, Real Reductive Groups, Volume I, Academic Press, 1988.

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On Multiplicity Free Actions

Chal Benson and Gail Ratcliff Department of Mathematics East Carolina University Greenville, NG 27858, U.S.A . Email: bensonfQmai1.ecu. edu, [email protected]. edu

Contents 1 Preliminaries 1.1 Algebraic groups 1.2 Regular functions 1.3 Algebraic groups as Lie groups 1.4 Structure theory 1.5 Rational representations 1.6 Highest weight theory 1.7 The contragredient representation 1.8 Decompositions and multiplicities 1.9 Group actions 1.10 Section 1 notes 2 Multiplicity free actions 2.1 Borel orbits 2.2 Quasi-regular representations 2.3 Maximal unipotent subgroups 2.4 S-varieties 2.5 Spherical pairs 2.6 Section 2 notes 3 Linear multiplicity free actions 3.1 Connectivity of G 3.2 Borel orbits 221

223 223 223 224 224 226 228 229 229 230 231 23 1 232 233 235 236 237 238 239 239 241

222

C. Benson and G. Ratcliff

3.3 Fundamental highest weights for a multiplicity free action 3.4 Section 3 notes 4 Examples of multiplicity free decompositions 4.1 GL(n) @ GL(m) 4.2 S2(GL(n)) 4.3 A2(GL(n)) 4.4 SO(n) x C x 4.5 GL(n)@GL(n) n2(GL(n)) 4.6 Section 4 notes 5 A recursive criterion for multiplicity free actions 5.1 GL(n) 5.2 GL(n) @ GL(n) 5.3 GL(n) @GL(n) A2(GL(n)) 6 The classification of linear multiplicity free actions 6.1 Irreducible multiplicity free actions 6.2 Decomposable actions 6.3 Saturated indecomposable multiplicity free actions 6.4 Non-saturated indecomposable multiplicity free actions 6.5 Completing the classification 6.6 Proof outline 6.7 Section 6 notes 7 Invariant polynomials and differential operators 7.1 Polynomial coefficient differential operators 7.2 Invariants in P’D(V) 7.3 A canonical basis for the invariants 7.4 The fundamental invariants 7.5 The algebra 7.6 Section 7 notes 8 Generalized binomial coefficients 8.1 The polynomials qx 8.2 The generalized binomial coefficients 8.3 Eigenvalues for operators in P’D(V)G 8.4 Examples 8.5 Section 8 notes 9 Eigenvalues for operators in P’D(V)G 9.1 Eigenvalue polynomials 9.2 A Harish-Chandra homomorphism for multiplicity free actions 9.3 Characterizing the eigenvalue polynomials 9.4 GL(n) 8 GL(n) yet again 9.5 Section 9 notes References

245 247 248 248 252 254 257 258 260 262 263 264 264 266 266 268 268 270 270 272 273 273 273 276 278 279 28 1 283 284 284 285 288 289 293 294 294 296 298 299 30 1 301

223

On Multiplicity Free Actions

1. Preliminaries Much of the literature on multiplicity free actions is set in the framework of algebraic groups. We begin by summarizing the basic definitions and results we require concerning such groups and their representations.

1.1. Algebraic groups The general linear group GL(n,C) can be viewed as an algebraic group. Letting gl(n,C) denote the space of n x n complex matrices, the group GL(n,C) can be identified with the zero set for the polynomial function p ( A , w ) = d e t ( A ) w - 1 on gl(n,C) x C.This determines the structure of GL(n,C) as an affine variety. One calls G a reductive complex (linear) algebraic group when 0 0

(linear) G is an algebraic subgroup of GL(n,C),and (reductive) C" is a direct sum of G-irreducible subspaces. The classical examples are

GL(n.C),S L ( n ,C>,O(n,C ) ,SO(n,C>,Sp(2n,C) and direct products of these groups. The torus (C")" is a direct product of copies of GL(1, C ) = Cx . A reductive complex algebraic group is connected (in the Zariski topology) if and only if it is irreducible as an algebraic variety. The classical examples are all connected except for O(n,C),which has two components. W e wall assume that our algebraic groups G are connected unless noted otherwise.

1.2. Regular functions

C [ G ]denotes the ring of regular functions on G. This is the coordinate ring of G as an affine variety. More concretely C[G] is the algebra generated by the matrix.entries of G and det-l.

A function f : G

---t

C is regular if and only if

f is the restriction of a

regular function on GL(n,C). So

@[GI2 C[GL(n,C ) ] / I ( G )where , I(G) = {f E C[GL(n,C ) ] : f(G) = 0). Examples 1.2.1: For g = [aij]E G L ( n , C )let z i j ( g ) = aij. Then 0 0

C[GL(n,C)]= C [ z i j , d e t - l ] , C [ S L ( nC)] , = C[zij], C[(C")"] = C[zll,2;1,. . . ,Znn,z;;].

C. Benson and G. Ratcliff

224

1.3. Algebraic groups as L i e groups As an algebraic group, G carries the Zariski topology. As a set of n x n complex matrices, G also has a subspace topology from g l ( n , C ) . In fact, G is a smooth submanifold of gl(n,C), viewed as a real vector space of dimension ( 2 ~ 2 )In ~ .this way G is seen as a (real) Lie group with Lie algebra g = { A E g l ( n , C ) : etA E G for all

t

E

W}.

Moreover g is closed under multiplication by i and hence is a complex Liesubalgebra of g l ( n , C). Alternatively one can define the (complex) Lie algebra g for G as g = { A E gl(n,C) :

f E I(G)

+ Af E I ( G ) }

where

When G is reductive, g is a complex reductive Lie algebra. This implies that

B = 2(B) e3 9’ where ~ ( g )denotes the center of g and the derived subalgebra 0’ = [g,g] is semi-simple, a direct sum of simple ideals. g’ is the Lie algebra of the commutator subgroup G’ = (GIG ) .

1.4. S t r u c t u r e theory A maximal connected solvable algebraic subgroup B of G is called a Borel subgroup. The following facts concerning such subgroups are well known: a 0

Any two Borel subgroups are conjugate in G. Given any Borel subgroup B , there is an opposite Borel subgroup Bwith the property that B-B is Zariski dense in G and contains an open neighborhood of I . B is the semidirect product B = H N of its commutator subgroup N = ( B ,B ) with a maximal torus H in G. The group N is a maximal unipotent subgroup of G. The Lie algebra b of H is a maximal abelian subalgebra of g. For Q E b*

let ga = {Y E g : [ X , Y ]= a ( X ) Y for all X E b}.

On Multiplicity Free A c t i o n s

225

Then

is the set of roots for g (relative to I)). Each root space g a is one dimensional and

g =b@

@ ,€A

There is a subset A+ of A , called the positive roots, such that N has Lie algebra

Now

0 0

A = A + U (-A+), b = b @ n is the Lie algebra of B, and N - has Lie algebra n- = gPa.

eaEA+

For each a c A+ there are elements which form an sl(2)-triple:

[Ha,X,] = 2X,,

X,

[ H a ,X-,] = -2X-,,

E go,

X-, E

[X,, X-a]

g-,

Ha E b

= Ha.

For each a E A we have a root reflection

sa : b*

-+

b*,

sa(X) =

x - (X,a)a

where

( & a )= X(Ha). The Weyl group W = W(g,q)is the subgroup of GL(b*) generated by { s a : a E A } . It is a finite reflection group that acts by permutations on the set A.

Example 1.4.1: The standard Bore1 subgroup in G = GL(n,C ) is

the group of invertible upper triangular matrices. We have

B, = H , N,

C. Benson and G. Ratclaff

226

where H , denotes the diagonal matrices in G L ( n , C ) and N , denotes the unipotent upper triangular matrices. The opposite Bore1 subgroup for Bn is

B, = H,Ni where B; and N; are the invertible and unipotent lower triangular matrices respectively. The Lie algebra ljn of H, is the set of all diagonal matrices. Letting gi E b* denote the functional Ei(diag(zl,.. . , 2,)) = Zi

one has roots A = {

~ i ~j :

i # j } and positive roots A+ = {

~ i ~j :

i <j).

For a = ~i - ~j E A+ we have

X, = Eij, X-, = Eji, Ha = Eii

- Ejj.

The root reflection s, satisfies s , ( E ~ ) = ~ ~ ( where k ) T E S, is the transposition that interchanges i with j . Thus the Weyl group is isomorphic to sn.

1.5. Rational representations

Definition 1.5.1: Let (a,V ) be a representation of G. (1) (a,V ) is said to be rational (or regular) if it is finite dimensional and its matrix coefficients 9

E(a(g)v),

E E v*,vE

v

all belong to @.[GI. ( 2 ) (u,V )is locally rational (or locally regular) if dim(V) = 00 and for any finite dimensional subspace F of V there is a u(G)-invariant subspace W with F c W c V for which alW is rational. h

We let G denote the set of equivalence classes of irreducible rational representations of G and sometimes write V, for the representation space of aE

e.

Note that subrepresentations of rational (or locally rational) representations are rational (resp. locally rational).

On Multiplicity Free Actions

227

Example 1.5.2: As G = C x is abelian, its irreducible representations are given by characters p : C x 4 C x . For such a character to be rational we require p E C[G] = @[z,l/z]. So p is hoIomorphic on C x and hence determined by its restriction to the unit circle T.One concludes that (CXT={p, : n E Z }

where p,(z) = 2 , . The character

gives a representation of Cx which is not rational. This example illustrates the:

Weyl Unitarian Trick Rational representations of G are determined by holomorphic extension from a maximal compact connected subgroup K of G (viewed as a real Lie group). It now follows from the representation theory for compact Lie groups that rational representations are completely reducible. Moreover, the Unitarian "rick establishes a bijection between and the set of unitary equivalence classes of irreducible unitary representations of K . So one can work entirely in the compact group setting, should one so prefer. Maximal compact subgroups for the classical groups GL(n.C), S L ( n ,C), O(n,C), SO(n,C), Sp(2n,C) are

k

U ( n ) ,sU(n),o(n,R), SO(n,a), Sp(2n) = U ( 2 n )n sp(an,C ) . Each rational representation (a,V )of G is smooth. That is, t H a(etx)v is a smooth map R -+ V for each u E V , X E 0. Thus we obtain a derived representation

of the Lie algebra 0 on V . (Note that we are using the same notation for the representation CT on both the Lie group and the Lie algebra.) When there is no ambiguity, we will denote the action of G (or g) on V by o ( g ) u = g . u (or a ( X ) u = X

. v).

C. Benson and G. Ratclaff

228

1.6. Highest weight theory Let B = H N be a Borel subgroup in G and (a,V) be a rational representation. By the Lie-Kolchin Theorem, there are non-zero u(B)-eigenvectors. That is, there are vectors v # 0 in V such that

u(b)v = $(b)v for all b E B , where ?I, : B and hence 0 0

--f

C x is a regular character. As N = (B, B ) , we have

$IN

=1

{v E V : v is a B-eigenvector} = V N ,the N-fixed vectors in V, and $ is determined by $ 1 ~E @.

Highest weight theory asserts that u is irreducible

d i m ( V N ) = 1.

h

For each u E G, there is a non-zero B-eigenvector v, E V,, unique up to scalar multiples. This is the highest weight vector for u. The corresponding character $ : B -+ Cx can be differentiated to give a functional X on the Lie algebra 6, with X(n) = 0. We have X . v , = X(X)v, for all X E 6. As noted above, X is determined by its value on the Lie algebra b of H . The functional X in b* is the highest weight for u. We can extend X to b (or 6-) by taking X(n) = 0 (resp. X(n-) = 0). Thus v, is, up to scalars, the unique vector in V with

X . v,

= X(X)v, for all

X

E

6.

Highest weight theory asserts, moreover, that is determined up to equivalence by its highest weight. Given a representation of G with highest weight A, we will denote the corresponding representation as V,.(Keep in mind that this all depends on the initial choice of a Borel subgroup.) For an element b in the subgroup B, we will denote the corresponding character by b H b X . The highest weights for G L ( n , C ) (with respect to the standard Borel subgroup) are n.

(diag(h1,. . . ,h,)

dihi : d l , . . . ,d, E Z with d l >_ dz 2 . . . >_ d,}.

H

i= 1

On Multiplicity Free Actions

229

1.7. The contragredient representation Given a representation of G on V, we define the contragredient representation of G on V' by: g . ((v) = ((g-'

. w) for 5 E V*, w E V .

If V has highest weight vector v with highest weight X with respect to some Borel subgroup B , then V* has a highest weight vector w* with weight -A with respect t o the opposite Borel subgroup B-. 1.8. Decompositions and multiplicities Let (p, W ) be a rational representation of G and u E sum decomposition

e. One has a direct

W=$W" U€G

of W into a-isotypic components

W" = c { V : V is a p(G)-invariant subspace of W with plV N u ) = C { T ( V u ):

T : V,

-+

W intertwines u with p}.

Then

W"

21 m(u,p)V,=

v, @ . . . €3 V" m(a>P)

as G-modules where the multiplicity m(a,p) of u in p is given by m(o,p) = dim(W")/dim(V,)= dim(HOm~(Vu1 W)).

(Homc(V",W ) is the space of linear maps V, P.

+

W intertwining u with

1

The multiplicity m(c1p ) can also be characterized using highest weight theory. Let B be a Borel subgroup of G and X be the highest weight for u. Then

m(a,p) = dim ( W B > A ) where wBJ = {W E

W

:

b.w

= b'w}

is the space of weight vectors in W with weight A.

230

C. Benson and G. Ratclaff

1.9. Group actions

We use the notation G:X to indicate that there is a rational action of G on a variety GxX This means G x X

-+ X

4

X,

X,

(g,x) Hg . x .

is a morphism of algebraic varieties satisfying

( g h ). IC = g . ( h . x) and e .x = x. The variety X may be affine or, more generally, quasi-projective. That is, the intersection of a closed with an open set in CP". It follows that i f f E @ [ X I the , regular functions on X ,then 2 H f(g-1

. x)

is a regular function. Thus we obtain a representation p of G on @ [ X I ,

( p ( g ) f ) ( z )= ( 9 . f)(.)

=

fb-' . .).

Lemma 1.9.1: T h e representation p of G o n C[X] is locally rational. Remark 1.9.2: Linear actions G : X will be our main concern. That is, X will usually be a vector space on which G acts by a rational representation. Then @[XIis the algebra of polynomials on the vector space X and one can give an easy proof of Lemma 1.9.1. Let ? k ( X ) denote the space of polynomials on X homogeneous of degree k. For any finite dimensional subspace F of C [ X ] we , have F c W where n

for n sufficiently large. Now W is a finite dimensional, G-invariant subspace and we have a rational representation. Now for

0

E

define the a-isotypic component C [ X ] "as:

@[XI"= C { T ( V u ): T

E

Homc(Vm, @ [ X I ) } .

Lemma 1.9.3: W e have a n algebraic direct sum

C [ X ]= @ C[X]". v€l3

231

O n Multiplicity Free Actions

Thus

C[X]

= @ m(a,@[X])Vg a&

as G-modules, where the (possibly infinite) multiplicity

C[X]) of

m(g,

(T

in

@[XIis m(a,@.[XI)= d i m ( H o m ~ ( V C[X])) a, = dim (C[X]'*'), where X is the highest weight for the representation a. We can now introduce our principal objects of study.

Definition 1.9.4: G : X is a multiplicity free action if m(u,@[XI)5 1 for each a 6

e.

Given an action G : X and an algebraic subgroup H of G, one has

H : X multiplicity free

G : X multiplicity free.

(1)

Indeed, if G : X fails to be multiplicity free, then some representation a E occurs in @[XIwith multiplicity greater than one. Thus all irreducible constituents of a l occur ~ with multiplicity greater than one. So H : X also fails to be multiplicity free.

1.10. Section 1 notes The material in this section is standard and there are many excellent references. One is the book [17] by Goodman and Wallach. See Chapter 1 and Appendix D in [17] for further details on foundational material concerning algebraic groups and Lie groups. See Section 11.3 of [17] for proofs of the assertions concerning Bore1 subgroups. Our treatment of the isotypic decomposition for @[XIfollows Section 12.1 in [17].

2. Multiplicity free actions This work is mainly devoted to the study of linear multiplicity free actions. In this section, however, we consider actions G : X in the more general context of algebraic varieties. Our main purpose is to describe some noteworthy non-linear examples.

232

C. Benson and G. Ratclaff

2.1. Borel orbits There is a simple criterion for multiplicity free actions.

Theorem 2.1.1: If a Borel subgroup B in G has a (Zariski) dense orbit in X then G : X i s multiplicity free. Proof: Suppose that B . IC, is dense in X. Let CT E occur in @[XI(that is m(a,C[X]) > 0) and let A be the highest weight for CT. Let f 1 , f i E @[XI be two B-highest weight vectors with weight A. One has

f j is regular and B . xo dense in X, we see that f j is completely determined by the value fj(x,). In particular, fj(x0) # 0 (as f j # 0) and we can write

As

So the space of A-highest weight vectors in hence m(a,@[XI)= 1.

C[X] is one

dimensional and 0

Since any two Borel subgroups are conjugate in G, the criterion in Theorem 2.1.1 does not depend on the choice of Borel subgroup B . Suppose that B . xo is dense in X. Since B is connected, it follows that X must be an irreducible variety and that B . x , is a Zariski open set. Thus X\(B.x,) is a closed set which contains no open subsets in X. We conclude that there is only one dense open B-orbit in X . Conversely, if G : X is a multiplicity free action with X an irreducible a f i n e variety then X contains an open (hence dense) B-orbit. We will prove this result for linear multiplicity free actions in the following section. (See Theorem 3.2.8 below.) One special case of the converse admits, however, a direct proof. This is the case where G is an algebraic torus G = A 2 (C")". In this case the Borel subgroup is A itself and one has the following.

Proposition 2.1.2: Let X be a n irreducible a f i n e variety and A be a torus. If A : X is multiplicity free then there is a n open (hence dense) A-orbit in X. Proof: One can choose weight vectors f l , . . . , f, in @[XIthat generate C[X] as an algebra. Since A : X is multiplicity free the weights { A i , . . . , A,}

O n Multiplicity Free Actions

233

for f 1 , . . . , fr must be linearly independent. Choose a point xo E X for which f j ( x o )# 0 for 1 5 j 5 r. Define a map p : @[XI+ @[A]by ( p f ) ( a ) = ( a . f ) ( x o )= f(a-l

We claim that p is injective. Indeed, for f = f

'

xo).

=f y '

'

. .f y r

one has

( p f ) ( a ) = am'X' . . . U r n F X , f ( X o ) = a r n X f ( x 0 ) , and hence (pf)(a) =

c

CmamXfm(xo)

m

for f = Cmcmfm E @[XI.If ( p f ) ( a ) = 0 for all a E A , then linear independence of the X j ' s implies that c, = 0 for all m. Thus every regular function on X is determined by its restriction to A . x o . It follows that A . xo is open and dense in X . 0 2.2. Quasi-regular representations

We continue to let G denote a reductive complex linear algebraic group. The left and right actions of G on X = G

-

0

g x = gx, and g.x=xg- 1

give rise to the left and right regular representations

.

L ( g ) f ( x )= f ( g - l x > ,and R ( g ) f ( 5 )= f (x g ) respectively

of G on @[GI.For any algebraic subgroup H of G we let

@ [ G / H= ] @ [ G ] R ( H )@[H\G] , = C[G]L(H) and define the left (resp. right) quasi-regular representation of G as the restriction of L to C [ G / H ](resp. R to @[H\G]).The representations L and R of G on @ [ G / Hand ] @[H\G] are equivalent via the intertwining operator

T : C [ G / H ]4 @[H\G], T ( f )= f, ( f ( x )= f ( x - l ) ) . In fact the homogeneous space G / H is a smooth quasi-projective variety with coordinate ring @[GIN]. If H is a reductive or normal subgroup then G / H is an affine variety. We remark that one can have @ [ G / H = ] @ even when d i m ( G / H ) > 0. This situation occurs whenever G / H is a projective variety, in particular when H is a Bore1 subgroup of G. In any case, the

C. Benson and G. Ratclzff

234

left action G : (G/H) is rational and gives rise t o the left quasi-regular representation. Similar remarks apply for H\G and the right quasi-regular representation. The isotypic decomposition for the quasi-regular representations is given by Frobenius Reciprocity:

Theorem 2.2.1: A s a G-module we have @[G/H] % @ dim(V$)V,. U€Gc

I n particular, C[G] "- @,dim(V,)V,.

Proof: Lemma 1.9.3 applies here since G : (G/H)is a rational action. It % V 3 . For this, one verifies that suffices t o show that Homc(V,, @[G/H]) @ :

v?

+

Homc(Vu,@[G/H]), @ ( E ) ( v ) ( g=) E(c(g-')v)

is an isomorphism with inverse

Corollary 2.2.2: The action of G o n G/H (or o n H\G free if and only i f dim(V/) 5 1 for all a E

e.

) is multiplicity

Note that if H1 and H2 are algebraic subgroups of G with H1 c H2 then V$ c V$. Thus if G : (G/H1)is a multiplicity free action then so is G : (G/H2). The proof of Theorem 2.2.1 shows that the a-isotypic component in @[GIfor the left regular representation is = {T(v) :

2,

E V,,T E Homc(V,,C[G])}

= { Q ( E ) ( v ) : v E VU,E E V,.} = : 21 E V,,[ E v;},

where mc,v(g)= O } = { a € A I -~r$!Ap}, one has

c

[=?J@

u=

gal

a€AL

c

ga.

aEAu

Letting u- = CaEAu g P a we have u c n, u-

c n-

and

g=peu- =I@u@u-.

We will make extensive use of the following map (which depends on h ) . Let

V, = { Z

E

V

:

h ( z )# 0},

a principal open set in V , and define

x = X h V,

4

g*

via

Lemma 3.2.1:

x is P-equivariant.

Proof: Note that the action of P on V preserves V,.Also P cts on * via the coadjoint action ( 9 .N X ) = f ( A d ( g - l ) X ) .

For g

E

P one has

Thus X(g-'

. z ) ( X )= (9-l . x ( z ) ) ( X )so x is P-equivariant.

0

On Multiplicity FTee Actions

243

Lemma 3.2.2: The stabilizer of X in P is Stabp(X) = L . Moreover, Stab,y(X) = { e } . Proof: Here X E Q* is regarded as a functional on all of g with X(n) = (0) = X(n-). For X E p one has X . X = 0 if and only if X[X,g]= ( 0 ) . Clearly g stabilizes X because [b, g] c n n-. Also for any a: E AL and any

+

y

=

+Cpe*

c p x , E 9, X([X,, Y ] )= c-,X(H,)

= c-,(X,

a ) = 0.

+ xaEaL

Thus X , stabilizes X for each a: E AL. So I= b g, stabilizes A. We have shown that L c Stabp(X). For the reverse inclusion, let

stabilize A. Then for

P E A,,

+,

X-pl = b p X ( H p ) = b p ( X , P).

But (X,P) > 0, and hence b p = 0. Thus Stabp(X) c L and the stabilizer of 0 X in U is trivial.

Proposition 3.2.3: The image of in g*.

x : V, -+g*

is X

+ p l , a single P-orbit

Proof: For all z E V , and X E p ,

Thus x ( z ) - X annihilates p and x(Vo)c X+p'. Note that dirn(X +p') = dirn(u).As U is unipotent and acts without stabilizer on X we conclude that U.XisbothopenandclosedinX+p'andhenceU.X=X+p'.As P = LU and L stabilizes X we also have P . X = X + p'. As x is P-equivariant we must have P . X = x(Vo). 0

Corollary 3.2.4: X = x ( z o )f o r some zo E V,. Let

c = Ch = x-'(X)

= (2 E

The group L acts on C because

v, :

x ( z ) = A}.

x is P-equivariant

Lemma 3.2.5: U x C E V, via ( g , z ) H g . z .

(4)

and L stabilizes A.

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C. Benson and G. Ratcliff

Proof: Given z E V, one has x ( z ) = g . X for some g E U in view of Proposition 3.2.3. Thus X = g-l . x ( z ) = x ( g - l . z ) , so 9-l . z E C. Now (9,g-I . z ) E U x C maps to z . To see that ( 9 ,z ) H g . z is injective, suppose that g . z = 9’. z’ for some g , g’ E U , z , z’ E C. Applying x gives g . X = g . x ( z ) = 9’. x(z’) = g’ . x and thus g-lg‘ stabilizes A. As that g = g‘ and thus also z = z’.

U acts without stabilizer on X it follows 0

Lemma 3.2.6: There is a unique parabolic subgroup P = Ph of lowest possible dimension. Moreover, P c Ph’ f o r all B-highest weight vectors h’ E C [ V ] .

Proof: Suppose that h E C [ V ]is a B-highest weight vector with prime decomposition h =pyl ...pyre The proof of Lemma 3.1.1 shows that Ph acts by a character on each irreducible factor p j . Thus

Ph = Ppl n . . . n pP,. Now assume that h, h’ E C [ V ]are two highest weight vectors and that Ph has minimal dimension. Let P I , . . . , p , be the prime factors of h and 41, . . . ,qs the prime factors of h’. Letting h” = p l . . .p,ql . . . qs one has

PhjJ = Ppl n ” ’ n Ppr n Pql n “ ’ n Pq8 = Ph n Ph,.


m. When n = m, the action SL(n) @I SL(n) fails to be multiplicity free. Indeed, det : M,,,(C) -+ C is a non-constant ( S L ( n , C )x S L ( n , C ) ) invariant polynomial. So C[Mn,,(cC)] contains two copies of the trivial representation of S L ( n ,C) x S L ( n ,C ) .

+

C. Benson and G. Ratcliff

252

0

When m = 1 this example reduces to the action of G L ( n , C ) on C" by the defining representation (or its twisted form, the contragredient representation on ( C " ) * ) . In this case the decomposition in Theorem 4.1.1 reduces to M

C [ z l , .. . , zn] = @pk(Cn). k=O

The action has rank one with fundamental highest weight (11)and zf is a highest weight vector in Z)k(Cn).

4.2. S2(GL(n)) Next we consider the action of GL(n,C) on the symmetric 2-tensors S2(C2) via the symmetric square of the defining representation. Identifying S2(Cn) with the complex n x n-symmetric matrices

Sym(n,C) = { A E Ad,,,(@)

:

At = A}

our action reads

g . u = g v g t.

(11)

As in the preceding example we prefer to twist the action by g This gives

H

(g-l)t.

g'u = (g-l)tug-l,

g.p(u) = p(gtug) for u E S y m ( n , C ) ,p E C [ S y m ( n , ~ ) (12) As for GL(n) @ GL(m),twisting ensures that all weights X = (XI,. . . ,A,) that occur in C [ S y m ( n C , ) ] are non-negative. Again we let

y 2 . . . 0).

If z E V is a A-highest weight vector then g . t = @z if and only if Sx = 0. If Sx = 0 for all highest weights X E a, then G acts by scalars on every weight space in V . In this case, we essentially have a torus action. In particular, G : V is multiplicity free if and only if H : V is multiplicity free. This happens if and only if the set 9 of weights for H : V is linearly independent. Suppose that X E 9 is a highest weight with Sx # 0. Let zo E V be a A-highest weight vector and let fo E V* be the corresponding (-A)-lowest weight vector normalized so that fo(zo)= 1. Let P = Pfoand C = Cfo be as in Equations 3 and 4. Now P- = LU- is the opposite parabolic subgroup to P = LU and P- = Pzo. From the definition of C one has that z E C if and only if f o ( z )# 0 and ( X . fo)(z)= - X ( X ) f o ( z ) for all X E g = p u-. But X . fo = - X ( X ) f o for X E p, so

+

C = { z E V : f o ( z )# 0 and (u- . fo)(z)= O}. Hence C is a open set in the subspace

w = (u-

. fo) I

of V . Recall that C is invariant under the action of the Levi component L of P . As U . C = V, = { z E V : f o ( t ) # 0} by Lemma 3.2.5, we see that there is an open B-orbit in V if and only if there is an open L n B-orbit in C. Equivalently, G : V is multiplicity free if and only if L : W is multiplicity free. We know, moreover, that I = I) CacA(L) ga where

+

A(L) = { a E A : (X,CY) = 0)

= A\Sx.

The positive roots for L are A+(L) = A+\Sx. Each of the root vectors { X - a : a E Sx} c u- acts non-trivially on fo, so the set of weights in

O n Multiplicity Free Actions

263

+

u- . fo is {-A a : a E Sx}. Thus the set of weights in W = (u- . ')of is *\(A - Sx). In summary, we have a recursive algorithm that begins with the pair

(A: = A f l 90= 9) where 0

0

A+ is the set of positive roots for G, and

9 c b* is the set of all weights for the representation of G on V , listed with multiplicity.

Given a pair (A:, 0 0

0

do the following:

For each highest weight X E 9 nlet SX = { a E A: : ( & a )> O}; If Sx = 8 for all highest weights X E then G : V is a multiplicity free action if and only if Q n is linearly independent; Otherwise, choose a highest weight X E Q n with Sx # 8 and apply the above steps to the pair (A:+ll 9n+l)= (A$\Sx, !Pn\(X - Sx)).

an,

Eventually all the SX'Sare empty and the algorithm terminates at the second step above. To illustrate this method, we revisit some of the examples described in Section 4. 5.1. GL(n)

Here G

= G L ( n , C ) acts

A$ = A+ = {

on V = Cn as usual. One has

~ i ~j

:

i <j } ,

90= 9 = ( ~ 1 , .. . , E ~ } .

We know e l = (II0, . . . I 0) is a highest weight vector in V with weight and so

s,, = {El

-

E2,.

.., E l

~ 1 ,

- En}.

The coordinate vector z1 E V * dual to el has parabolic subgroup P = Pz, with Lie algebra p spanned by {Ell}U {Eij : i > 2). The Levi component is L = GL(1,C) x GL(n - 1,C). The nilpotent u- = Span{Elj : j 2 2) gives u-21 = Span(z2,. . . zn} and hence W = (u-21)' = Cel. Thus we have

AT = A$\S,, = {

~ i ~j

: 2

5 i <j},

91 = Q O \ { E ~ ~ .

..,

E ~ = } (~1).

Since S,, = 8, the process terminates. One concludes that G : V is multiplicity free since 91 is a linearly independent set.

C.Benson

264

and G. Ratcliff

In practice, it is not necessary to identify L and W at each stage in the induction. We illustrate this in the next example. 5.2. GL(n) C3 GL(n) Here G = GL(n,C) x GL(n,C) acts on V = Cn @ C n . Now

A , + = A + = { E ~ - E: ~I ~ ~ < ~ < ~ ) U { E : - E :; i ~ i < j i ~ ) and : 1 Ii I n , 1 Ij I n } .

90=9= {Ei+E; X

= ~1

+ E:

is the only highest weight in @o. One has

SEIfE; = { E l - E 2 , . . . , E l - E n }

u {&; - E'2,. . . , &'1 - E k } .

Thus

A ; = { E ~ - E ~:

~ < Z < ~ < ~ } U ( E : - E ;

:

25i<j’for all dominant weights A. Proof: For

D E P V ( V ) G ,b E B

and h a A-highest weight vector, we have

b . ( D h ) = D ( b . h ) = D(b’h)

= bXDh.

Thus PD(V)Gpreserves the space C[V]”>X of A-highest weight vectors. and consider Next let h l , h2 be two A-highest weight vectors in @[V] the finite dimensional G-invariant space

X

= Span(G . h i )

+ Span( G . h ~ ) .

Since S p a n ( G . h l ) and S p a n ( G . hz) are equivalent as G-modules, there is some T E E n d c ( X ) with T(h1)= hz. By Lemma 7.1.1 there is an operator D E P D ( V ) with D]X= T . Now Dh E P D ( V ) G and

D‘hl =

L

IC. D(k-’ . h i ) d k

=L

k . T ( K 1. h l ) d k

=

J,T ( h 1 ) d k

(as k-l . hl E X )

(as T is G-invariant)

= T(h1) = hz.

This shows that PD(V)Gacts irreducibly on C[VIByX

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C. Benson and G. Ratclzff

Theorem 7.2.3: G : V is a multiplicity free action if and only i f PD(V)G is abelian. Proof: Let G : V be a multiplicity free action and

@[VI= @ P A X€A

be the decomposition of C[V]into pair-wise inequivalent G-irreducibles. Schur’s Lemma ensures that any operator D E PD(V)G must preserve each PA and acts on PA as multiplication by some scalar. It follows that any two operators in PD(V)Gcommute. Conversely, suppose that PD(V)Gis abelian. As PD(V)G acts irreducibly on C[VIBgXwe must have dirn(C[V]BiX) 5 1 for all dominant weights A. Hence G : V is multiplicity free. 0 Thus when G : V is multiplicity free, both algebras @[V@ V*IGand PD(V)Gare abelian. Although S and o are not algebra maps, they induce algebra isomorphisms

@[VI3 V*]G2 gr(@[VG3 V*]G)2 gr(PD(V)G) between the associated graded algebras. Concretely, this means that although one generally has p ( z , a)&, a) # ( p q ) ( z ,a), the operators p ( z , a ) q ( z , a ) and ( p q ) ( z , d ) have the same top degree terms. Here “degree” in PD(V) is defined using the filtration PD(‘)(V) from Section 7.1. In particular, zadP has degree la1 [PI.

+

7.3. A canonical basis for the invariants Suppose that G : V is a (linear) multiplicity free action. The trivial representation of G occurs in @[V]on Po(V) = @, the constant polynomials. As the representation of G on @[V] is multiplicity free, it follows that @[VIG= C. That is, there are no non-constant G-invariants in @[V]. Because of the connection with G-invariant differential operators it is, however, of interest to study the G-invariants in @[V@ V * ] . Let

@[V] = @PA XE A

denote the multiplicity free decomposition of @[V] under the action of G. Then

@[V@ V ” ]= @[V] @ @[V]* = @ PA X,X’EA

P;,,

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O n Multiplicity Free Actions

where the subspaces

PA8 P:, are G-invariant in @[V@ V * ] Thus .

@[V@ V*]C=

@

(PA @ PX*,)".

X,X'EA

But

H ( p +-+< ( p )f). by Schur's Lemma. The first isomporphism is given by f The element in PA 8 Pi that corresponds to IpA under the isomorphism PA 8 Pi "= Horn(P~, PA)is

c dx

px

=

fj

8 fj'

(18)

j=1

where dx = dirn(Px) and {fj : 1 5 j 5 d x } is any basis for PA with dual basis {f;}. Thus we have shown that

@[V@ V*]G= $(PA 8 P;)" = @@px. XEA

XEA

So {FA1 X E A} is a basis for @[V@V*IG. As Equation 18 does not depend on the choice of basis { fj} for PA,the basis {pX I X E A} for @[V@ V*IG is canonical. Applying Wick quantization we obtain a canonical basis for PV(V)G. We call the polynomials pX the unnormalized canonical invariants. To achieve some simplification in formulae to be derived below, we also introduce the (normalized) canonical invariants 1(dx = d i m ( P x ) ) . dX In summary we have proved the following. PA = -PA,

Proposition 7.3.1: { p ~: X E A} and { p ~ ( z , a ): X E A} are canonical vector space bases f o r @[V@ V*IGand PD(V)G respectively.

7.4. The fundamental invariants Now let r be the rank of the multiplicity free action G : V and

A' = { A i , . . . , A T ) be the set of fundamental highest weights. Recall that A mrAT I rn E W}. (See Proposition 3.3.1.)

= (rnlX1

+ .. . +

280

C. Benson and G. Ratclzff

Definition 7.4.1: The fundamental invariants for G : V are (71,. . . , r,.} where rj = PA,.

For X E A let 1x1 E N denote the degree of homogeneity of PA. That is, PA c Plxl(V). Then the canonical invariant p x is homogeneous of degree 214. For any weights p , v E b* we will write

when v - p is a sum of positive roots.

Lemma 7.4.2: For any A, p E A there are values c,

= CA,,,,

f o r which

PAP, = c c u p w , U

where the sum is over all v E A with 1v1 = (XI # 0.

+ 1pl and v 3 X+p.

Moreover,

CA+,

Proof: The product pxp, is G-invariant and belongs t o P l ~ l + l ~ l ( V8) PIAl+l,l(V*). As the p,’s form a homogeneous basis for CIV @ V*IG we conclude that PAP, =

c

CUP,

l4=I4+IP

for some values c,. Let { fj},”L, and {h}j$’l be bases of weight vectors for PA and Pp so that f l , hl are highest weight vectors. We know that all other weights in an irreducible representation space precede the highest weight in the partial ordering defined above. We have

The @[V]-components f i h j in this sum are weight vectors with weights X i + p j 3 X+p. It follows that c, = 0 unless v + X+p. Moreover, the term f l h l @ f,’h; contains the (A + p)-highest weight vector f l h l . We conclude that CA+, # 0. 0

Theorem 7.4.3: C[V @ V*IG = C [ y l , . .. ,r,.]. That is, @[V@ V*IG is a polynomial ring freely generated b y the fundamental invariants.

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O n Multiplicity Free Actions

Proof: Given m E N', let X = mlXl+. . . + mrXr E A. Lemma 7.4.2 shows that

y" = yl"'

...

y r T

= P?

-P?

= axpx

+CaYPY Y+X

for some coefficients ay with ax # 0. As { P A } is a basis for @[V@ V*IG,we 0 conclude that {y" I m E N ' } is also a basis for @[VCB V*IG.

Corollary 7.4.4: PD(V)G is a polynomial ring freely generated by ( 0 3 = yj(z,d) : 15 j 5 r}. Proof: From Theorem 7.4.3 we see that { y " ( z , d ) : m E W} is a basis for the vector space PD(V)G.Also, given m = ( m l ,. . . , m y ) ,

D"

=

07' ...OFT= ~ 1 ( ~ , d ) ".'. . y r ( ~ , 6 ' ) " '

+

differs from ym(z,d) by an element of PD(21xl-1)(V), where X = mlX1 . . . m,.X,.. By induction on degree in PD(V) we conclude that {D" : m E W} is a vector space basis for PD(V)G.Thus PD(V)G= @ [ D 1 ,.. . , D,].0 7.5. The algebra

@[VR]~

An alternative viewpoint on @[V@ V*IGwill prove useful. Let K denote a maximal compact connected Lie subgroup of G and (., .) be any K-invariant positive definite Hermitian inner product on V . The conjugate-linear vector space isomorphism

v + v*,

w H w* = ( . , 2 1 )

is K-equivariant (but not G-equivariant). In view of the Unitarian Trick we obtain an algebra isomorphism

@[VR]K = C[V @3 V ] K E @[V@ V*]K= @[V@ V*]G.

(19)

v

Here denotes V with the conjugate complex structure and VR is the underlying real vector space for V . Introducing coordinates ( 2 1 , . . . , zn) on V with respect t o an orthonorma1 basis, one has C [ V ]= @ [ z l , .. . , zn]. This polynomial ring also carries an inner product, namely (P, 43 = ( P ( W (0) = ( m p ) (011

the so-called Fischer i n n e r product. Here p ( d ) = p ( & , . . . , an) for p = C , c,P, ij(z) = C , c z " .

p ( z 1 , . . . , zn) and for q ( z ) =

C. Benson and G. Ratclzff

282

Thus

( z a ,zp)F = &,,pa! = 6%P a l ! .. . a,! for multi-indices a = (a1,.. . ,an),/3 = (PI,. . . ,p,).

Lemma 7.5.1: The subspaces {PA : X E A} in @[V] are pair-wise orthogonal with respect to the Fischer inner product. Proof: This follows from the fact that K c U ( V ) and U ( V ) preserves (., .)F.This can be seen using an alternative formula for the Fisher inner product: 1 ( p , q)F = /p(z)~e-~’lZdz&. (20) In ( 2 0 ) , n = d i m @ ( V ) ,1 . ~ 1 =~ ( z , z ) and “dz&” denotes Lebesgue measure on Vw normalized using (., .). We see that (k . p , k . q)F = ( p ,q)F for k E V ( V )via a change of variables in ( 2 0 ) , since both 1 . ~ 1 and ~ dzdZ are V ( V ) invariant. To establish (20) it suffices to verify that

J

z a ~e-lz12 ’ d z o= ~ Ir, 6,,, a!.

For this, use polar coordinates

zj

The integral in 0, is zero unless

= rjez’,

aj

to write

= a;, in which case one has

Using isomorphism ( 1 9 ) we can regard the canonical invariants p~ as elements of @[VjIK = @[zl,.. . , z,, z1,. . . ,Z,lK. We have

where { e j } is any orthonormal basis for PA (with respect to (., .)F).Note that each p~ E @[VwlK is real valued and non-negative. We later require the formula

On Multiplicity Free Actions

283

Indeed C,x,=k dxpx(z,Z) = C, le(z)I2 where e ranges over an orthonorma1 basis for ? k ( v ) obtained by concatenation of orthonormal bases for {PA : 1x1 = k}. The sum is, however, independent of the basis and we can use {za/&J : la1 = k} to compute

as stated. On @[Vi]= product

@[z1,.

. . , zn,Z1,. . . ,Zn]

we consider the Hermitian inner

(P, 4 ) . = ( P P , 3if)( 0 ) = (.(a, B ) P ) (0). This “doubled Fischer inner product” is determined by (zaZ@,za’z@’)*= ba,a’bp,p’a!p!.

Proposition 7.5.2: {px I X E A} is a n orthogonal basis f o r @[ViIKwith respect t o the inner product (., .)*. Moreover ( p x , p x ) * = l/dx. Proof: Let {ej}, {fj} be (., .)F-orthonormal bases for PA, P,. Using Lemma 7.5.1 we compute (PhPP)* = ( P x ( a , @ , ( z , m O )

7.6. Section 7 notes

Theorems 7.2.3, 7.4.3 and Corollary 7.4.4 are from [24]. An action whose invariants form a polynomial ring is said to be coregular. The coregular actions for simple groups are classified in [27] and [48].

284

C. B e n s o n and G. Ratcliff

The Fischer inner product is also called the Fock inner product, especially in connection with Equation 20. Fock space 3 is the Hilbert space This can be identified as the completion of C[V] with respect to (.,.),. space of holomorphic functions on V square integrable with respect to the Gaussian measure e-IZ12dz& [14].The inner product (., .)* can be regarded as the restriction of (., .),@(., .)pfrom F@F*to C [ V @ v ]2 C[V]@(C[V]*. One can identify F @ F with the space of Hilbert-Schmidt operators on F. Now (., )., induces the Hilbert-Schmidt norm. There is also a connection with the Berezin star product, as explained in [I]. 8. Generalized binomial coefficients

We continue to assume that G : V is a multiplicity free action. As in the previous section, {px I X E A} are the canonical invariants. We view these as living in cc [Vw]K .

8.1. The polynomials qx Let A = a

= dl81

+ . . . + anan and consider the operator T : C[Vi]+

@[VwI, A

( T p ) ( z , ~=) ( e p ) ( z ,-z) = e-

A

( p ( z ,-z)).

Note that T is an involutive automorphism. Indeed, writing ( M p ) ( z , t )= p ( z , -F) one has

T

= M o e a = e P A o M = T-’.

(22)

Definition 8.1.1: Let qx = T(px)= (-l)lxle-Apx for each X E A. The two formulas in the definition for qx agree because p x ( z , -F) = (-1)’xlpx(z,F).

Lemma 8.1.2: { q x : X E A} is a vector space basis for CC[VwIK.

Proof: First note that qx E (C[Vw]is K-invariant because px is K-invariant and A is a U(V)-invariant operator. Moreover qx = (-1)”lpx

+ rx

where px E Pzlxl(Vw) and rx is of lower degree. As { p x } is a basis for WQI~, so is { q x } . 0

O n Multiplicity Free Actions

285

8.2. The generalized binomial coefficients

As qx belongs to @[Vi]",it can be written as a finite linear combination of the canonical invariants p,.

Definition 8.2.1: The generalized binomial coefficients A, u E A via

[t]are defined for

The proof of Lemma 8.1.2 shows that

So in fact qx =

c

[;I.,

(-l)lVl

l,lllxl

=

(-l)lx'px

+

c

(-l)IYI[t]P,.

I,I j. Hence

+

+

+

+

det[(vi n - i 1 vj

+n -j ) ]= n ( v i+n i

-

i)!

On Multiplicity Free Actions

301

The denominator in st(v) is the Vandermonde determinant in the variables v + p. This gives det[(vi

+ n - i 1 n - j)]= n ( v i - vj - i + j ) , i<j

so st(v) = H ( v ) as claimed.

0

9.5. Section 9 notes

The Harish-Chandra isomorphism first appeared in [19]. The reader can find a modern treatment in Section V.5 of [29]. One reference for facts concerning integrality and integral closures, used in the proof of Theorem 9.2.1, is the text [a] by Atiyah and Macdonald. The results in this section are due to Knop. A considerably more general version Theorem 9.2.1 was proved in [30]. This asserts that the center of the ring of invariant differential operators for any smooth aEne G-variety is a polynomial ring, canonically isomorphic t o the ring of invariants for a finite reflection group. Knop calls W, the little Weyl group. This group is given explicitly in [31] for each saturated indecomposable multiplicity free action in the classification from Section 6. In most cases, W, coincides with N = Stab,(A). Theorem 9.3.1 was conjectured by Sahi, who proved a special case in [46]. Shifted Schur polynomials are due to Okounkov and Olshanski [42,43]. The proof of vanishing is taken from [42], which includes many remarkable properties for these functions. Recent work of Knop yields the eigenvalue polynomials e, for many other multiplicity free actions. For the actions GL(n,@): S2(@"),GL(n,@): R2((Cn),SO(n,@)x CX : @" and E6 x Cx : @27 this results in shafted Jack polynomials with various parameters. See [32,33]. A multiplicity free action G : V is said to be a Capelli action when the map 7r : 2 U ( g ) --f PD(V)Gin (30) is surjective. In [24] it is shown that the irreducible multiplicity free actions G' x ex : V in Table 3 are all Capelli actions except for

G' = Sp(2n) 8 SL(3),Spin(9),Eg References 1. D. Arnal, 0. Boukary Baoua, C. Benson, and G. Ratcliff, Invariant theory f o r the orthogonal group via star products, J. Lie Theory 11 (2001), 441-458. 2. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, Mass., 1969.

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3. C. Benson, J. Jenkins, R. Lipsman, and G. Ratcliff, A geometric criterion for Gelfand pairs associated with the Heisenberg group, Pacific J. Math. 178 (1997), 1-36. 4. c. Benson, J. Jenkins, and G. Ratcliff, Bounded K-spherical functions o n Heisenberg groups, J. F'unct. Anal. 105 (1992), 409-443. 5. C. Benson and G. Ratcliff, A classification for multiplicity free actions, J. Algebra 181 (1996), 152-186. 6. -, Combinatorics and spherical functions o n the Heisenberg group, Representation Theory 2 (1998), 79-105. 7. __ , Rationality of the generalized binomial coefficients f o r a multiplicity free action, J. Austrl. Math. SOC.(Series A) 68 (2000), 387-410. 8. M. Brion, Classification des espaces homogbnes sphe'riques, Compos. Math. 63 (1987), 189-208. Spherical varieties: a n introduction, Prog. Math., vol. 80, pp. 11-26, 9. -, Birkhauser, Basel, 1989. 10. M. Brion, D. Luna, and T. Vust, Espaces homogbnes shpe'riques, Invent. Math. 84 (1986), 565-619. 11. C. Chevalley and R. Shafer, The ezceptional Lie algebras F4 and EG,Proc. Nat. Acad. Sci. Amer. 36 (1950), 137-141. 12. H. Dib, Fonctions de Bessel sur une algbbre de Jordan, J. Math. Pures Appl. 69 (1990), 403-448. 13. J. Faraut and A. Koranyi, Analysis o n symmetric cones, Oxford University Press, New York, 1994. 14. G. Folland, Harmonic analysis in phase space, Princeton University Press, New Jersey, 1989. 15. -, A course in abstract harmonic analysis, CRC Press, Boca Raton, 1995. 16. R. Gangolli and V.S. Varadarajan, Harmonic analysis of spherical functions o n real reductive groups, Springer-Verlag, New York, 1988. 17. R. Goodman and N. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, New York, 1998. 18. V. Guillemin and S. Sternberg, Multiplicity free spaces, J. Differential Geom. 19 (1984), 31-56. 19. Harish-Chandra, O n some applications of the enveloping algebra of a semisimple Lie algebra, Trans. Amer. Math. SOC.70 (1951), 185-243. 20. S. Helgason, Diflerential geometry, Lie groups, and symmetric spaces, Academic Press, New York, 1978. 21. ___ , Groups and geometric analysis, Academic Press, New York, 1984. 22. R. Howe, Remarks o n classical invariant theory, Trans. Amer. Math. SOC. 313 (1989), 539-570. 23. -, Perspectives o n invariant theory: Schur duality, multiplicity-free actions and beyond, Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995. 24. R. Howe and T. Umeda, T h e Capelli identity, the double commutant theorem and multiplicity-free actions, Math. Ann. 290 (1991), 565-619.

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25. K. Johnson, O n a ring of invariant polynomials o n a hermitian symmetric space, J. Algebra 62 (1980), 72-81. 26. V. Kac, S o m e remarks o n nilpotent orbits, J. Algebra 64 (1980), 190-213. 27. V. Kac, V. L. Popov, and E. B. Vinberg, S u r les groupes lin aires alg briques dont l'alg bre des invariants est libre, C. R. Acad. Sci. Paris S r. A-B 283 (1976), A875-A878. 28. T. Kimura, Introduction t o prehomogeneous vector spaces, Transl. Math. Mono., vol. 215, Amer. Math. SOC.,Providence, Rhode Island, 2003. 29. A. Knapp, Lie groups beyond a n introduction, Progress in Math., vol. 140, Birkhauser , Boston, 1996. 30. F. Knop, A Harish-Chandra homomorphism for reductive group actions, Annals of Math. 140 (1994), 253-289. 31. ___, S o m e remarks o n multiplicity free spaces, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 514, pp. 301-317, Kluwer Acad. Publ., Dordrecht, 1998. 32. ___ , Construction of commuting difference operators for multiplicity free spaces, Sel. Math., New ser. 6 (2000), 443-470. 33. -, Semisymmetric polynomials and the invariant theory of matrix vector pairs, Representation Theory 5 (2001), 224-266. 34. F. Knop and S. Sahi, Difference equations and symmetric polynomials defined by their zeroes, International Math. Research Notes 10 (1996), 473-486. 35. B. Kostant and S. Sahi, The Capelli identity, tube domains and the generalized Laplace transform, Advances in Math. 87 (1991), 71-92. 36. M. Kramer, Spharische untergruppen in kompakten zusammenhangenden Liegruppen, Compos. Math. 38 (1979), 129-153. 37. M. Lassalle, Une formule de binhme ge'ne'ralise'e pour les polynhmes de Jack, C . R. Acad. Sci. Paris, SQrieI 310 (1990), 253-256. 38. A. Leahy, A classification of multiplicity free representations, J. Lie Theory 8 (1998), 367-391. 39. I. G. Macdonald, Symmetric functions and Hall polynomials, second edition, Clarendon Press, Oxford, 1995. 40. I. V. Mikityuk, O n the integrability of invariant Hamiltonian systems with homogeneous configuration spaces, Math USSR-Sb. 57 (1987), 527-546. 41. A. Okounkov and G. Olshanski, Shifted Jack polynomials, binomial formula, and applications, Math. Res. Letters 4 (1997), 69-78. 42. -, Shifted Schur functions, St. Petersburg Math. 9 (1998), 239-300. Shifted Schur functions II. The binomial formula for characters of 43. -, classical groups and its applications, Amer. Math. SOC.Transl. Ser 2 181 (1998), 245-271. 44. V. L. Popov, Stability of the action of a n algebraic group o n a n algebraic variety, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 367-379. 45. B. Brsted and G. Zhang, Weyl quantization and tensor products of Fock and Bergman spaces, Indiana Math. Journal 43 (1994), 551-583. 46. S. Sahi, T h e spectrum of certain invariant differential operators associated t o Hermitian symmetric spaces, Lie Theory and Geometry (J. L. Brylinski, ed.), Progress in Math., vol. 123, Birkhauser, Boston, 1994, pp. 569-576.

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47. M. Sat0 and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1-155. 48. G. Schwarz, Representations of simple Lie groups with regular rings of invariants, Invent. Math. 49 (1978), 167-191. 49. F. Servedio, Prehomogeneous vector spaces and varieties, Trans. Amer. Math. SOC.176 (1973), 421-444. 50. T.A. Springer, Invariant theory, Lecture Notes in Math., vol. 585, Springer Verlag, New York, 1977. 51. R. Stanley, Some combinatorial properties of Jack symmetric functions, Advances in Math. 77 (1989), 76-115. 52. E. B. Vinberg, Complexity of actions of reductive Lie groups, h n c t . Anal. and Appl. 20 (1986), 1-11. 53. ___ , Commutative homogeneous spaces and co-isotropic symplectic actions, Russian Math. Surveys 56 (2001), 1-60. 54. E. B. Vinberg and V. L. Popov, O n a class of quasihomogeneous a f i n e varieties, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 749-764. 55. H. Weyl, T h e classical groups, their invariants and representations, Princeton University Press, Princeton, N.J., 1946. 56. Z. Yan, Special functions associated with multiplicity-free representations, unpublished preprint.

Multiplicity-Free Spaces and Schur-Weyl-Howe Duality

Roe Goodman Rutgers University Department of Mathematics 110 Relinghuysen Rd Piscataway N J 08854.8019. USA E-mail: [email protected]

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Representations and duality . . . . . . . . . . . . . . . . . . . . . . . 1.1. Representations of algebraic groups . . . . . . . . . . . . . . . 1.2. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Reductive groups and isotypic decompositions . . . . . . . . . 1.4. Multiplicities and duality . . . . . . . . . . . . . . . . . . . . . 2 . Proof of duality theorem and examples . . . . . . . . . . . . . . . . . 2.1. Densitylemmas . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Proof of duality theorem . . . . . . . . . . . . . . . . . . . . . 2.3. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Schur-Weyl duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Commutant of GL(n) action on tensors . . . . . . . . . . . . . 3.2. Highest weight theory . . . . . . . . . . . . . . . . . . . . . . . 3.3. Duality and N-fixed vectors . . . . . . . . . . . . . . . . . . . 4 . Commutant character formulas . . . . . . . . . . . . . . . . . . . . . 4.1. Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Frobenius and determinant character formulas . . . . . . . . . 4.3. Proof of Frobenius character formula . . . . . . . . . . . . . . 4.4. Proof of determinant character formula . . . . . . . . . . . . . 5 . Character formulas for Schur-Weyl duality . . . . . . . . . . . . . . . 5.1. Frobenius formula for 6 k characters . . . . . . . . . . . . . . . 5.2. Determinant formula for 6 k characters . . . . . . . . . . . . . 5.3. Schur-Weyl duality and GL(k)-GL(n) duality . . . . . . . . . 305

307 307 307 308 309 311 313 313 315 316 320 320 321 326 329 329 329 330 331 332 332 334 336

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R . Goodman

6 . Polynomial invariants and FFT . . . . . . . . . . . . . . . . . . . . . 6.1. Invariant polynomials . . . . . . . . . . . . . . . . . . . . . . . 6.2. Invariants of vectors and covectors . . . . . . . . . . . . . . . . 6.3. Polynomial FFT for GL(n) . . . . . . . . . . . . . . . . . . . . 6.4. Polynomial FFT for the orthogonal group . . . . . . . . . . . 6.5. Polynomial FFT for the symplectic group . . . . . . . . . . . . 7. Tensor invariants and proof of FFT . . . . . . . . . . . . . . . . . . . 7.1. Tensor invariants for GL(V) . . . . . . . . . . . . . . . . . . . 7.2. Proof of polynomial FFT for GL(V) . . . . . . . . . . . . . . . 7.3. Tensor invariants for orthogonal and symplectic groups . . . . 7.4. Proof of polynomial FFT for orthogonal and symplectic groups . . . . . . . . . . . . . . . . . . . . . . . . . 8. Weyl algebra and Howe duality . . . . . . . . . . . . . . . . . . . . . 8.1. Duality in the Weyl algebra . . . . . . . . . . . . . . . . . . . 8.2. Howe duality for orthogonal/symplectic groups . . . . . . . . . 8.3. Howe duality for GL(k) . . . . . . . . . . . . . . . . . . . . . . 9. Harmonic duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Harmonic polynomials . . . . . . . . . . . . . . . . . . . . . . 9.2. Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Decomposition of harmonic polynomials . . . . . . . . . . . . . . . . 10.1. O ( k ) Harmonics ( k odd) . . . . . . . . . . . . . . . . . . . . . 10.2. O ( k ) Harmonics ( k even) . . . . . . . . . . . . . . . . . . . . . 10.3. Examples of harmonic decompositions . . . . . . . . . . . . . . 11. Symplectic group and oscillator representation . . . . . . . . . . . . . 11.1. Real symplectic group . . . . . . . . . . . . . . . . . . . . . . . 11.2. Holomorphic (coherent-state) model for oscillator represent at ion . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3. Bargmann-Segal transform . . . . . . . . . . . . . . . . . . . . 11.4. Real (oscillatory-wave) model for oscillator representation . . . 11.5. Analytic vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Dual pair S p ( n , R)-O(k) . . . . . . . . . . . . . . . . . . . . . . . . . 12.1. Decomposition of H 2 ( M n x k ) under Mp(n, W) x O ( k ) . . . . . 12.2. Square-integrable representations of Sp(n,R) . . . . . . . . . . 13. Brauer algebra and tensor harmonics . . . . . . . . . . . . . . . . . . 13.1. Centralizer algebra and Brauer diagrams . . . . . . . . . . . . 13.2. Generators for the centralizer algebra . . . . . . . . . . . . . . 13.3. Relations in the centralizer algebra . . . . . . . . . . . . . . . 13.4. Harmonic tensors . . . . . . . . . . . . . . . . . . . . . . . . . 13.5. Decomposition of harmonic tensors for Sp(V) . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

338 338 339 341 341 343 344 344 345 347 350 351 351 355 357 358 358 360 364 365 371 374 375 376 383 387 389 391 394 394 399 402 402 405 407 409 410 414

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Dedicated to the memory of Irving E. Segal, who introduced m e to the beauties and mysteries of representation theory. Introduction The unifying theme of these lectures is the duality between the irreducible representations occuring in a linear group action and irreducible representations of the commuting algebra relative to this action. This notion of duality in representation theory was introduced by Schur a century ago, and it has developed into an important tool with many applications. In keeping with the tutorial aspect, I have tried to tell the story starting from the beginning and including complete proofs of all the major results (at several points I refer to the lectures of Benson-Ratcliff in the present volume for details). Of course, this limits the scope of the lectures to the more classical parts of the theory: Schur-Weyl-Brauer duality for finitedimensional representations, and Howe duality between finite-dimensional and infinite-dimensional highest-weight representations. Substantial parts of these lectures are based on joint work with Nolan Wallach and I would like to acknowledge his contributions to my understanding of representation theory. I would also like to thank Eng-Chye Tan and Chen-Bo Zhu for inviting me to give these lectures and for their wonderful hospitality.

1. Representations and duality 1.1. Representations of algebraic groups

Assume that G c GL(n, C ) is an algebraic group (defined by a set of polynomial equations in the matrix entry functions). We denote by Aff (G) the commutative algebra of regular functions on G (the restrictions to G of polynomials in the matrix entry functions x i j and det-l). Let ( p , L ) be a representation of G on a complex vector space L. If L is finite-dimensional, then we say that p is regular (rational) if the representative functions g H tr(p(g)E), for E E End(L), are regular. Every regular function on G arises as such a representative function. When L is infinite dimensional, we say that p is locally regular if for all x E L there is finite-dimensional G-invariant subspace M containing x so that ( p (M , M ) is a regular representation. The most fundamental tool in representation theory is Schur’s Lemma: I f E and F are irreducible, finite-dimensional representations of

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a group

G, then dim HomG (E,F ) =

c

1 i f E r F 0 ifEFF

(HomG(E,F) denotes the space of linear transformations T : E F that intertwine the G actions on the two spaces). To prove Schur's Lemma, observe that the null space and range of T are G-invariant subspaces, so T must be either zero or bijective, with the first case holding if E F. When E F and S,T are two nonzero intertwining maps, take X t o be an eigenvalue of S P I T .Since S-IT - X I commutes with the action of G on E and has a nonzero null space, it must be zero. --f

1.2, Examples (1) Let ( T , V ) be any regular (finite-dimensional) representation of G. We denote by P ( V )the algebra of complex-valued polynomial functions on V . Define a representation of G on P ( V ) by p(g)f(v) = f(.rr(g)-'v)

for

f E P ( V )and g

E

G.

Since the G action is linear, it commutes with the (Cx action on V by scalar multiplication, and we have the direct-sum decomposition into finitedimensional G-invariant subspaces

P ( V )= @ P W )

7

k>O

where P k ( V )is the space of homogeneous polynomials of degree k. The action of G on each of these spaces is regular, so the representation p is locally regular. Furthermore, the G action preserves the multiplication on

P(V). (2) With (n,V ) as above, we can take the full tensor algebra

I ( V )= @ v @ k k>O

with G action p ( g ) ( v l 8 . . . 8 vk) = T(g)vl 8 .. . @ T(g)vk. Since G leaves invariant each subspace V@'", the representation p is locally regular. As in the previous example, the G action preserves the (noncommutative) multiplication on I ( V ) .

(3) Let X c Cm be an affine algebraic set (the zero set of a family of polynomials) and suppose that there is a regular G action on X

GxX

--t

X,

( g , X )H g . z .

Multiplicity-Free Spaces and Schvr- Weyl-Howe Duality

309

Set L = Aff (X) (the restriction to X of the polynomial functions on C"). Let G act on L by p(g)f(z) = f(g-l . x). We can prove that this representation is locally regular as follows. Given f E Aff(X),set Vf = Span{p(g)f : g E G}. The function (9, z) H f(g-' . x) on G x X is regular, and Aff(G x X) = Aff(G) 8 Aff(X), there are regular functions qhi on G and Qi on X so that

k=l In particular, V f C Span{$k} is finite-dimensional, so we can choose 91,. . . , gr in G such that the functions fi = p(gi)f give a basis for V f . Now choose points x1 , . . . , xk in X so that the evaluation functionals d,, are a basis for V f *Since . r

M g ) f i , ~ z j )= P(ggi)f(xj) = C 4 k ( g g i M ( x j ) , k= 1

we see that the representation of G on V f is regular. Thus ( p , L ) is locally regular.

1.3. R e d u c t i v e groups a n d isotypic decompositions

A complex algebraic group G is called reductive if every finite dimensional regular representation decomposes as a direct sum of irreducible representations (this property is equivalent to every G-invariant subspace of a regular representation having a G-invariant complementary subspace). The classical groups are reductive: 0 0

0

0

the general linear group GL(n, C ) of invertible n x n complex matrices the special linear group SL(n,C ) of n x n complex matrices of determinant one the orthogonal group O ( C n , u ) of n x n matrices preserving a nondegenerate symmetric bilinear form u ( z , y ) = xtBy on C", where B is a symmetric invertible n x n matrix (defining equation gtBg = B ) the special orthogonal group SO(@", w)of orthogonal matrices of determinant one the symplectic group Sp(Canlw)of 2n x 2n matrices preserving a nondegenerate skew-symmetric bilinear form w ( x ,y) = zt J y on Can, where J is a skew-symmetric invertible 2n x 2n matrix (defining equation gtJg = J )

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Finite groups are shown to be reductive by the method of averaging over the group. The proof that classical groups are reductive can be carried out analytically by integrating over a compact real form (Weyl’s unitary trick see [16, Theorem 2.4.7]), or algebraically by using a Casimir operator (see [16, Theorem 2.4.51). Direct products of reductive groups are reductive, and the quotient of a reductive group by a closed normal subgroup is reductive (this is obvious). An algebraic group is reductive if and only if its identity component is reductive. Assume G is reductive, and let G be the equivalence classes of irreducible finite-dimensional regular representations of G. For each A E 6 fix F A )in the class A. Let A* be the equivalence class of a representation (rA, the contragredient representation on the dual space ( F A ) * . Given a locally regular representation ( p , L ) of G, set h

L ( A )= X V

(sum of all V

c L such that plv

S

FA).

Call L(x) the A-isotypic component of L. Define Spec(p) = {A E L(x) # 0) (the G-spectrum of ( p , L ) ) .

Proposition 1.1: L = $AESpec(L) L(A)

:

(algebraic direct s u m ) .

Proof: We first verify that the sum is direct. Suppose, for the sake of contradiction, that L ( A )n L ( p )# 0 for some A , p E 6 with A # p. Then there exists a G-invariant subspace W # 0 so that W c L ( x )n L ( p )and dim W < 00. Since G is reductive, W = Vl @ . . . @ V,, where each V, is an irreducible G-module. Hence V, ? F A and also 5 2 F P , a contradiction. To see that L is the sum of its isotypic components, set LO= L ( x ) .If LO# L , then there exists a nonzero x E L\Lo. But x is contained in a finite dimensional G invariant subspace W that is the direct sum of irreducible G-invariant subspaces. Hence W c Lo, a contradiction. 0

Corollary 1.2: There is a linear projection x H xb f r o m L onto the space LG of G-fixed vectors. We now turn to the G-module structure of the isotypic components of a representation L. Denote by Homc(FX,L ) the vector space of all linear maps T : FA---t L that intertwine the G actions on these spaces. This is the space of covariants of type A.

Theorem 1.3: If ( p , L ) i s a locally regular representation of a complex reductive algebraic group G , then

~r

@I E

X S p 4 p )

~

F

~

,

Multiplicity-flee Spaces and Schur- Weyl-Howe Duality

311

where E X = HomG(FX,L ) and G acts by 18 p on each summand. In particular, the multiplicity of X in p is the dimension of the space of covariants of type A. Proof: Let T E HomG(FX,L ) be a nonzero intertwining operator. Then T is injective, by Schur’s lemma. Conversely, if W c L ( x )is a G invariant irreducible subspace, then there is an intertwining map T so that W = T(FX).This implies that the map T 8 v H Tv from E X 8 F A to L ( x )is surjective. It remains to prove that the map E X@ F A4 L ( x )is injective. Suppose v3 E FAand Tj E E X satisfy C j T j v j = 0. We may assume that {vj} is linearly independent. Fix a decomposition

L ( x )= @ Fi ,

Fi

S

FA

a

This defines G-invariant projections

Pi : L ( x )+ FA,and by assumption

By Schur’s lemma, PiTj = cij I for some cij E CC, so we conclude that cij = 0 for all i, j , by the linear independence of {vj}. Hence Tj = 0 for all j . 0

1.4. Multiplicities and duality One says that L is multiplicity-free as a G module if dim E X = 1 for all X E Spec(p). In this case L is uniquely determined as a G-module by its spectrum. For a detailed analysis of such representations when L = P ( X ) and X is a vector space or afine variety with regular G action see the lectures by Benson-Ratcliff in this volume. In these lectures we will study representations (p, L ) that are not multiplicity free. We want to determine 0 0

c,

The spectrum Spec(p) c The multiplicities rnx = dim E X , Explicit models for the multiplicity spaces E X

Let EndG(L) be the algebra of linear transformations on L that commute with the G action. There is a natural representation of this algebra on each multiplicity space EX.Indeed, if A E EndG(L) and T E E X ,then the linear map A o T : FA-+ L also commutes with the G action on L , and

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312

hence is an element of E X .Following the ideas of I. Schur, H. Weyl and R. Howe, the unifying theme in our approach will be

Hidden Symmetry: Study the spaces E X as modules for good subalgebras of EndG (L). The term hidden symmetry comes from applications of representation theory to quantum mechanics in cases where the geometric symmetries such as rotation invariance do not suffice to explain the multiplicities in the energy spectrum. In some cases, one can find a larger symmetry group containing G and extend the representation of G to a representation of this larger group on L that is multiplicity free. In other cases the hidden symmetries are given by a Lie algebra of differential operators commuting with the G action (see [30]). When L is infinite-dimension (for example, when L = Aff (X) with X an affine G variety), then End(L) is too big to deal with purely algebraically. In the context of unitary representations on a Hilbert space, one uses the von Neumann algebra of bounded operators that commute with G. In our algebraic setting we shall assume that L is of countable dimension and that we have a subalgebra R c End(L) that satisfies (i) R acts irreducibly on L (ii) R is invariant under G, relative to the action Ad(g)T = p(g)Tp(g)-', and the representation Ad of G on R is locally regular

In case dim L < 00 we take R = End(L) L 8 L* and these conditions are always satisfied. When L = P ( X ) with X a smooth afine G variety, we take R = D ( X ) , the algebraic differential operators on X (see Agricola [l]). In particular, if X is a vector space with linear G action, then D(X) is the Weyl algebra P D ( X ) of differential operators with polynomial coefficients, which we will examine in detail in Section 8. Fix R satisfying the conditions (i) and (ii) and let

RG = {T E R : Ad(g)T = T for all g E G} (the commutant of p(G) in R).

Theorem 1.4: Each multiplicity space E X is an irreducible RG module. Furthermore, if A , p E Spec(p) and E X 2 EP as an RG module, then

x = p.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

313

In the next section we will prove this theorem.a At this point we derive some consequences. The following corollary plays a fundamental role in our approach to Howe duality.

Corollary 1.5: Let u be the representation of RG on L. Then ( a l L ) is a semisimple RG module, and each irreducible submodule E X occurs with finite multiplicity dim FA. When L is finite-dimensional then R = End(L), and from the inequivalence of the representations E X together with Schur's lemma we obtain the classical Double Commutant Theorem.

Corollary 1.6: If dim L < co and B = EndG(L), then Span{p(G)} consists of all linear transformations on L that commute with B. Corollary 1.7 (Duality Correspondence): Let Spec(a) denote the set of equivalence classes of the irreducible representations of the algebra RG that occur in L. Then the map F A -+ E X sets up a bijection between Spec(p) and Spec(u).

2. Proof of duality theorem and examples 2.1. Density lemmas Lemma 2.1 (Dixmier-Schur): Let L be a vector space over C of countable dimension. Let R c End(L) be a subalgebra that acts irreducibly on L . Suppose A E End(L) commutes with R. Then A = XI for some X E C. Proof: Suppose that A is not a multiple of the identity. Since R acts irreducibly] Schurls lemma implies that A - X I is invertible for all X E C. Hence for every nonzero polynomial p ( x ) in one variable the operator p ( A ) is invertible (factor p(x) into linear factors). Thus there is an algebra homomorphism from the field C ( x )of rational functions in one variable into End(L) given by p ( x ) / q ( z )H p(A)q(A)-'.Fix a nonzero vector v E L. Then the linear map r(x) ++ r(A)v is injective from C(z) to L. But C ( x ) has uncountable dimension as a vector space over C , since the functions {(x - A)-' : X E C} are linearly independent, a contradiction. 0 Assume now that L has countable dimension as a complex vector space and that R c End(L) is a subalgebra that acts irreducibly on L. "See [16,Theorem 4.5.121 for the case that presented here is due to Agricola [l].

R is a graded algebra; the generalization

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314

Lemma 2.2 (Jacobson): Let X be any finite-dimensional subspace of L. Then every f E Hom(X, L) is of the fonn rIx for some r E R.

Proof: Let (51,. . . , z},

be a basis for X . Define n copies

n copies

L e t R a c t o n L ( " ) b y r . [ y l , . . . , y , ] = [ ryl, ..., r y , ] € o r r ~ R a n d y i E L , and extend f to a linear map f(") : X ( n ) -+ L(") by

f ( n ) [ ~..l ,

YnI =

. )

[ ~ ( Y I.).,. )f ( ~ n ).I

Denote by M = R . z(n)the cyclic R submodule generated by d"). Define Li c L(n) to be the vectors that have arbitrary entries from L in the ith place and are zero in the other positions. Pick a maximal subset I c { 1,.. . ,n} with the property that the sum N=M+CLi iEI

is direct. Then N is an R submodule of L("). Since R acts irreducibly on L, the R modules N n L j are either zero or L j for each j . But if N n L j = 0, then the sum N Lj would be direct, contradicting the choice of I . Hence N= proving that M has an R-invariant complement. Thus there is a projection P : L(n) + M that commutes with the action of R.We can write

+

1

n

L j=1 wherepij E End(L). Since P commutes with R on L(n),the transformations pij all commute with R on L . Hence pij E CI by Lemma 2.1 and [pij] is a matrix of scalars. Now calculate

[

n

1

f ( " ) P [ Y l , . . . , ~ n l= C P l , f ( y j ) , . . . l ~ P n j f ( Y j )= ~ f ( n ) [ Y l l . - , Y n l . j=1

j=1

Hence f(") commutes with P . Since dn)E A4 we have f(n),(n)

= f( n ) p , ( n ) = p f ( n ) & )

E

M.

Thus there exists T E R so that f ( n ) z ( n = ) r d n ) . Since basis for X , this implies that f = T I X .

,.. .

(21

z,}

is a 0

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

315

Corollary 2.3 (Burnside): If dim L < KI then R = End(L). Now let ( p , L ) be a locally regular representation of G with dim L countable. Assume that R c End(L) satisfies conditions (i) and (ii) stated before Theorem 1.4. Lemma 2.4: Let X c L be a finite-dimensional G invariant subspace. Then RGlx = HOmG(X,L).

Proof: Let T E HomG(X, L). Then by Lemma 2 . 2 there exists r E R such that T I X = T . Since G is reductive, condition (ii) implies that there is a projection r H rh from R -+ RG.But the map R -+ Hom(X,L) given by y H ylx intertwines the G actions, since X is G-invariant. Hence T = T b= ~hlx. 2.2. Proof of duality theorem

Take X E Spec(p) and let Zx c L(x) be any irreducible G-submodule. Given f E L, we denote by U f = RGf the cyclic RG module generated by f . We write @[GIfor the group algebra of G (the formal finite linear combinations of the elements of G). (a) If 0 # M

c L(x) is an RG-module,then M n Zx # 0.

To verify this, take 0 # m E M and set X = Span{p(G)m}. Then d i m X < 00 and X c L(x). Hence there exists T E HomG(X, Zx) with T m # 0. By Lemma 2.4 there exists r E RG with T [ X = T . Then r m = T m E M n Zx.

(b) If 0 # f E ZA then U f rl Zx

= Cf.

Take u = r f E U f n Zx. Since Zx = Span{p(G)f}, we have

rZx = Span{p(G)rf} = Span{p(G)u} c Zx Thus rlzX E EndG(Zx) = C I by Schur’s Lemma. So u = r . f E C f , proving (b). (c) I f f E Zx is nonzero, then U f is an irreducible RG-module. Indeed, if 0 # M C U f is an RG-submodule, then 0 # M n Zx and (b). Thus f E M and hence M = U f , which proves (c).

c Cf by (a)

(d) Let f i , . . . , f d be a basis of Zx. Set Mi = U f i . Then the sum is direct and Mi S Mj as RG modules.

cld_lMi

We have Span{p(g)lz, : g E G} = End(Zx) by Corollary 2.3. Thus for each i there exists an element ui E @[GIsuch that p ( u i ) f j = 6ij fj. Suppose

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316

mi

E Mi and

xi

m i = 0.

There exist

ri

E

i

i

RG SO that

m i = Ti f i .

Hence

i

for j = 1,.. , , d . This proves the first statement of (d). For the second, apply Corollary 2.3 again to obtain uji E @[GI such that p ( u j i )fi = fj. Since Mi and Mj are irreducible by (c), the map p ( u j i ) : Mi 4 Mj is an RG-module isomorphism, by Schur’s Lemma. (e) Let

Mi

d

be as in (d). Then L(x) = @i=l M i .

Recall that L(x)is the sum of all irreducible G-submodules of L that are in the class A. Thus it is enough to show that if Wx is such a submodule then d

i=l

Take a G isomorphism T : ZA -+ Wx. Then Lemma 2.4 furnishes r E RG such that r = T on Zx.Hence Wx satisfies (l),which proves (e). The first assertion of Theorem 1.4 now follows from (c), (d), and (e). To prove the second assertion, it suffices to prove the following.

(f) Let f A and f, be nonzero vectors in irreducible G subspaces Zx and 2,. Suppose U f , 2 U f , as RG-modules. Then A = p . Let T : UfA -+ U f , be an RG-module isomorphism. Let X be a finitedimensional G-invariant subspace containing f x and Tfx. There is a projection operator PA : X -+ L(x) onto the A-isotypic component of L , and Lemma 2.4 furnishes r E RG such that T I X = PA. Thus r . fx = fx so we have

T fx = T r fx = rT fx = PATfx E L(x). Since T is an RG module isomorphism, it follows that U f , f, E L ( x ) ,and so we conclude that p = A.

c L(x).Hence O

2.3. Examples (1) (Product groups) Let H and K be reductive complex algebraic groups, and let G = H x K be the direct product algebraic group, where Aff (G) Aff (H)@Aff(K) under the natural pointwise multiplication map. We can use the duality theorem to prove that = x k : Every irreducible regular representation ( L , p ) of G is given by

L

=M @N

,

p(h, Ic) = a ( h )8 ~ ( k )for h E H and Ic E K

(2)

Multiplicity-Bee Spaces and Schur- Weyl-Howe Duality

317

where (a,M ) is an irreducible representation of H and (7,N ) is an irreducible representation K. To prove this, suppose first that (p,L) is defined by (2). Then Corollary 2.3 implies that End(L) is spanned by the transformations { p ( h , Ic) : h E H , k E K } and hence E n d c ( L ) = @ I ,showing that L is irreducible. Conversely, given an irreducible regular representation ( p , L ) of G, use Theorem 1.4 (with R = End(L)) to decompose L as a K-module:

@ EX@PFX. (3) X€i2 Set a ( h ) = p ( h , l ) and ~ ( k =) p(1,Ic). Since a ( h ) commutes with ~ ( k )H, acts on each E X by some representation p X .We claim that EXis irreducible under H . To prove this, note that L=

@ End(EX)@ I . (4) X€R Given T E EndK(L), we know by Corollary 2.3 that T is a linear combination of the transformations a(h)T(k).Under the isomorphism (4) the K-invariant transformations only act on EX.This proves that EndK(L) is spanned by { a ( h ) : h E H } , and hence EXis irreducible under H by Theorem 1.4. Thus each summand in (3) is an irreducible G module, by the earlier argument, so there can be only one summand. EndK(L)

(2) (Multiplicity-free representations of product groups) Suppose ( p , L ) is any locally regular representation of G that is multiplicity-free. By Example (1) the isotypic decomposition of L under H x K is of the form

L=

@ E"@F@

(5)

P)EA

(0%

where A c E x 2 and E" is the irreducible H-module of type a , while F P is the irreducible K-module of type p. Set a = p l and ~ 7 = p l ~ . Then Spec(a) is the projection A + G, whereas Spec(.r) is the projection A k .In general A is not determined by these projections. If both of these projections are injective, we say that the representation p sets up a duality correspondence between Spec(a) and Spec(-r). Clearly such representations of G must be very special, and in these lectures they will play an important role. The next example is the most familiar of them. ---f

( 3 ) (Two-sided group action) Let K be any reductive complex algebraic group. Set G = K x K and L = Aff(K). Define the representation p of G on L by P ( Z , y)f(k) = f(.-lW

for k , 2 , y E K .

R. Goodman

318

= From Example (1) we know that x I?. Consider Aff(K) as a K-module relative to the right translation action p(1,Ic) and apply Theorem 1.3:

Aff(K) =

@ E X@ F A X€il

Here K acts on E X = HomK(FX,Aff(K)) by p ( k , 1) o T,where T : F A -+ Aff (K) intertwines the action of K on F A with the right translation action of K on Aff(K). We claim that E X 2 F A * . To prove this, define a map E X 4 FA* (a special case of Frobenius reciprocity) by

T

+-+

T^ E F ~ ,'

@,u) = ( ~ v ) ( i ) for

EFA.

This map obviously intertwines the action of K. It is injective, since (Tv)(l) = 0 for all IJ E F A implies

(Tv)(lc)= (T7rX(Ic)v)(l)= 0 for all Ic E K , and hence T = 0. The map is surjective, since w* E FA* defines T E E X by

(Tv)(Ic)= ( V * , 7 r X ( I c ) W ) . A

Clearly T = v*.Thus the decomposition (6), relative to the action of K x K , is Aff(K) 2

@ FA*8 F A Z @ End(FX). A&

XE

-

ii-

This shows that Aff(K) is multiplicity free as a representation of K x K and there is a duality correspondence X X*. (4) (Harmonics on the zero-sphere) Let G = 0(1) = {fl} acting on C , and take L = P(C). In this case

E = {F+, F - }

(trivial, signum).

The G-isotypic decomposition of L is thus

L = L+ @I L-

(even polynomials

@

odd polynomials)

and each component has infinite multiplicity. We apply the duality philosophy to explain the multiplicities by finding operators on L that commute with G. Let P D ( C ) be the polynomial coefficient differential operators on P(C). Then one has

Multiplicity-Free Spaces and Schur- Weyl-Houe Duality

319

+

(a) The operators A = ( d / d ~ ) multiplication ~, by x 2 , and x ( d / d x ) 1 / 2 (shifled Euler operator) commute with G and span a Lie algebra 0‘ E d(2,c ) in P D ( c ) . (b) The Lie algebra 0’ generates the commutant PP(@)Gof G. The proof of (a) is an easy calculation. The proof of (b) follows by considering the symbol f (x,

Examples

(1) The group Gk has two one-dimensional representations: the trivial representation and the sign representation. The corresponding subspaces of @."C" are the symmetric tensors S k ( C n )and (if n 2 k ) the skew symmetric tensors C". Hence these subspaces must be irreducible GL(n, C )

Ak

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328

modules, by Schur-Weyl duality (this is also easy to verify directly). The symmetric tensor eFk is N-fixed with weight kal , while the skew-symmetric . . &k (when k 5 n).Thus tensor el A ' . ' A ek is N-fixed with weight in the duality correspondence,

--

(trivial, C) = (Jk1,dk1)

(sgn, C) = ( a [ l k ]E,[ l k ] )

+

(Jk1,s'(c")) ,[lk],

AkC")

if n 2 k .

When k = 2 and n 2 2 this gives the complete decomposition of B 2 C n : under 6 2 x G L ( n , @ ) :

a3C" for n 2 3. There are three partitions of 3, giving the

(2) Consider decomposition

B3 2 { E[3,01 @"

@

> {,9[2,11 Fl2j11> {E [ ' I ~ , ~A]

S3((cn)

@

@

@

@

under 6 3 x G L ( n , @ ) .Here the representation E[2i1]of dimensional standard representation on C3/C[l, 1,1].

6 3

3Q.n}

is the two-

We can view Schur-Weyl Duality as a method to construct representations of G' = 6 k from representations of G = G L ( n , @ )via the Theorem of the Highest Weight. Here we take the representations of G as the known objects, and the representations of G' as the unknown objects.c The relative size of n (the rank of G) and k then determines which representations of GI we get this way.

n 2 k: All partitions of k have a t most k parts, so all representations of 6 k occur in B~C" in this case. n 5 k: Only those representations Of 6 k occur in @" that correspond to partitions of k with a t most n parts.

mk

To make this method effective, we will develop character formulas for the representations of 6 k in the next two sections, based on the celebrated Wegl character formula for G L ( n ,C ) . CTherepresentations of 6 k can be constructed directly by group-theoretic and combinatorial methods. Special elements of the group algebra C [ G k ] (Young symrnetrizers) project tensor space onto irreducible representations of GL(n, C) - the so-called Weyl modules - see [16,Sec. 9.31.

Multiplicity-Ree Spaces and Schur- Weyl-Howe Duality

329

4. Commutant character formulas 4.1. Characters Let G be a connected complex reductive algebraic group. Then G contains a maximal algebraic torus H and a maximal connected solvable subgroup H N (semidirect product), where N is the unipotent radical of H N . In fact, one can always embed G into GL(n,C) so that H consists of the diagonal matrices in G, and N the upper-triangular unipotent matrices in G, just as in the case of GL(n, C ) treated in Section 3. Let l~ = Lie(N) and n = Lie(N). The irreducible representations of the torus H are given by h H h X , where X is in the weight lattice P c b* of H . By the Theorem of the Highest Weight (which is proved for G along the same lines as in Section 3 for GL(n, C ) ) , the irreducible regular representations of G are parameterized by the set P++ of dominant weights determined by the choice of N . For X E P++ let (xX,FA)be the irreducible representation of G with highest weight A. Let ( T , V) be a finite-dimensional rational representation of G. Set B = Endc(V). From Theorem 3.7 V decomposes under the joint action of G and B into a multiplicity-free direct sum

Here g E G acts by 1@ d ( g ) and b E B acts by d ( b ) @I 1 on the summands in (11).We may take E X = V(X)N(the space of N-fixed vectors of weight X in V ) with d ( b ) the natural action of b E B on this space. Finding the spaces V ( X ) Nexplicitly is usually difficult. An easier problem is to calculate characters. For X E P++ we write

4.2. Frobenius and determinant character formulas

We now obtain two formulas for the characters xx that only involve the full H-weight spaces in V . Let be the weights of Ad(H) on n and p = $ CaEa+ a. Let W = NormG(H)/H be the Weyl group of (G, H). Set

D ( h )=

sgn(s) hS’P

for h E H

SEW

(the Weyl denominator). Here s signum character on W .

H

sgn(s) = det(Ad(s)Ir,) is the usual

R. Goodman

330

Theorem 4.1 (Generalized Frobenius Formula): For X E P++ and b E B one has

xx(b)

=

coeficient of hx+p in ~ ( htrv(x(h)b) )

(12)

(where h E H ) . Theorem 4.2 (Generalized Determinant Formula): For X E P++ and b E 23 one has

xx(b) =

c

sgn(s) trV(X+p-s.p) ( b ) .

(13)

SEW

I n particular,

4.3. Proof of Frobenius character f o r m u l a

For X E P++ we write xx(g) = tr(xx(g)) for g E G . (the character of the representation with highest weight A). We note from (11) that trv(x(g)b) =

c

xX(g)Xx(b)

for g E G and b E

B.

(15)

XESpec(?r)

By the Weyl character formula (WCF), we have

D(h)x x ( h ) =

c

sgn(s) hS’(’+”)

for h E H .

SEW

Using the WCF in (15) we can write

D ( h )t r v ( 4 h ) b ) =

c c

(16)

Sgn(s) ~ x ( b h) S ’ ( x + p ) .

XESpec(.rr) sE W

Due to the shift by p, the map (s, A) H s . (A + p ) from W x P++ P is injective.d Hence the character h H hX+p only occurs once in (16), and has 0 coefficient xx ( b ) , as claimed. --f

In the case of G L ( n ,C ) the partition X

+ p has all parts of dzfierent sizes.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

4.4. Proof of

331

determinant character formula

For the proof of Theorem 4.2, we need the following consequence of the WCF (which is, in fact, equivalent to the WCF).

Lemma 4.3: Let m x ( p ) = dim F X ( p )for p E P and X E P++ (the multiplicity of the weight p in F A ) . Then for X,p E P++ one has

C sgn(s) m , ( ~+ p - s . p ) = 6Xp . SEW

Proof: Write the Weyl denominator as an alternating sum over W of the characters h H hs'p. Multiplying this sum by x p , we get

D(h)X'l(h)=

c{

SEW

sgn(s)m,(v) hvfS'P}

for h E H .

YEP

In the inner sum make the substitution v the right becomes

--f

v

+ p - s .p; then the sum on

On the other hand, the WCF asserts that the coefficient of h X f p in 0 D ( h ) f ( h )is d ~ ,when Alp E P++.

Proposition 4.4 (Outer Multiplicity Formula): Let L be any regular G module. For X E P++ let rnultL(X) be the multiplicity of the representation F A . Then multr,(X) =

sgn(s) d i m L ( X + p - s . p ) . SEW

Proof: For v E P we have dimL(v) = right side of (17) is

PEP++

(17)

cPEp++ multr,(p) m,(v). Hence the

SEW

But the inner sum is b ~ by , Lemma 4.3, which proves (17).

0

Proof of Theorem 4.2: Let b E B. Then b has a Jordan decomposition b = bs b,, where b, is semisimple, b, is nilpotent, and be is a polynomial in b. Hence b, E B and X x ( b ) = X x ( b , ) . So we may assume b is semisimple. For E C and X E P++ define

+


. . * I aT, b l , b l .~. . 3

1

b,Z

I bT, I

cl> c1 I

C l l ' ' ' I cT3

"

\-

r1 singles

pairs

~2

r3

> cr3 >

cT3 I ' '

'1

triples

where a,, b,, cZr. . . range from 1 to n. Hence

er = e a 1 8 . . . @ e a r l @ e r @ . . . @ e Frz2@ e r @ . . . @ e F r t 8 . . .

0

The lemma follows. For h = diag[xl,. . . , xn] define

p , (x)= tr% ( p k ( h ) ) = xi

+ . + x; s .

( j t h power sum).

Then Lemma 5.1 implies that k

k

trF,(Pk(h)) = n t r V . ( p k ( h ) ) " = n p ~ ( x ) .' ~ ,=1

j=l

Hence from Theorem 4.1 and (18) we obtain the Frobenius character formula:

Theorem 5.2: Let s E C(lT12T2 . . . I c r k ) and X E Par(k, n ) . Then

c1 }

x x ( s ) = coeficient ofx'+p[nl in D ~ ( ~ Hc p)j ( x l T j where

,qn] = [n- 1,n - 2,.

,,

,1,0] and D n ( x ) = n,,,<jln(xi

-

xj).

Examples

+

(1) Suppose s = ( 1 , 2 , . . . , m ) ( m 1 ) .. . ( I c ) is a single m-cycle with k - m fixed points. Then

x x ( s ) = coefficient of

xx+plnl

in

(21

+ . . . + Z,)~-~(Z;" + . . . + xr)

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334

We call a monomial x;' ...x? strictly dominant if a1 > a2 > . . . > a,. For partitions X with two parts and cycles of maximum length m = Ic, the strictly dominant terms in this formula are z : + l - zfz2. Hence for s = (1,2 ,.", Ic),

x x ( s )=

{ -:

for X = [k - 1,1], for X = [k - j , j ] with j

> 1.

'

(2) Consider the group 6 3 , which has three conjugacy classes: C(13) = {identity), C(112') = {(12), (13),(23)) and C(3l) = {(123), (132)). As we noted at the end of Section 3, the three representations of (553 are (the trivial representation), a[2y1](the two-dimensional standard representation), and o [ l ~ l ~(the l l signum representation). To calculate the character of a[2)11 by the Frobenius formula, we let z = [ X I , 521 and expand the polynomials ~ 2 ( x ) p(x13 l =

x;'+ 223x2 + . . .

Dz(x)p1(x)pz(z)= z;' + * . * DZ(X)p3(X)

3 = 214 - 21x2

+

' ' '

where . . . indicates non-dominant terms. By Theorem 4.1 the coefficients of the dominant terms in these formulas furnish the entries in the character table for 6 3 . We k i t e x x for the character of the representation a x . For example, when X = [2,1] we have X + p = [3,1],so the coefficient of x;z2 in D 2 ( ~ ) p 3 (gives ~ ) the value of X [ ~ , J on ] the conjugacy class C(3l). Table 1 gives all the characters, where the top row indicates the number of elements in each conjugacy class, and the rows in the table give the character values for each irreducible representation. Table 1. Character table of

5.2. Determinant formula for

6 k

63.

characters

We next apply Theorem 4.2 to obtain an alternating sum formula for the characters of 6 k . For this, we need to identify the weight spaces V ( Y )as

Multiplicity-Fkee Spaces and Schur- Weyl-Howe Duality

6k-modules. Here v = [vl,. . . ,v,] with vi 2 0 and v1 have already seen that V ( v )= Span{er : p(1) = v}.

Lemma 5.3: Let 6, = G,, x a 6k-module.

+ . . . + u,

335

=

k. We

. . . x G,, c 6 k . Then V ( v )cx e[6,/6:,]as

Proof: If I is a multi-index such that p(1) = u, then there is s that

---

E 6 k so

s . I = [ l ,..., 1 , 2, . . . ,2 , 3,.", 3 ,...1 . v1

m

v3

0

Since uk(s)el = es.r, the lemma follows.

From the lemma we see that ~ ( s acts ) as a permutation matrix on V ( v ) ,and hence trV(y)(uk(s))= #{fixed points of s on 6 k / G y } . The Weyl group for G is 6, and acts on the weight lattice P as permutations of the coordinates of the weights. Applying Theorem 4.2 and using Lemma 5.3, we obtain the following character formula.

Theorem 5.4: Let X

E Par(k,n)

and s E 6 k . Then

x x ( s ) = C s g n ( t ) #{fixed points of s on

6k/6X+pjnl--t.p,,l).

t

Here the sum is over all t E 6,such that all the coordinates of X+p[,]-t.p[,l are nonnegative. In particular,

In Theorem 5.4 44 = [n- 1, n - 2 , . . . , 1 , 0 ] and

is the multinomial coefficient (with the usual convention that it is zero if any entry in u is negative). The dimension formula can be written as a determinant and then reduced to Vandermonde form. This gives the following product formula for the dimension of the representation E X that is analogous to the Weyl dimension formula for the representation FA.

Corollary 5.5: Let X E Par(k,n). Then d i m E X= &Dn(X

+ p[,l).

R. Goodman

336

A partition X = [XI,. . . , An] can be represented in terms of its Ferrers diagram: a left-justified array of boxes, with X i boxes in the ith row (counting from the top down). Each box in the diagram has a hook length: the total number of boxes to the right and below the given box (including the box itself). We can then fill each box with its hook length. For example, X = [4,3,1] E Par(8,3) has Ferrers diagram and hook lengths

F’rom Corollary 5.5 one obtains (by induction on the number of columns of A) the Hook Length Formula

where hij(X) is the hook length of the i j box in the Ferrers diagram of A. By way of comparison, the Weyl Dimension Formula for GL(n, C)can be written as

(see [16, Sec. 9.1.4, Example #9]). For example, for on B 8 C 3 we have p[31 = [2,1,0]and

(58

x GL(3,C) acting

@I F/;j”.’l is an irreducible subspace of @*C3 of dimension Thus E[4,3i1] (70) * (15).

5.3. Schur-Weyl duality and GL(k)-GL(n) duality There is another model for the irreducible representations of 61, that comes from the identification of (5k with the Weyl group of GL(k,C). Let X = Mk,, (k x n complex matrices) and let GL(lc, C)x GL(n, C ) act on P ( X ) by p(gi, g z ) f ( x ) = f(gi2gz)

for gi E GL(k, c) and gz E GL(n, @) .

This representation is multiplicity free and decomposes as

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

337

with the sum over all partitions p with a t most min{k,n} parts (see [16, Sec. 5.2.41 or the article by Benson-Ratcliff in this volume). Let H k C GL(k,C) be the maximal torus of diagonal matrices, and embed 6 k c GL(k, C)as the permutation matrices. If we only consider the together action of the subgroup Norm ( H k ) 2 Gk K H k of GL(k, C)on with the right action of GL(n,C), then

x

Y

k summands

Hence

P(X)r S(C")8 . .. @ S(Cn) k factors

as a representation of Norm ( H k ) x GL(n, C).Here 6 k acts by permuting the tensor factors, while h = diag[zl,. . , ,xk] E Hk acts by multiplication by xj on the j t h factor. The weight space decomposition of P ( X ) relative to the H k action is thus

P ( X ) ( p )% Sml(C") @ . . . @ Smk(C") for p = [ m l ,. . . , mk] .

(22)

Here 6 k acts by permuting the factors in this decomposition while GL(nC) acts as usual on each copy of C".In particular, the weight detk = [l,1,. . . ,1]is fixed by 6 k and the corresponding weight space is P(X)(detk) % sl(@") €9.. * @ sl(@")= (Cn)@'" with the usual commuting actions of 6 k and GL(n, C). On the other hand, if we calculate this weight space using (21), we see that

(Cn ) @k 2

@

f'i\,)(detd @ F $ )

XEPar(k,n)

as a module for Gk x GL(n, C).Invoking Theorem 3.8 we conclude: For all

X E Par(k,n),

as a $k

module, with the action of

6 k

coming from its embedding into

GL(k, C). Examples (1) Take X = [ l , .. . , I] E Par(k). Then the representation Pi\,, of GL(k,C)

is Ak C k , on which GL(k, C)acts by g H det(g). This shows once again that EXis the sgn representation of 6 k .

R. Goodman

338

(2) Now take X = [k].Then the representation F h ) of GL(lc, C ) is Sk(Ck)Z Pk((Ck)*).The detk weight space is one-dimensional and spanned by the monomial X I . . ' x k , which is fixed by 6 k . Again we see that E['] is the trivial representation of 6 k . 6. Polynomial invariants and FFT

6.1. Invariant polynomials

Let G be a reductive linear algebraic group. Recall from Section 1that given a regular representation (T,V )of G, we have a locally regular representation p of G as automorphisms of the commutative algebra P ( V ) of complexvalued polynomial functions on V :

p ( g )f(v)

= f(g-'v)

for f E P ( V ) and g

E

G

(here we write g v for 7r(g)v when the action 7r is clear from the context). Since G acts by automorphisms of P ( V ) , the space J' = P ( V ) Gof Ginvariant polynomials is a subalgebra of P ( V ) .Thus we can consider P ( V ) as a module for 3 under the action of pointwise multiplication, which commutes with the G action. Then in the isotypic decomposition

P ( V ) = @P(V)(A)

x€a each summand is invariant under J . By Corollary 1.2 there is a projection f H f b from P ( V ) onto J , with degfb 5 d e g f . If f E P ( V ) and cp E 3' then

(PfIb= cpfb

+

(Decompose f = f b . . . into isotypic components; then q f = c p f b is the isotypic decomposition of cpf .)

(23)

+

Theorem 6.1 (Hilbert-Hurwitz): J' is finitely generated as an algebra over @.

Proof: Let J+ = {f E 3 : f(0) = 0) and write R = P ( V ) . Since R is a polynomial ring in dim V generators, the Hilbert basis theorem implies that the ideal R3+ is finitely generated as an R module: there exist q j E J'+ such that n

339

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

Furthermore, since ,7+ is invariant under the C x action on R (f(v) H f (Cv) for E Cx ), we may take each 'pj to be homogeneous of some degree d j 2 1. We claim that {'pj} generate 3 as an algebra over C. Let f E ,7 be of degree d and assume inductively that all polynomials in 3 of degree less than d are polynomials in 91,. . . ,9,.We can find fj E R so that f = C jfjcpj. Now project onto ,7 and use (23):

v i , 211,'

>urn)=

( ~ 5vj) ,

(the contraction of the ith covector with the j t h vector). There is a natural action of L on Mkxm with GL(k,C) acting by left multiplication and GL(m, C) acting by right multiplication, Hence L acts . map p intertwines the two L actions. on P ( M k x m ) GThe

Multiplicity-Ree Spaces and Schur- Weyl-Howe Duality

341

6.3. Polynomial F F T for GL(n)

The FFT for G L ( n , C ) is the assertion that the method just indicated to construct invariants furnishes the complete algebra of polynomial invariants. Theorem 6.2: Let G = GL(n,C). T h e n the m a p p* is surjective. Hence the km quadratic polynomials q5ij = p*(xij) with 1 5 i 5 k and 1 5 j 5 m give a set of basic invariants for P ( M k x n @ Mnxm)G . After discussing tensor invariants in the next section we shall show there how this theorem is an immediate consequence of Proposition 3.1. At this point we observe that the image of p consists of all k x m matrices x with rank(x) 5 min(k, m,n). This gives rise to the following dichotomy: (1) If n 2 min(k,m), then p is surjective. Hence p* is injective and

P(Mkxn @ hfnxm)GL(n'@) P(Mkxm) is a polynomial algebra with km generators. One says that h f k x m is the algebraic quotient of V*'"@ V" by GL(n,C). The representation of L on P ( M k x m ) is multiplicity-free (see [16, Theorem 5.2.71 or the article by Benson-Ratcliff in this volume). (2) If n < min(k,m) then Ker(p*) # 0. The group L acts on Ker(p*), and from the multiplicity-free decomposition of P ( M k x m ) under L one finds that Ker(p*) is a determinantal ideal generated by ( n4- 1) x ( n 1) minors. Thus P ( M k x n CB Mnxm)GL(niC) is the algebra of regular functions on the associated determinantal variety. This is the Second Fundamental Theorem (SFT) for GL(n,C) invariants (see [16, Theorem 5.2.151 for the complete statement).

+

6.4. Polynomial F F T for the orthogonal group

We next consider the full orthogonal group relative to the bilinear form B ( z ,y) = xty on V = Cn: G = O ( n , C ) = ( 9 E G L ( n , C ) : gtg =I}. Since V 2 V * as a G-module, via the form B , it suffices to consider the invariants of k vector arguments P ( V k ) G= P ( M n x k ) G ,where G acts on Mmxk by left multiplication. Define a map 7

: MnXk 4

SMk

(k x k symmetric matrices),

~ ( x =) x t x

R. Goodman

342

For g E G we have r ( g x ) = xtgtgx = r ( x ) . Hence T*

:P(sMk)+ P(Vk)G

as in the case of GL(n, C). In particular, if we take f = xij (the ( i ,j ) matrix entry function on S M k ) , then T*(Xij)(VI,.

.. ,Uk)

= VZVj t

(the inner product of the ith and j t h vectors). The map r intertwines the right action of the hidden symmetry group L = GL(k,C) on Mnxk. Here the action of L on SMk is given by x H bxbt (for b E L ) . Theorem 6.3: Let G = O(n,C). Then the map T * is surjective. Hence the k(k 1)/2 quadratic polynomials 8ij = r * ( x i j ) with 1 5 i 5 j 5 k give a set of basic invariants f o r P(Mnxk)G.

+

Proof for the case n 2 k: There is a natural G-equivariant embedding Mnxk C M n x n ; just add n - k columns of zeros on the right to make x E Mnxk into an n x n matrix. Hence we may assume that k = n . Now 0 see [16, Proposition 4.2.61 for the proof.e We shall complete the proof for the general case n < k after discussing tensor invariants in the next section. Here we observe that the image of r consists of all k x k symmetric matrices x with rank(x) 5 min(k,n). This gives rise to the following dichotomy:

(1) If n 2 k, then 7 is surjective. Hence T * is injective and P ( M n x k ) G P ( s M k ) is a polynomial algebra with k(k + 1)/2 generators. One says that SMk is the algebraic quotient of Adnx,+by O ( n ,C).The representation of L on P(hfn,k)G is multiplicity-free (see [16, Theorem 5.2.91 or the article by Benson-Ratcliff in this volume).

(2) If n < k then Ker(r*) # 0. From the multiplicity-free decomposition of P ( s M k ) under L one finds that Ker(.r*) is a determinantal ideal generated by ( n 1)x (n 1) minors. Thus P ( M n x k ) Gis the algebra of functions on the associated symmetric determinantal variety. This is the Second Fundamental Theorem (SFT) for O(n, C)invariants (see [16, Theorem 5.2.171 for

+

+

the complete statement). eThe proof is by induction on n and can be viewed a s an algebraic group version of the Q R factorization for M , and the Cholesky Decomposition for SM,. This result is associated with a particular partial compactzfication of the symmetric space GL(n, @ ) / O ( nC). ,

Multiplicity-he Spaces and Schur- Weyl-Howe Duality

343

6.5. Polynomial FFT for the symplectic group Now consider the symplectic group G = Sp(Cn,R), where n = 2p is even and

Here I p is the p x p identity matrix. Thus G is the subgroup of GL(n,@) defined by g t J g = J . Since (Cn)* 2 @" via the form 0, it suffices to consider the invariants of k vector arguments P ( V k ) ) G= ? ( M n x k ) . Define a map : Mnxk

-+

AMk

(k x k skew-symmetric matrices), y ( x ) = x t J x .

For g E G we have y ( g x ) = xtgt J g x = y(z). Hence

y* : P ( A M k ) -+ p ( V k ) ) G as in the case of O(n,C). In particular, if we take f entry function on AMk), then

= xi?

(the ( i ,j ) matrix

y*(xij)(vl,"' ,vk) = fl(vi,vj) (contraction of the ith and j t h vectors by 0). The map r intertwines the right action of the hidden symmetry group L = G L ( k , C ) on Mnxk with the action of L on AMk given by x H bxbt (for b E L ) .

Theorem 6.4: Let G = Sp(Cn,0). Then the map y* is surjective. Hence the k(k - 1 ) / 2 quadratic polynomials wij = y * ( x i j ) with 1 5 i < j 5 k give a set of basic invariants f o r P(Mnxk)G. Proof for the case n

2 k: The

same citation as for the orthogonal

0

case.

We shall complete the proof for the general case n < k after discussing tensor invariants in the next section. Here we observe that the image of y consists of all k x k skew-symmetric matrices x with rank(x) 5 min(k,n). This gives rise to the following dichotomy:

2 k, then y is surjective. Hence y* is injective

and P(Mn,k)G E ?(Ahfk) is a polynomial algebra with k(k - 1 ) / 2 generators. One says that AMk is the algebraic quotient of Mnxk by Sp(C", 0).The representation of L on P ( h & x k ) Gis multiplicity-free (see [16,Theorem 5.2.111 or the article by Benson-Ratcliff in this volume).

( 1 ) If n

R. Goodman

344

(2) If n < k then Ker(y*) # 0. From the multiplicity-free decomposition of P ( A M k ) under L one finds that Ker(y*) is generated by a set of Pfafian polynomials of degree n/2 1. Thus P ( I I ~ , , ~is)the ~ algebra of functions on the associated skew-symmetric Pfafian variety. This is the Second Fundamental Theorem (SFT) for O ( n ,C ) invariants (see [16, Theorem 5.2.181 for the complete statement).

+

Summary: For a classical group G (general linear, orthogonal, symplectic), the G-invariant polynomial functions of vectors and covectors are generated by all the possible G-invariant contractions of vectors and covectors. 7. Tensor invariants and proof of FFT 7.1. Tensor invariants f o r GL(V)

We turn now from consideration of invariant polynomials to the general case of invariant tensors. Let V = C" and consider the mixed tensor space V@m8 V*Bk as a GL(V) module. For E (Cx the element i=l

p(g)Mip(g-') = C g i j M j . j=1

(34)

+

The set {Gr(M"DP) : la1 IpI = k } is a basis for Grk(PD(V)).Thus the nonzero operators of filtration degree k are those of the form

+

with cap # 0 for some pair a,@with JaI 1/31 = k (note that the filtration degree of T is generally larger than the order of T as a differential operator). If T in (35) has filtration degree k then we define the symbol of T to be the polynomial o(T)E P k ( V@ V') given by

Lemma 8.1: The symbol map gives a linear isomorphism P V ( V ) P ( V @ V') as GL(V)-modules.

Z

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

353

Proof: Using (33), one shows by induction on k that any monomial of degree Ic in the operators D1,. . . ,D,, M I , . . . , Mn is congruent (modulo P D I , - ~ ( V ) to ) a unique ordered monomial MOD0 with Icy1 = Ic. Hence o ( T )= o(S)if Gr(T) = Gr(S). Thus (T gives a linear isomorphism

+

Grk(PD(V))

P k ( V @ V*)

Since p(g)PDk(V)p(g-') = PDk(V), there is a representation of GL(V) on Grk(PD(V)) for each k. From (34) we see that Di transforms as the vector ei under conjugation by GL(V), while Mi transforms as the dual vector e t . Since GL(V) acts by algebra automorphisms on Gr(PD(V)) and on P ( V @V*), this implies that (36) is an isomorphism of GL(V) modules. Now compose these maps with the canonical quotient maps P'Dk(v) -+ Grk( P D ~ ( v ) ) . 0 We can now obtain the general Weyl algebra duality theorem:

Theorem 8.2: Let G be a reductive algebraic group acting regularly on V. Then there is a multiplicity-free decomposition P(V)E

@

EX@@,

(37)

X€C(V)

as a module under the joint actions o ~ P D ( Vand ) ~ @[GI.Here C ( V ) c 6?, FAis an irreducible regular G-module of type A, and E X is an irreducible module for P V ( V ) G that uniquely determines A.

Proof: We apply Theorem 1.4, with L = P ( V ) and R = P D ( V ) . Note that L is the direct sum of the finite-dimensional G-invariant subspaces Lk = P k ( V )of homogeneous polynomials of degree k . The action of G on each LI, is regular, so L is a locally regular G module. We shall show that R satisfies conditions (i) and (ii) of Theorem 1.4. Let 0 # f E P ( V ) be of degree d. Then there is some Q E W" with ( Q (= d such that 0 # D"f E C. Given any g E P ( V ) ,let M g E P D ( V ) be the operator of multiplication by g . Then g E @MgDaf. This proves that R acts irreducibly on P ( V ) (condition (i)). The algebra R is the union of the finite-dimensional G-invariant subalgebras PDk (V), and the action of G on FDI,(V) is regular by Lemma 8.1. Hence R also satisfies condition (ii). 0 To use Theorem 8.2 effectively for a particular G-module V we need a more explicit description of the algebra PD(V)G. The following result is a first step in that direction.

R. Goodman

354

Theorem 8.3: Let {$I,. . . , $J,} generate the algebra P(VCBV*)~. Suppose Tj E PD(V)G are such that a(Tj) = $j for j = 1,.. . , T . Then {TI,.. . , T,} generates the algebra PD(V)G.

Proof: Let 3 C PD(V)G be the subalgebra generated by T I ,. . . ,T,.. Then P D o ( V )=~ CI c 3. Let S E PD,(V)G have filtration degree k. We may assume by induction on k that P D ~ - I ( Vc) 3. ~ Since a(S) E P k ( V ~ V * ) G by Lemma 8.1, we can write

where ~ j ~ . .E. ~CC., Set

Although R is not unique (it depends on the enumeration of the Tj), we have a ( R ) = a ( S ) since 0 is an algebra homomorphism. Hence R - S E PDk-I(V) by Lemma 8.1. By the induction hypothesis, R. - S E 3,so we have S E 3. 0

Corollary 8.4: (Notation as in Theorem 8.3) Suppose T I , .. . ,T, can be chosen so that

g’ = Span(T1,. . . ,T,.} is a Lie subalgebra of PD(V)‘.

Then in the canonical decomposition (37) the spaces E X are irreducible modules for the Lie algebra g‘, and X is uniquely determined b y the equivalence class of E X as a 0‘-module. Hence there is a bijection (duality correspondence)

W )* W)

1

where A(V) is the set of irreducible representations ofg‘ that occur in P ( V ) . Proof: The representations of the Lie algebra g’ are the same as the representations of the universal enveloping algebra U(g’). Let p’ : U(g’) -+ End(P(V)) be the representation associated to the action of g’ as differential operators. The assumption on T I ,. . . , T, implies that p’(U(g’)) = PD(V)G. Hence the irreducible PD(V)G-modulesare the same as irreducible 8’-modules. 0 Remark: Theorem 8.2 is also valid when V is any smooth connected affine G-variety. Here we take R = D(V) to be the ring of algebraic diferen-

355

Multiplicity-Fkee Spaces and Schur- Weyl-Howe Duality

tial operators on V and use Theorem 1.4.9 The algebra D(V)G in this case (G connected, reductive), has been studied by Knop [25]. He proves that its center 3G(V) is a polynomial ring in rankG(V) generators, where rankG(V) = dim Bx - dim N z for a generic point z E V (here B is a Bore1 subgroup of G with nilradical N ) . Furthermore, ItD(V)G is a free module over 3G(V) (this is a generalization of results of Kostant [26] for the case V = G, with G acting by left multiplication). The representation theory of D(V)G seems to be unknown, in general, although special cases have been studied by I. Agricola, F. Knop, T. Levasseur, G. Schwarz, J. Stafford, and others. 8.2. Howe duality f o r orthogonal/symplectic groups We now determine the structure of PD(V)G when G is an orthogonal or symplectic group and V is the sum of n copies of the fundamental representation of G. Using the First Fundamental Theorem of classical invariant theory, we will show that the assumptions of Corollary 8.4 are satisfied. This will give the Howe duality between the (finite-dimensional) regular representations of G occurring in P ( V ) and a set of irreducible representations of the dual Lie algebra 8'. Let w be a nondegenerate bilinear form on C kthat is either symmetric or skew symmetric, and let G c GL(lc,C) be the isometry group of w .Thus G is the (complex) orthogonal group when w is symmetric, and G is the (complex) symplectic group when w is skew (and k even). Let V = (C'))". Then

--

P ( V @ V*) = P( C k63 * * . @ ck@

(P)* @ . . . @ (Ck)*) .

n copies

n copies

Hence if T E P D ( V ) then the symbol of T is a polynomial function

f(x1,.. . ,xn,J1,. . . ,&)

,

where xi E Ck,

&

E (Ck)*.

From Section 6 we know that the algebra of G-invariant polynomials P ( V @ V*)G is generated by three types of quadratic polynomials: evaluation of w on two vectors: evaluation of W * on two covectors: contraction of vector-covector:

. . , xn,tl,. . . ,&) . . ,En)

pij(x1,.. . ,xn,&,.

= w(xi,xj) = w*(&, [ j )

cij(x1,.. . ,x,,&,

= (xi,&)

rij(x1,.

. . . ,&)

gSee [l];the smoothness assumption on V is essential here, since the action of D(V) on Aff(V) can fail to be irreducible when V is not smooth.

R. Goodman

356

where 1 5 i , j 5 n and w* is the form on (Ck)* dual to w . There is a canonical GL(V)-module isomorphism d from P(V*) E S(V) to the algebra of constant-coefficient differential operators on V. The linear span of the quadratic invariant polynomials above furnish symbols for the following Lie algebras of G-invariant differential operators:

p- = Span(multip1ication by rij : 1 5 i , j 5 n } p+ = Span(differentiati0n by Aij = a ( p i j ) : 1 5 i , j 5 n } t = Span{Eij + $ ~ i :j 1 5 i , j 5 n> Here it is convenient to identify V with M n x k , with G acting by right multiplication. If xi denotes the ith row of x E Mnxkrthen k

Eij = xi . VXj=

d 1 xir -. r=l axji"

The operators Eij , which correspond to vector-covector contractions, commute with the right action of all of GL(lc,C) (Eij is the classical polarization operator). Obviously [p-,p-] = 0 and [p+,p+] = 0. An easy calculation shows that lelP,I

The choice of shift tation relation.

f&j

= P,

9

[P-,P+I = I!.

for the operators in t arises from the last commu-

p- + t + p + . Then 5' is a Lie algebra and it generates the associative algebra P v ( h f n x k ) G . Furthermore, Theorem 8.5: Set 5' 5,

~

=

{ sp(n,@) 50(2n,@)

The subalgebra t representation

S

when w is symmetric when w is skew

5K(n,@) acts o n

p(j?fnxk)

b y the differential of the

of K = GL(n,@) (replace K by its two-fold cover if Ic is odd).

Proof: The first statement follows from Corollary 8.4. The other parts are 0 easy calculations (see [16, Sec. 4.51 for details). We call g' the Howe dual of g = Lie(G) associated to the representation of 5 on V. Notice that the correspondence between 5 and g' interchanges orthogonal and symplectic Lie algebras. There is an asymmetry between g and g', however. The action of 5 on P ( V ) is by vector fields (corresponding

Multiplicity-Free Spaces and schur- Weyl-Howe Duality

357

to the representation of G on V ) ,whereas the subalgebras pk of g' act by second-order differential operators and multiplication by quadratic polynomials, which do not come from a geometric action on V . We will show in Section 11 how to exponentiate the action of a real form of g' on P ( V ) to a unitary representation of an associated real Lie group on a Hilbert-space completion of P ( V ) . 8.3. Howe duality for GL(k)

--

Now consider G = GL(k,@) acting on

v = ck@ . . . €9 ck€9 (C"*@ ' . . €9 (C"* p copies

q copies

In this case the symbol of T E P D ( V ) is a polynomial function

f(x1,.. . , x p , v 1 , .. . , 7 7 q , F l , . . . , E P , Y l , .. . , Y q ) , where [XI,. . . , xP,q1,.. . , vq] E V and [(I,. . . , Q , Y I , . . . ,yql E V * (xi,y j are vectors in C k and &, vj are covectors in ( C k ) * ) Theorem . 6.2 asserts that the algebra of G-invariant polynomials on V @ V* is generated by contractions of a vector with a covector. Now there are four possibilities for contractions: (1) vector and covector in V : (xi,q j ) for 1 5 i 5 p and 15j.59

(2)

vector and covector in

V*:

(yj,

ti)

for 1 5 i

< p and

I l j < q

(3) vector from V , covector from V * : (xi,&) (4) covector from V , vector from V * : (yi,733)

We can identify V with Ad(,+,)

k

for 1 5 i , j 5 p for 15 i , j < q

if we make g E G act on the right by

Here xi is the ith row of x and v3 the j t h row of v. Contractions of type (1) and ( 2 ) furnish symbols for the G-invariant operators

p- = Span{ multiplication by rij = (xi,vj)) p+ = Span{ differentiation by Aij} , where

Aij

= V,,

. V,

=

8 C" T=l

dxiT

%jT

for 1 < i < p a n d 1 < j 5 q .

R. Goodman

358

The linear span of contractions of type (3) and (4) furnishes symbols for the G-invariant operators t = Span{E$)

+ ~ d i :j 1 5 i,j L p)

+ $6ij

Span{,$)

:

1I i,j

I q) ,

where E$) is the polarization operator for the x variables and Ei:) for the 77 variables. By the same argument as in Theorem 8.5 we conclude that

P V ( V ) Gis generated by

+ e + p+ .

g’ = p-

These subalgebras have the commutation relations

P, P*I

= P*

[P-,P+l c e

I

*

+

In this case g’ is isomorphic to gI(p q , C ) , with t 3 gI(p, C) @ g ( ( q , C). The action of e on P(M,,k @Mpxk)is the differentialof the representation

p(g,h)f(x,7 ) = (det g det h)-”’f(g-’x,.h-’7) for (9,h ) E K = GL(p, C ) x GL(q, C). (We must replace K by the two-fold covers of each factor when n is odd). 9. Harmonic duality

9.1. Harmonic polynomials

Let G be O ( C k , w ) ,Sp(Ck,w),or GL(lc,C) acting on V = M n x k on the right. In the case of GL(Ck) the first p rows of 2 E Mnxk transform as vectors, whereas the remaining q rows transform as covectors. From Section 8 the Howe dual to G is, respectively, g’

sp(n, C), so(2n,C), or gI(p

+ q1C )

with p

+q =n.

We will assume that p > 0 and q > 0 in the third case.h With G fixed, the spectrum C ( V ) of G on P ( V ) only depends on n (or the pair p,q in the third case); we can thus denote it by E(n) (or C ( p ,4)). From the dual point of view if we fix g’, then the set A(V) of irreducible representations of g‘ that occur in P ( V ) only depends on k ; we can thus denote it as A ( k ) . The general duality theorem gives a bijection E ( n ) H R(lc). We now show how to express this bijection in terms of harmonic duality. hIf p = 0 or q = 0 then e = g’ and the modules E X8 F A that occur in the decomposition of P(V)are finite-dimensional, This is the well-known GL(n)-GL(k) duality (see the lectures of Benson-Ratcliff in this volume).

Multiplicity-FI-ee Spaces and Schur- Weyl-Howe Duality

359

In all cases there is a triangular decomposition g1 = p- @ e e 3 p+

.

Here C is the Lie algebra of the reductive group K (a two-fold cover of GL(n, C ) or GL(p,C ) x GL(q, C ) in general). The representation of K on P ( V ) is the natural representation associated with the left multiplication action of GL(n, C ) on V tensored with the one-dimensional representation g

H

(det g)-'/'

or

(g, h ) H (det g det h)-'/'.

Let 6 denote this character, viewed as a weight of the maximal torus of K . The subalgebra p- acts by multiplication by G-invariant quadratic polynomials, whereas p+ acts by G-invariant constant-coefficient Laplace operators {Aij}. We define the G-harmonic polynomials to be

ni,j

'FI = P ( v ) ~ =+

Ker(Aij)

Since Ad(K)p+ = p + , the space 'FI is invariant under the reductive group K x G. In this section we will show that 'FI gives a multiplicity-free duality pairing between irreducible representations of K and G; furthermore, the decomposition of Fl generates the decomposition of P ( V ) under g1 and G. Let U ( k ) c GL(k,C) denote the unitary group. Then KO= K n U ( n ) is a compact real form of K . We assume that the bilinear form w is chosen so that Go = G n U ( k ) is a compact real form of G and w is real on R k . Define an inner product on Mnxk by

(x I y) = tr(y*x) for x , y E M n x k (y* = g t ) . This inner product is invariant under U ( n )x U ( k ) ,acting by left and right multiplication, hence it is invariant under KOx UO.We set 11z11' = (z I x). Let f(x) = C, caxa be in P ( V ) , where LY is a multi-index and xa = as usual (xij are the matrix entry functions on i k f n x k ) . Define the constant-coefficient differential operator

a(f)

is an algebra isomorphism from P ( V ) to the constantThen f H coefficient differential operators on V that is equivariant relative to the action of U ( k )x U ( n ) .Set g*(z) = s(z)for g E P ( V ) .If g ( x ) = C , daze, then

(a(f)g*)(o)=

c

a! c a d , .

a

(38)

R. Goodman

360

We define (f I g ) = ( a ( f ) g * ) ( O ) .From (38) we see that this is a positive definite Hermitian inner product on P ( V ) ,called the Fischer inner product. We note that

( f g I h) = (f I d(g*)h)

(39)

for all f , g , h E P ( V ) . The Fischer inner product has the following analytic definition. Denote Lebesgue measure on V by d X ( z ) , where we identify V with RZnkvia the real and imaginary parts of the matrix coordinates.

Lemma 9.1: For f , g E P ( V ) one has

( d = dim@V = nk).

Proof: See the lectures of Benson-Ratcliff in this volume.

0

9.2. Main theorem

We now apply the Weyl algebra theorem from Section 8 to obtain multiplicity-free decompositions of the harmonic polynomials and the entire space P (V).

Theorem 9.2 (Harmonic Duality): (1) The space 'H of G-harmonic polynomials on V decomposes under K x G into mutually orthogonal subspaces (relative to the Fischer inner product) as 'H = @

&'(")+6@3"

oEC(V)

Here C ( V ) c is the spectrum of P ( V ) as a G module, 3' c H ' is an irreducible G-module of type 0, and c 'H is an irreducible finite-dimensional K-module with highest weight r(c)+ 6. In particular, every irreducible representation of G in P ( V ) is realized in the harmonic polynomials. (2) Set E 7 ( u ) f 6= P ( V ) G.&'(')f6. ule and

Then E'(")+' is an irreducible g' mod-

P ( V ) = @ E'(")+6@3" UEC(V)

is an orthogonal decomposition of P ( V ) (relative to the Fischer inner product) under the mutually commuting actions of g' and G. (3) The map c H ~ ( 0from ) C(V) free as a K x G module.

---f

k is injective. Thus 'H is multiplicity-

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

361

Proof: Since 3-t is an invariant subspace for the reductive group K x GI x and multiplicity function m : -+ { 1 , 2 , . . .} there is a subset I? c such that

2

3-t g @

@"(PP)

@

&P+b

@

30

(40)

(p,+r

with K x G acting trivially on the multiplicity spaces @ " ( P i u ) . Indeed, we first consider 3-t as a locally-finite K x G-module relative to the natural left-right action on V = M n x k (omitting the determinant twist from the K representation) and use Proposition 1.1and Example 1 in Section 2.3. Then tensor with the character det-"' of K to shift the highest weights from p, to p, b. To prove that r = { ( T ( o ) , o ) : D E C ( V ) } and ~ ( T ( o ) , D = ) 1, we need to examine the action of 0' on P ( V ) in more detail. Let J' = P ( V ) G be the G-invariant polynomials, and let 3. be the homogeneous polynomials of degree j in 3. Then J'j = 0 for j odd, and ( p - ) j acts by multiplication by 3 2 . on P ( V ) . Since the bilinear form w is real on R k , we have J * = J and 3-t* = 3-t. Let J+ = {f E J' : f ( 0 ) = 0). We claim that

+

7-P = J+.P ( V )

(41)

(orthogonal complement relative to the Fischer inner product). Indeed, if

f E ,7+ . P ( V ) and h E 3-t then a(f)h = 0 by definition of 'HI and thus f Ih. Conversely, if h IJ+ . P ( V ) then for all f E P ( V ) and g E J+, 0 = (fg I h ) = (f I a ( g * ) h ) . Hence d ( g ) h = 0 for all g E ,7+,so we have h E 3-t. We can now determine the general structure of the irreducible 0'modules in P ( V ) . The commutation relations in 0' can be expressed as

P+P-

c P-P+ + e

I

eP-

c P-(e

+ 1)

in the universal enveloping algebra U(0'). Hence by induction, one has

P+(P-)"

C

(P-)"P++(P-)"-l(e+l)

for all integers m 2 1. Thus if 2 then (42) implies that

p+(p-)".

2 c (p-)"-'

e(P-)"

C

(P-)"(t+l)

(42)

c 3-t is any P-invariant linear subspace,

.Z and t ( p - ) " . 2 c (p-)" .Z

(43)

for all m 2 1. (a) L e t & c 3-t be a n y t-irreducible subspace. S e t E = J .&. T h e n E is a n irreducible 0'-module and & = E n 3-t.

R. Goodman

362

Indeed, (43) implies that E is invariant under g’. Also every f E E is of the form m

f=xgjhj

whereOfgjEJ2jandhjEE.

(44)

j=O

Suppose F c E is a nonzero g‘-invariant subspace. Take f E F so that the integer rn in (44)is minimal. Then (43) implies that p + f = 0. Hence f E 3-1. Thus m j=1

Since the left side is in J+.P(V), it must be zero by (41). Hence we conclude that f E &. But t acts irreducibly on &, so U(0)f = & and thus F = E . The same argument shows that E n 3-1 = &, completing the proof of (a). (b) Let E c P ( V ) be an irreducible g’module. Set & = E n 3-1. Then & is an irreducible t-module and E = J .&. Note that the action of p+ on P ( V ) lowers the degree of polynomials, so & # 0. If 0 # 3 c & were a proper t-submodule, then J .F c E would be a proper irreducible g’-submodule by (a). Hence & must be irreducible as a t-module and E = J . &, proving (b). (c) Let & and 3 be t-invariant subspaces of 3-1. Assume that & I3 (relative to the Fischer inner product). Set E = J ‘ & and F = J .3. Then E I F . By (41) we have the orthogonal decompositions

E

=&

@J+.&

F = 3@ J+ .3.

Thus E I 3 and F I&, so we only need to verify that Now

J+ . & I J+ .3.

(J+ I Jt . 3)= (E I 8(J+)J+. 3 ) . ’

But 8(J+)Jt . 3 proving (c).

c F since 3 is &invariant. Hence & I 8(J+)Jt . 3,

We now complete the proof of the theorem. It is clear from the integral formula for the Fischer inner product (Lemma 9.1) that Go and KO act by unitary operators on P ( V ) , hence the decomposition (40) of X is orthogonal relative to the Fischer inner product because Go and KOhave the same finite-dimensional invariant subspaces in P ( V ) as G and K , respectively (see [16, Sec. 2.4.41). Also, since K is connected, a finite-dimensional subspace of P ( V ) is invariant under K if and only if it is invariant under 0.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

363

Let ( p , a) E I’ occur in (40). By (a), (b) and Theorem 3.4 we know that the irreducible 8’-module E’”+6= ,7 . €’”+6 uniquely determines p. On the other hand, Theorem 8.2 asserts that P ( V ) is semisimple as a 8‘-module, with the 8’ multiplicity spaces being irreducible regular G-modules corresponding bijectively to the associated 8’-modules. Hence J? is determined If we call this projection C and write the elements by its projection onto of I’ as (~(a),cr), then the map a H .(a) is injective. The multiplicities m ( ~ ( c ) ,= c )1 for all a E C, since otherwise (a) and (c) would imply that 3“ is paired with more than one copy of an irreducible 8‘ module, contradicting Theorem 8.2. Finally, (b) implies that C = C ( V ) ,since P ( V ) is semisimple as a g’-module 0

e.

Remarks: (1) Theorem 9.2 was obtained by Howe in his influential paper [21] (which circulated as a preprint for more than a decade); his proof used an argument based on a filtration by finite-dimensional subspaces and the classical double commutant theorem, instead of Theorem 8.2. Knowing that the decomposition of the harmonics is multiplicity free simplifies the task of finding harmonic highest weight vectors, as we will see in Section 10. (2) The shift by S in the highest weights for K in the harmonic decomposition would appear to be a minor nuisance. In fact, it plays an important analytic role. For T E P D ( V ) let T * denote the adjoint of T relative to the Fischer inner product:

If we write T in polarized form as T = Cj f j a ( g j ) with fj, g j E P ( V ) ,then we see from (39) that T* = Cjg,*a(f,’). Hence ( p * ) * = pr and t* = t. It follows that := {T E 8’ :

T* = -T}

is a real form of 8’.We will show in Section 11 that the irreducible representation of gb on the space E X ,with X = .(a) 6, can be integrated to a

+

unitary representation .TT’ of a (noncompact) real group Gb with Lie algebra gb. The shift by 6 controls the rate of decay a t infinity on Gb of the matrix entries of d. We will show in Section 12 that for lc large enough (relative to n),the representations .TT’ are square-integrable (recall that 6 = lc60, where So is a fixed weight of 8’).

(3) The injective map a ++ is called the theta-correspondence (more precisely, the local theta-correspondence over R) because of the connection between the oscillator representation and theta-functions (see [5]), There

364 h

R. Goodman

are many recent papers devoted to the problem of understanding the thetacorrespondence from a geometric orbit perspective (see [19, Chap. 121 for a survey). 10. Decomposition of harmonic polynomials

We now turn to the explicit determination of the harmonic duality from Section 9 when G is the orthogonal group and g' the symplectic Lie algebra (for the other two cases, when G is the symplectic or general linear group, see [22]and [S]).It is convenient to take G as the orthogonal group O ( C k w) , for the symmetric form w ( x ,y ) = X t C k y on C k , where

when

when

Here Il denotes the 1 x 1 identity matrix. This choice of w ensures that the diagonal matrices in G give a maximal torus. Also G is a self-adjoint matrix group (invariant under g H g * ) , so the subgroup Go = G n U ( k ) is a compact real form of G, and w is real on the real matrices, as we assumed in Section 8. In accordance with the block decomposition of c k , we write elements z E Mnxk as

z = [ x y ] whenk=21,

z=[x y

t ] whenk=21+1,

(45)

where x , y E Mnxl and t E @". Define the map

P : Mnxk

-+

SMn,

P(Z)

= zCkzt.

From Theorem 6.3 we know that the algebra of G-invariant polynomials on Mnxk (relative to right G-multiplication) is generated by the matrix entries of 0:

P(Z)P,

=

EL1

(YPS

xqs

+ XPS Yqs)

c",=, (yPsxqs + xPsy q s ) + tptq a(Ppq)

when Ic = 21, when k

= 21

+ 1.

(46)

We denote by Apq = the corresponding constant-coefficient differential operators, as in Section 9.1. The space of G-harmonic polynomials is

Multiplicity-he Spaces and Schur- Weyl-Howe Duality

365

Denote by 'Ft(j) the G-harmonic polynomials that are homogeneous of degree j . The space IH is invariant under GL(n, C)x G with the action ~ ( hg ), f ( z )= f(h-l.9)

for h E GL(n, C ) and g E G .

(Note that we have omitted the factor (deth)-k/2 that occurs in Theorem 8.5, so IT is single-valued on GL(n, C)even when k is odd). From Theorem 9.2 we know that 'H decomposes under the representation IT as a multiplicity-free direct sum' 'Ft

=

@ E'(")

@ FU.

UTEC

We will now determine C and the duality correspondence c H ~ ( 0 )The . key point is to find generators for the algebra W N n x Nof harmonic highest weight vectors, relative to a Borel subgroup B, x B c GL(n,C) x G. Here B, = D,N, is the upper-triangular subgroup of GL(n, C ) (D,the diagonal matrices, N , the unipotent upper-triangular matrices), and B = H N is a Borel subgroup of G. The fact that 'Ft is multiplicity-free under GL(n, C )x G will play a crucial role.

Notation: We denote by ~j the character diag[al,. . . , a,] ++ aj of D,. We write N$+ for the integer ptuples X = [ml, . . . ,m,] with ml 2 m2 2 . . 2 mp 2 0. Set 1x1 = m l + . . . + m , and define the depth of X to be the smallest integer i such that mi > 0 (if X = 0, set depth(0) = 0). +

10.1. O ( k ) Harmonics (k odd) Assume that k = 21+1 is odd. Then G = G ox {H}, where Go = SO(C',w) is the identity component of G. We fix the Borel subgroup B = H N c Go as follows. The maximal torus H consists of the diagonal matrices h = diag[zl,. . . ,zi,zF1, . . . ,zcl,11, xi E C x .

The unipotent radical N has Lie algebra n consisting of the matrices with block decomposition

[: O

b -at -Ct

O

,

a E Mixi strictly upper-triangular, b = -bt € M i x i ,C E C ~ .

'The irreducible GL(n, C)-module

&r(u)

I$::',

in the notation of Section 3.

(47)

R. Goodman

366

The weights of H are parameterized by Z1. For h E H and X = [ml,. . . , ml] E Z'we set hX = xy' . . . xy' for the corresponding character of H . The irreducible representations of G remain irreducible on restriction to Go and (? is parameterized as {d>'where }, X E.N'++- is the highest weight ~ l . G = G I U G-1, where for G o ,E = f l , and 7 r X l ' ( - I ) = ~ ( - 1 ) lThus h

21 = {(A,

h

1) : X E

,

N:+}

G-1 = {(A, -1) : X

E

PI;+},

(see 116, Sec. 5.2.21).

+

Theorem 10.1: (G = O(Ck,u), k = 21 1) Let C be the spectrum of G on the G-harmonic polynomials 'Ft C P ( M n x k ) . (a) Assume k 5 n. Then C

=

2 and

hence C does not depend on n

(G-stable range). (b) Assume 1

< n < k. Then (?I c C and C n (?-I

= {(A, -1) :

k - n 5 depth(X) 5 l }

(unstable range: C depends on k and n ) . (c) Assume n I 1. Then C n

2-1= 8 and

C n (?I = {(A, 1) : depth(X) 5 n} .

Thus C does not depend on k (GL(n)-stablerange). The duality correspondence is given as follows: Let X = [ml,. . . , md, 0 , . . . , O] E Pi$+ have depth d with 0 5 d 5 min{l,n}. Then '

-

[o,. . . ,o, -md,. . . , -ml]

for u = (A, 1) E

c n El,

n-d

.(a)

=

[O,. . . , O , -1,.

. . -1, - m d , . . . , -ml]

-A n-k+d

k-2d

for u = (A, -1) E

cnC-l.

Remark: The parameter E for the representation diE is determined by the corresponding GL(n)highest weight .(A, E ) , since left multiplication by -In on Mnxk is the same as right multiplication by -Ik. Hence 'Ft is also multiplicity-free as a module for GL(n,C)x Go (this property will be used in the proof). The first step in the proof of Theorem 10.1 is to find a set of generators for the joint eigenfunctions of B, x B in 'H. Just as in the case of GL(n) x

Multiplicity-free Spaces and Schur- Weyl-Howe Duality

367

GL(k) duality (see the article by Benson-Ratcliff), the general strategy is to take appropriate minor determinants. By (46) the operators Apq are given in coordinates as

The minors of z = [x y t] are linear functions of each column of the matrix components x, y, t. If the minors are chosen to depend only on x or to be linear in t , then they will obviously be harmonic. If they depend on both x and y, then interchanging a n x column for a y column will change the sign of the minor but not change the action of the operators Apq= Aqp, so once again the minor will be harmonic. We now proceed to carry out this program. Let p 5 n and q 5 1. For u E Mnxl define p x q submatrices Lp,q(u)and Rp,q(u) of u by

For t E @" and j 5 n define

qj)=

[tnI:"]

EP - j

(the bottom j entries o f t ) . Let z = [ x y t ] E M n x k as in (45). Define f j ( z ) = detLj,j(x) for 15j 5 min(1,n).

If n

2 1 + 1 then we also define

Sj(Z)

=

i

+

I

for j = 1 1 , det [ L j , l ( 4 t ( j ) det [Lj,l(x) Rj,j-~-~(y) tcj)] for 1 2 I j I min{n, Ic}.

+

Lemma 10.2: (a) Let 1 5 j 5 min{l,n}. Then B, x B eigenfunction of weight ( p ,v), where p=

-&,-j+l

- . . . - E~

and

fj E

> 1 and let 1 + 1 5 j 5 min{n, Ic}. Then g j a B, x B eigenfunction of weight (ply), where

(here y = 0 zfj = I c ) ,

- . . . - E,

fj

is a

v = E ~ + . . . + E ~ .

(b) Assume n

p = -&n-j+l

' H ( j ) and

and y = E I +

9

.

.

E 'H(j)

+&k-j

and g j is

R. Goodman

368

Proof: Assume 1 i j 5 min{l, n}. It is clear from (48) that Apqf j ( z ) = 0 for 1 5 p , q 5 n, since f j ( z ) only depends on x. The diagonal matrices in Bn and B act on f E P(hfnxk)by with a E Dn, b E D1 .

f(x,y , t ) H f (a-'zb, a-lyb-l, a-'t),

(49)

+

Since f j involves columns 1 , .. . ,j and rows n - j 1,.. . ,n of x, it has weight ( p , v ) as stated (the columns of x transform as vectors under H , whereas the rows of x transform as covectors under D,).To verify that f j is fixed under the left action of N,,note that u E N, acts by x H ux.Since u is unipotent upper triangular, this action transforms the ith row of x by adding multiples of rows below the ith row, so it fixes f j . To verify that f j is fixed under the right action of the unipotent radical N = exp n of B , we observe from (47) that N is generated by the subgroups N A , N B , NC consisting of the matrices

[:

0 (a;-1

:j [:0' :] [: 11

0

I

I b O

'

'

1 -2cc

YCt

t

1

(50)

respectively, where a is upper-triangular unipotent, bt = -b, and c E C1. The elements of the subgroup N A act by [x y t] H [xu y t ] ,while the subgroups N B and Nc fix x. Hence the action of N transforms the ith column of x by adding multiples of columns to the left of the ith column, so f j is invariant under N . This proves part (a) of the lemma.

+

Now assume n > 1 and 1 1 5 j 5 min{k,n}. Then g j ( z ) is a linear function of t and does not depend on the variables yrs for s 5 k - j . Hence by (48) we have A p q g j ( z ) = 0 if min{p, q } 5 k - j . If min{p, q } 2 Ic - j 1, then

+

+

Fix s with Ic - j 1 5 s 5 1. If the column #s of x and column #s of y are interchanged in the determinant defining g j ( z ) , then g j ( z ) changes sign. Hence the function

likewise changes sign since the differential operator is symmetric in the variables x and y. But h ( z ) is of degree zero in the variables xps,xqs,yps,

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

369

yqs since g j ( z ) depends linearly on each variable. Hence h ( z ) = 0. This proves that g j ( z ) is G-harmonic. Since gj involves columns 1,. . . ,1 of z and columns k - j + 1,.. . ,1 of y , we see from (49) that gj transforms under H by the weight

7 = (El

+ + ' ' '

El)

- (&k-j+1 + ' * ' + E l )

= &1

+ + &k-j ' ' '

.

+

Since gj involves rows n - j 1,.. . , n of 2 , it transforms under D, by the same weight p as does fj. It is clear that g j is fixed under the left action of N,. To verify that gj is fixed under the right action of N , it suffices to check the action of the matrices in (50). These give the transformations

respectively. The determinant defining g j involves all the columns of z and t . Since the columns of zcct, zc, and tct are linear combinations of the columns of z and t , it is clear that these transformations fix g j . 0

Corollary 10.3: Let m = [ m l ,. . . ,m,] E NI'++, where r = min(1,n). Assume that m has depth d and set X = [m,0,.. . ,O] E N :+. Define q m = flml-mz . . . md-1-md fT" (when m = 0 set cpo(z) = 1). fd-1 (a) pm is a G-harmonic polynomial, homogeneous of degree Iml. Thus cpm(-z) = (-l)lmlcpm(t) for t E M n x k . Furthermore, cpm is a B, x B eigenfunction of weight ( a ,A), where

-

a = [ 0,. .. ,O,

-md,..

. ,- m l ]

n-d

(when m = 0 take a = 0).

+

(b) Suppose n > 1 and n - k d 2 0.For m # 0 , define $m = v m g k - d l fd (when m = 0 set $0 = g k ) . Then $m is a G-harmonic polynomial, homogeneous of degree JmJ+ Ic - 2d. Thus ?lm(-,z) = - ( - l ) ~ m ~ $ m ( zfor ) z E M n x k . Furthermore, $ J is~ a B, x B eigenfunction of weight @,A), where /? = [ 0 , . . . ,0, -1,. . . , -1, -md,. . . , -ml]

-

vn-k+d

k-2d

(When m = 0 take ,6 = [ 0 , . . . , 0,-1 n-k

k

R. Goodman

370

Proof: Since p m ( z ) is a function of IC alone (where z = [z y above), it is clear that pm is G-harmonic. For the same reason, A p q $m = ( p m / f d ) A p q g k - d

t] as

=0.

Thus we see that $m is also G-harmonic. The other assertions are immediate consequences of Lemma 10.2 0

Now we turn to the proof of Theorem 10.1. A B, x B joint eigenfunction generates an irreducible subspace under the action of GL(n) x Go by Theorem 3.5.Since the space of harmonic polynomials on Mnxk is a multiplicity-free GL(k,cC) x Go module, it follows that a Bk x B eigenfunction is uniquely determined (up to a scalar multiple) by its weight and parity. If m E PI$+ has depth d 5 n, then from Corollary 10.3 we see that the right translates of pm under G span an irreducible space of type (m, l), while the right translates of $m under G span an irreducible space of type (m, -1). When k 5 n,then the conditions n > 1 and n - k d 2 0 in part (b) of Corollary 10.3 are automatic. Thus every irreducible representation of G occurs in ‘H in this case, as asserted in part (a) of the theorem. To prove parts (b) and (c) of the theorem, assume that Ic > n. Let f E ‘H be a B, x B eigenfunction. Define a polynomial f on MkXk by

+

-

We claim that f is G-harmonic. Indeed, if min{p, q } 5 k - n then Apqf = 0 since does not depend on the variables zpq for p 5 k - n. On the other hand, if min{p, q } > k - n then

f

A P q f ( z )= Ap!qif(z’’) =0

+

+

(where p’ = p - k n and q‘ = j - k n),since f is G-harmonic. To see that Tis a Bk x B eigenfunction, write b E Bk as

where Then ?(b-lzb’) = f(S-lz”b’) for b’ E B . Since f is B, x B eigenfunction, it follows that is a Bk x B-eigenfunction. Furthermore this shows that Bk weight p of ?is of the form

7

-

p = [ 0 , . . . , O , a,, k-n

. . . ,all

with a, 2

. . . 2 a1 .

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

371

Thus by part (a) we know that ?is a multiple of either vrn or qrnfor some m E N!++of depth d 5 n, since f is homogeneous. If 1 < n < k,then vm is defined for all m E N\+,but $rn is only defined when the depth d of m satisfies k - n 5 d 5 1. This implies part (b) of the theorem. If n 5 1, then (pm is defined for all m of depth d 5 n, but in this case $rn is never defined. This implies part (c) of the theorem. The formula for the map 7 follows from the formulas for Q and p in Corollary 10.3. 0 10.2. O ( k ) Harmonics (k e v e n )

We now assume that k = 21 is even. We take the Bore1 subgroup B c G whose Lie algebra consists of the matrices with block decomposition (block sizes 1 x 1) upper-triangular,

(If k = 2 then b = 0 and B 2 C " ) . Let N c B be the unipotent radical (the matrices as above with a upper-triangular unipotent). Recall that O(C',,W)= Go x ( 1 , s )

where Go = SO(C',u) is the identity component and s E G is the reflection interchanging the basis vectors el and e21 and fixing all other basis vectors ei. Since s normalizes B it acts on the characters of B. Let X = [rnl,. . . ,ml] E Nk+.If rnl # 0, then s . X # X (since s changes rnl to -ml). In this case there is a unique irreducible G representation 7rITx>Osuch that

(where p ) . If

7rp

rnl

denotes the irreducible Go representation with highest weight

= 0, then there are two irreducible representations

of G whose restriction to Go is &E

7rA.

7rA+ ( E

= fl)

They are related by

= det @7rAi-'

and labeled so that 7rA+(s)acts by E on the Go highest weight vector (see [16, Sec. 5.2.21). Thus can be written as a disjoint union = G-1 U u el,where

G*I

z

e

A

h

=

{7rTTX+ :

depth(X) < 1, E = fl} , Go = {7rAi' : depth(X) = 1 , =~ 0 ) .

R. G o o d m a n

372

Theorem 10.4: (G = 0 ( C k l w ) ,k = 21)

the G-harmonic polynomials 7-1 C (a) Assume k 5 n . Then C = range).

e and thus C does not depend on n (G-stable

(b) Assume 1 < n < Ic. Then

C ne - 1

Let C be the spectrum of G on

P(kfn,k).

c C, whereas

U

{(A, -1) : Ic - n 5 depth(A) < 1 )

=

(unstable range: C depends on k and n ) . (c) Assume n = 1. Then C (d) Assume n

=

61U eo.

< 1. Then C = {(A, 1) : depth(X) 5 n} C 6:.

Thus C does not depend on k when n 5 1 (GL(n)-stablerange). The duality : N correspondence is given as follows: Let A = [ml,. . . ,m d , 0 , . . . , 01 E + have depth d with 1 5 d 5 min{ 1 , n}. Then

r(o)=

I

--

[O, . . . , O ,

-md,.

. ., - m ~ ]

for o = (A, E ) E En (El uEo),

n-d

[O, . . . , 0 , -1,. . . , -1, n-kfd

-md,.

. . ,-ml]

for

(T

= (A, -1) E

c nE d 1 .

k-2d

To prove the theorem, we will find a set of generators for the joint eigenfunctions of B, x B in 7-1. For u E M n x l , p 5 n , and q 5 1, define ) Rp,q(u) as in the proof of Theorem 10.1. In this case matrices L p , q ( uand we write z = [ x y ] as in (45) and we define fj(z)

If n 2 1

= detLj,j(x)

for 15 j 5 min{l,n}.

+ 1 then we also define

g j ( z ) = det [ L j , l ( x ) Rj,j-l(y)

]

for 1

+ 15 j

Lemma 10.5: (a) Let 1 5 j 5 min{l,n}. Then B, x B eigenfunction of weight ( p , v ) , where - . . . - E,

p = -&n-j+l

Furthermore fj(zs)

=

and

fj(z) if j < 1, where s

+

fj

5 min{k,n}. E

7-1(j)

and

is a

v = ~ l + . . . + ~ j .

E

G is the reflection el

+

(b) Suppose n 2 1 1 and take 1 1 5 j 5 min{n, k } . Then g j E g j is a B, x B eigenfunction of weight ( p l y ) , where p = -Enpj+1

fj

- . . . - cn and y = E I + . ' .

+

H

e21.

'Fl(j)

and

&k-j

(here y = 0 if j = k). Furthermore g j ( z s ) = - g j ( z ) , with s E G as in (a).

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

373

Proof: Essentially the same as the proof of Lemma 10.2. Note that in this case the unipotent radical N of B is generated by the transformations

z

+-+

[ z a y] , z

H

[z y + z b ] ,

0

with a upper-triangular unipotent and bt = -b.

Corollary 10.6: Let m = [ m l ,. . . ,m,] E PI:+, where r = min(1,n). Assume m has depth d and set X = [m,O,. .. ,0]E N\+. Define (Pm = f;"'-"". . f dm- dl - l - m d f r d (when m = 0 set cpo = 1).

-

(a) cpm is a G-harmonic polynomial, homogeneous of degree \mi. Furthermore, vrn is a B, x B eigenfunction of weight (a,A), where

a = [ 0 , . . .,0 , -md,. . . ,-m1]

,

n-d

(when m = 0 take a d < 1.

=

0). Set

cpk(z)= cpm(zs). Then cp&

=

vm

when

+

(b) Let n > 1 . If d < 1 and n - k d 2 0 define Gm = (Pm g k - d l f d (when m = 0 set $0 = g k ) . Then qbrn is a G-harmonic polynomial, homogeneous of degree Im(+ Ic - 2d. Furthermore qm is a B, x B eigenfunction of weight (D, 4, where

--

p = [ 0 , . . . ,o, -1,. . . , -1, n-k+d

(when m = 0 take

. . , -ml]

k-2d

p = [ O , . . . , 0 , ,1, .:. , -y). Set $&(z) n-k

$"m = -$lrn.

-md,.

= $m(zs).

Then

k

Proof: This follows from Lemma 10.5 by the same arguments as in the proof of Corollary 10.3. 0 To prove Theorem 10.4, assume first that n 2 k. By Corollary 10.6 the functions cpm are defined for all m E N$+.If m has depth 1 then the right G-translates of ym span an irreducible subspace of type (m,O). If m has depth less than 1 then $m is also defined. In this case the right G-translates of (Pm span a G-irreducible subspace of type (m, I), whereas the right Gtranslates of $m span an irreducible subspace of type (m,-l). Thus we get all irreducible representations of G in X ,as asserted in part (a) of the theorem. The argument when n < Ic proceeds as in the proof of Theorem 10.1 by lifting harmonic B, x B eigenfunctions from M n x k to M k x k . Note that

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374

prnis defined for all m of depth d i min{n, l } , whereas $, when n > 1 and k - n I d < 1. We omit the details.

is only defined 0

10.3. Examples of harmonic decompositions

+

(1) Assume n 5 1 and k = 21 1 or 21, so that we are in case (c) of Theorem 10.1 or cases (c) and (d) of Theorem 10.4. The restrictions to S O ( k ) of the representations in C are the class n representations of SO(k) - those that have a vector fixed under the subgroup S O ( k - n). This follows from the branching law (see [16, Sec. 8.11). In this case the harmonic polynomials on M n x k decompose under GL(n) x S O ( k ) as

Here = [-mn,. . . , - m l ] and U X is the irreducible S O ( k ) module with highest weight X (when n = 1, k = 21 is even and ml # 0, then ItXis the sum of the irreducible representations with highest weights X and s A). For n = 1, (51) is the classical spherical harmonic decomposition and gives the decomposition of polynomials restricted to the sphere S O ( k ) / S O ( k - 1).For n > 1 (51) gives the decomposition of polynomials restricted to the Stiefel manifold S O ( k ) / S O ( k- n). This decomposition was obtained by Gelbart [12] and Ton-That [32] before Kashiwara and Vergne [22] worked out the general case that we have presented here. ( 2 ) Now assume 1 < n < k , so that we are in case (b) of Theorems 10.1 and 10.4. The decomposition of the harmonics in this case was obtained by Strichartz [29]. For example, let n = 2 and k = 3. Then we have the decomposition

of the harmonic polynomials on M 2 x 3 . Here Vim] denotes the irreducible SO(3) representation with highest weight m ~ 1The . B2 x B harmonic eigenfunction p(m) ( 2 ) = x y generates the summand & [ o ~ - m @] The B2 x B harmonic eigenfunction $ ( m , ( ~ = ) z F - ' ( x l t 2 - 2 2 t 1 ) generates the summand & [ - ' ~ - ~ l @ Vlrnl. Here we write

=

[;:

Y1

tl

y2

t2]

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

375

(3) Let TI = 3 and Ic = 3 so that we are in case (a) of Theorem 10.1. Then we have the decomposition

of the harmonic polynomials on M3x3 as a module for GL(3) x SO(3). The B3 x B harmonic eigenfunction ~ ( ~ ) = ( xz a) generates the summand The B3 x B harmonic eigenfunction (2) = xT-'(zzt3 z&) generates the summand E[o*-ly-m] @ l/""]. Here we write &[0103-ml@V[m1.

[; ;

z1 y1 tl

z=

b]

For m = 0 the function ~ ( o ) ( z=) det z generates the one-dimensional summand &[-11-1,-11 @ ),J"31. Let C ( m ) denote the one-dimension representation g H (detg)m of GL(3). Let wl = [1,0,0] and w2 = [1,1,0].Then the GL(3) representations occurring in 'FI are c(-'),

&[o,o>-"l

c(-")

@ &""2

for all m 2 0, and

for all m 2 1. 11. Symplectic group and oscillator representation

We now turn to the functional-analytic aspects of the harmonic duality decomposition in Theorem 9.2 (recall Example 4 in Section 2.3). If we replace the complex group G by its compact real form Go = GnU(V) then the finite-dimensional representations 3"remain irreducible under Go and the action of Go is unitary relative to the Fischer inner product. We would like to have a similar picture for the dual representations E X (where X = T ( ( T )+ 6). At the Lie algebra level it is clear that to obtain a unitary representation, we should take the real form gb of g' that acts by skew-hermitian operators relative to the Fischer inner product. The analytic problem is to construct a unitary representation of an associated real Lie

R. Goodman

376

group Gb on the completion of P ( V ) , and to describe its action on the Hilbert space completions of the infinite-dimensional spaces EX. We will construct Gb as a subgroup of the rnetaplectic group Mp(nk, W) (the two-sheeted cover of the real symplectic group Sp(nk,W)). The associated unitary representation will be the restriction to Gb of the oscillator representation of the metaplectic group. This representation already appears in the harmonic decomposition as a Lie algebra representation by elements of degree 2 in the Weyl algebra. However, when we try to exponentiate it to a unitary group representation, we encounter the conflict between the particle and the wave description of quantum mechanics; the representation has a simple description (the holomorphic model) relative to the maximal compact subgroup KO U(n) of Sp(n,W), and another simple description (the real-wave model) relative to the maximal parabolic subgroup P E GL(n,IW) K SM,(R) of Sp(n,R). In both descriptions KOn P 2 O ( n ) acts geometrically, but some of the remaining group elements act in a more subtle way. Thus it will be necessary to consider two matrix forms of the real symplectic group and the intertwining operator (the Bargmann-Segal transform) that relates the two versions of the oscillator representation. 11.1. Real symplectic group

Let Sp(n, C) be the subgroup of GL(2n, C) that preserves the skew-form n

n(5,Y) = C ( x i ~ n +-i zn+iYi) i=l

on C2,. Thus g E Sp(n,C) if and only if gtJng matrix transpose and J , is the matrix

=

J,, where gt denotes

We can also describe Sp(n,C) as the fixed-point group of the involution 7 : g H J n ( g t ) - l J l l on GL(2n,C). The Lie algebra sp(n,C) of Sp(n, C) consists of all X E M2n such that J,X + X t J , = 0. These matrices have block form

X =

[” ] C -At

withAEM,andB,CESM,

Here we use the notation Mn for the n x n complex matrices and S M , for the n x n symmetric complex matrices.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

377

The real symplectic group Sp(n,R) = Sp(n,C) n GL(2nlR). Its Lie algebra 5p(n,W) consists of all the real matrices in 5p(n,C).

Maximal Compact Subgroup. A fundamental technique for studying a unitary representation of a real reductive group such as Sp(n,R) is to restrict the representation to a maximal compact subgroup, under which the representation space decomposes as the (Hilbert-space) direct sum of multiples of irreducible (finite-dimensional) subspaces. The real orthogonal group O ( k ) c U(k) is the subgroup of real unitary matrices. Since Sp(n, C)and Sp(n, R) are invariant under the map g H g * , the groups Sp(n) = Sp(n, C)n U(2n) and Sp(n, R)n U(2n) = Sp(n, R)n O(2n) are maximal compact subgroups of Sp(n,C) and Sp(n, R), respectively (see [24, Proposition 1.21). The subgroup of diagonal matrices in Sp(n) is a maximal torus in Sp(n). However, the subgroup of diagonal matrices in Sp(n,R)n O(2n) is finite and is not a maximal torus in Sp(n,R). Hence it is convenient to replace Sp(n,R) by an isomorphic real form Go so that the diagonal matrices in Go n U(2n) comprise a maximal (compact) torus in Go. Define

and let cr be the conjugation (conjugate-holomorphic involution) cr(g) = In,n(g*)-lln,non GL(2n,C). The fixed-point set of cr is the real form U(n, n)of GL(2n, C). Set

Then J;'In,, = Kn,so it follows that (TT = T C J .Hence cr leaves Sp(n,C) invariant and its restriction to Sp(n,C)defines a conjugation of Sp(n, C) which we continue to denote as cr. If g E Sp(n,C) then o ( g ) = m ( g ) = KngK,. In terms of the n x n block decomposition, cr acts by

CJ[t;]=

D C [B A ] '

Define Go = { g E Sp(n, C) : a(g)= g } . Then Go is a real form of Sp(n,C). Its Lie algebra go = Lie(G0) consists of all matrices X E 5p(n,C)such that a ( X ) = X . In terms of the block decomposition, go consists of the matrices

A* = -A, B

= Bt

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378

Lemma 11.1: The subgroup KO= Go n U ( 2 n ) is a maximal compact subgroup of Go and consists of all matrices

[t 11,

withA e U ( n ) .

Hence KO U ( n ) and the subgroup of diagonal matrices in KOis a mmimal compact torus of Go. Proof Since a(g*)= a(g)* for g E G L ( 2 n , C ) , the group Go is invariant under g H g*. Hence KO is a maximal compact subgroup of GO. Write g E G L ( 2 n , C ) as

["C

'=

"1

D

with A,B,C,D E M,.

Then g E U ( n , n ) if and only if g*I,,,g written as

A*A - C*C = I,,

= In,,. This condition can be

B*B - D*D = - I n ,

A*B

- C*D = 0 .

(53)

On the other hand, g E U ( 2 n ) if and only if g*g = Iz,. This condition can be written as

A*A+C*C=I,,

B*B+D*D=I,,

A*B+C*D=O.

(54)

If g E U ( 2 n )n U ( n , n ) then from (53) and (54) we have C*C = 0 and B*B = 0. Hence B = 0 and C = 0, so A*A = I, and D*D = I,. Thus U ( 2 n )n U ( n , n ) = U ( n ) x U ( n ) . But if g E S p ( n , C ) is in block-diagonal form, then

0

Hence g E KOif and only if A E U ( n ) . Define an involution 6 on Sp(n, C)by

O(g) = In,ngIn,n (note that 6 is an inner automorphism of Sp(n, C)). If g E Go then (St)-' = J n g J i l and 3 = K,gK,. Hence (g*)-' = J n S J i ' = JnKngKnJi'. Since J,K, = I,,,, it follows that B ( g ) = (g*)-l

for g E Go.

Thus the maximal compact subgroup KO is the fixed-point set of 6' in Go. Its complexification is

K = ( 9 E Sp(n,C) : O(g) = 9 ) .

Multiplicity-he Spaces anu Schur- Weyl-Howe Duality

379

Note that if g E GL(2n, C), then O(g) = g if and only if

[; i] ,

g=

a,dEGL(n,C).

If g E K , then in this block decomposition d = ( u t ) ) - ' . Hence K via the homomorphism aH

[;

GL(n, C)

.

(a;-']

The complexification of the Lie algebra t o of KO is the Lie algebra t Z gI(n,C) of K . The involution O gives a decomposition of 5p(n,C). The f l eigenspace of 8 on 5p(n,C) is t, whereas the -1 eigenspace is

We have 5p(n,C)

=t@p

with commutation relations

[t,t] c t

1

[t,PI

c P , [P, PI c t

'

The center of t is spanned by I,,, and t = CI,,, @ [t,t], with the derived algebra [t,t] 2 4 7 2 , C). The f l eigenspaces of adI,,, on p are

P + = {O [ ~B o] :BESM,},

P - = { [ ~0

0] 0 :CESMn}.

These subspaces are invariant under t and have the commutation relations

[P+,P+l

=0,

[P-,P-I

=0,

[P+,P-I c t .

Thus there is a triangular decomposition

5p(n,C) = p- @ t @ p + (as we already noted in Section 8). The conjugation u interchanges p+ and

p - , since g([:

:]

:I)=[;

for B E SMn. We can describe these decompositions in terms of root spaces as follows (see [16, Sec. 2.3.11).The complexification T of TOis a maximal (algebraic) torus in Sp(n, C) and has Lie algebra

t ={X

= diag[zl,. . . , z,

-21,.

. . , -x,]

: xj E C } .

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380

The set of roots @ = @(g,t) of t on g is f ~ fi~j for 1 5 i,j 5 n, where E ~ ( X=) xi for X E t as above. We have @ = Q C u an,where Qc={+(&i-&j)

:l 0 such that

v,W c V" be the subspace for which (62) holds. Ur,o v,W of analytic vectors for g is invariant under g. Let

The space V" =

Theorem 11.7 (Nelson): Suppose there exists an r > 0 so that V,W is dense in V . Then the representation of 00 integrates to a strongly continuous unitary representation on V of the simply-connected Lie group Go with Lie algebra 00. Proof: (Sketch) Define a norm on g by then

( 1 c,"=, ciXi(I = c,"=,

Icil. If X E g

llXn41 5 IlxlInPn(v). , to V by the exponential series Hence the operator e x is defined from VW provided llXll < r. The map e x p X H e x defines a local representation of the complex Lie group germ corresponding to g (the rearrangement of the exponential series needed for the Campbell-Hausdorff formula is justified by convergence of (62); see [3] or [18]).The operator e x is unitary for X E go, since X is skew-Hermitian, and the local representation extends to a strongly continuous unitary representation of Go on W . 0

-

Suppose there is an element HO E g such that pl(v) 5

11~1+ 1 llHovll

for all v E V .

(63)

Let A = m a x l l i g II[Ho,Xi](I(the norm of adHo on 8). Note that A = 0 if and only if HO is in the center of 6 , and this case is of no interest here. So we assume A > 0.

Theorem 11.8: Every analytic vector for HO is an analytic vector for g. More precisely, if

then w E

v,W

for all r < min{A-l,A-'(l

- e-A5)}.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

393

Remark: If s can be arbitrarily large in (64) (one says that w is an entire vector for Ho in this case), then v E V,W for all r < A-l. Note that this upper bound for r is controlled by the non-commutativity of g and it is finite if A # 0. In general v is not an entire vector for g (see [15] for more precise results along this line). Proof: Let y3 be any of the basis elements X i . Then the a priori estimate (63) implies that

I IYm+1 Ym .

' *

y14

I L I lYm . . Yl vI 1 + I p o y m . . . Yl v )I '

for all t~ E V". Now m

HOYm"*Yl = Ym"'Y1HO

+ CY,"'Yk_1[HO,Yk]Yk+l

'**Y1,

k=l and by definition of p m and the constant A we have IlY,... [Ho,Yk]...Y1wlI 5 Ap,(w). Hence

+

IIYm+1...YlvII L I l y m * . . y ~ H o ~ I I(l+mA)p,(~)

I pm(H0v) + (1+ mA)pm(v). Since this holds for any choice of Yl , . . . , Ym+l,it implies pm+i(v) 5 pm(&w)

+ (1+ mA)p,(w)

for all w E V .

(65)

Now fix w E V" and set am,n = p,(H;w). Replacing w by Hgw in (65), we see that the sequence {am,n} satisfies the recursive inequalities

+

am+l,n 5 am,n+l+ (1 mA)am,n. To estimate the rate of growth of am,n, we introduce the majorant sequence bm,n defined by b ~= ,aO,n ~ for all n and b , + ~ , ~= bm,n+i

+ (1+ mA)b,,,

for all m 2 0, n

Clearly am,n 5 bm,n for all m,n. Consider the generating function

The recursion for bm,n implies that (as a formal series)

20 .

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394

We assume that the series f ( y ) = cp(0,y) converges for y = s. The Cauchy problem for this analytic first-order p.d.e. is easily solved by the method of characteristics, and one obtains cp(z, y) = (1 -

f (y - A-' log(1- A x ) ) .

(the analytic solution must agree with the formal solution since the line z = 0 is non-characteristic). Setting y = 0, we see that the series for p(z,0)

converges absolutely for 1x1 5 r provided

r < min{A-',A-'(l Since a,o

- e-As)}

5 bm,O this proves the theorem.

0

Corollary 11.9: Suppose Ho E 0 and V has an (algebraic) basis consisting of eigenvectors for H o . Then V c VW , for all T < A-' and hence VW , is dense in V for all T < A-l. Thus the representation of 00 integrates to a strongly continuous unitary representation of Go on V .

-

Proof: If HOV = Xu, then the left side of (64) is e3IxI, and hence is finite for all s > 0. By the remark after Theorem 11.8 this implies that (62) holds 0 for all r < A-l. Now apply Theorem 11.7.

12. Dual pair Sp(n, R)-O(k) The oscillator representation has many applications to analysis and physics (see [8]and [20], for example). Here we apply it in the context of unitary representation theory and highest weight representations (see [6] for more on this point). To determine which of the representations that occur in the decomposition of the oscillator representation are square-integrable, we apply Harish-Chandra's criterion to the explicit formula for the &correspondence that we calculated in Theorems 10.1 and 10.4. In particular, we show that all the square-integrable highest-weight representations of Sp(n,IR) occur in the duality correspondence with O(2n) (this was first proved by Gelbart [ll]). 12.1. Decomposition of H 2 ( M n x k ) under Mp(n, IR) x O ( k ) Let G = O(lc,C) = { g E GL(lc,C) : ggt = I } and let G' = Sp(n,C) c GL(2n, C ) be the symplectic group relative to the skew-form with matrix Jn as in Section 11.1. Define a skew form 0 on Mznxk by

n ( w , Z) = tr(wtJnz) for W, z E

M2nxk.

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395

Then R is nondegenerate. We embed G' x G into SP(MZnxk,R) as follows. Let g E G and h E G'. Then R(hwg, hzg) = tr(wt(htJnh)zggt) = R(w, z ) since ggt = I and htJnh = J,. Hence we have an injective regular homomorphism L x R : G' x G + sp(M2nxk, 0) given by R(g)z = zg-',

L(h)z = h~ for g E G, h E G', z E M2nxk.

We identify M2nxk with Cznk by the map

z = [ Z I , . . . , Zk]

H

5=

[

EC=2nk zk

where z j E Czn is the j t h column of z . It is easy to check that

R(z, W ) = z t J n k ' & , so Sp(MZnxk,52) becomes Sp(nk,C)under this identification. Thus we will view z either as a 2n x k matrix or a vector in Cznk,whichever is more convenient for the calculation at hand. Define a hermitian form on Mznxk by

( z ,W ) = tr(w*In,nZ) 7

where In,n is the matrix in Section 11.1. w e have (z,w) = '&*Ink,nkZ,so when z E M z n x k is identified with Z E CZnk,the form ( z ,w) becomes the one used in Section 11 to define the group U ( n k ,n k ) . Thus we will denote the isometry group of this form as U ( n k ,n k ) . If g E U(n, n)then (gZ99W) = tr(W*g*In,ngz)= (z, W) since g*ln,,g = Thus the left multiplication homomorphism L : GL(2n,C) -+ GL(Mznxk) carries U(n,n) into U ( n k , n k ) .If h E U(k) then (zh, wh) = tr(w*ln,nZhh*)= ( z , W) . Furthermore,

[;I

h=

[f]

.

Hence the right multiplication homomorphism R : GL(k, C ) GL(Mznxk) carries U(k) into the maximal compact subgroup U(nk) x U(nk) of U ( n k ,n k ) . -+

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396

Let Go = G n U(k) = O ( k ) be the compact real form of G, and let Gb = G'nU(n,n ) 2 Sp(n,R) be the real form of Sp(n, C)as in Section 11.1. Let KO = Gb n U(2n) 2 U(n) be the maximal compact subgroup of Gb. Then the embedding L x R : Sp(n,@) x O ( k , C ) -+ Sp(nk,(C) gives an embedding of the real forms Gb x Go

-

Sp(nk,CC) n U ( n k ,n k ) 2 Sp(nk,R)

and carries the maximal compact subgroup K O x Go into the maximal compact subgroup Sp(nk, C)r l ( U ( n k ) x U ( n k ) )of Sp(nk, C)n U ( n k ,n k ) .

If u E U(n) and ko

=

ti

[o

GI pair (k0,g) E KOx Go acts on

is the corresponding element of K O ,then the M2nxk

by

We now calculate the restriction of the oscillator representation w(nk) to L(K0) x R(Go) in the holomorphic model on P ( V ) ,where V = Mnxk. Let (k0,g) E KO x Go. F'rom (66) and Theorem 11.3 we see that cdnk)(L(ko)R(g))f(s)= (det u)-"'(det

g)"/2f(u-1sg)

for f E P ( V ) (67)

(note that the determinant of the map s H u-lzg is (det u)-'((det g),). If k and n are both even, formula (67) defines a representation of KOx Go. For the general case, let c Mp(n, R) be the two-sheeted cover of KOand let be the lift of R(G0) to Mp(nk,R). Then (67) gives a single-valued unitary representation of x In the following we shall simply drop the factor (det g ) n / 2 from (67) to make the representation single-valued on Go. is essential for extending the representation However, the factor (det from K O to Mp(n, R). Let 6 denote the differential of this character of Let gb be the Lie algebra of Gb, The complexification of gb is g' =

KO

eo

G.

&.

sp(n,C). We now calculate the action of t ~ ( ~ ~ ) ( L (Let g ' )X) .= Y=

[::]

with b, c E SM,. Then

[::]

and

Multiplicity-Free Spaces

and Schur- Weyl-Howe Duality

397

(where L ( b ) x = bx for x E M n x k ) . The quadratic form on V associated with L ( b ) is

Hence

) the operator of multiplication by -;Qh(.,. Since p* and c d n k ) ( L ( Y )is generate g', these operators determine a ( n k ) ( g ' ) . F'rom Theorem 8.5 we conclude that the algebra P D ( V ) Gis generated by z d n k ) ( g ' ) . We recall some notation that was introduced earlier. Let 'FI denote the space of G-harmonic polynomials on M n x k and let C c be the spectrum of G on 7-f. Let the map T :

C + A c Z+ :

be as in Theorems 10.1 and 10.4. Let T' c 7-f be an irreducible G-module in the class (T.Let &'(")f6 c IH be the irreducible finite-dimensional kmodule with highest weight T ( U ) 6, as in Theorem 9.2. Let V = M n x k . Consider the unitary representation of Mp(n,W) x O ( k ) on W2(V), where Mp(n,W) acts by the restriction of the oscillator representation m(nk)and O ( k ) acts geometrically by right multiplication on V . For D E C let IET(u)f6 be the closure in W2(V) of the 8'-irreducible subspace P ( V ) .

+

Theorem 12.1: The spaces Er(u)+6,f o r r~ E C, are irreducible and mutually inequivalent unitary representations of Mp(n,W). Furthermore, W2(V) decomposes as a multiplicity-free Halbert space orthogonal s u m

under the action of Mp(n,R) x O ( k ) .

Remark: When k is even the character u H (det u)-'12 occuring in the oscillator representation is well-defined on U ( n ) and ET(u)+6gives an irreducible unitary representation of Sp(n, W). Proof: The key point is the following density result:

R. Goodman

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(*) Suppose E c W2(V) is a closed subspace that is invariant under Mp(n,IR). Set EO = E n P ( V ) . Then EO is dense in E and is invariant under 8’. To prove this, let f(x) = C , c , P be in W2(V) and set

fq(x)=

C c,xa

for q = 0 , 1 , 2 , . . . .

la154

Then f, E P ( V ) and it is clear from (38) that Now take

/If

-

fqll

-+

0 as q

-+

m.

as in the proof of Theorem 11.3. Since

we see from (67) that d n k ) ( H o )acts by -i(j define

+ 3) on P j ( V ) .Hence if we

then Pq is a bounded operator on W2(V) and Pqf = fq for q = 0 , 1 , 2 , . . . . Since E is closed and invariant under Mp(n, R), we know that E is invariant under the one-parameter unitary group t H a ( n k ) ( e x p t H o )Hence . PqE C E for q = 0 , 1 , 2 , . . . . This shows that Eo = P ( V )n E is dense in E. To prove that EO is invariant under g’, take cp E EO and E E l . If X E gb, then

+

(dnk)(exptX)cp1 +) = o for all t E

IR ,

(70)

since E is invariant under Mp(n,IR). But since cp E P ( V ) ,the left side of (70) is an analytic function o f t for It( near zero by Corollary 11.9. Taking the derivative in t and setting t = 0, we conclude that

(dnk) (x>cp 1 $) = o

for all

+ E E‘- .

Hence d n k ) ( X ) c pE E , completing the proof of (*) To prove the theorem, first observe that if E c ET(u)f6is a proper closed subspace that is invariant under Mp(n, R), then by (*) the space Eo is invariant under g’. Hence EO= 0 by the irreducibility of & T ( “ ) + 6 . But EO is dense in E , so E = 0, showing that lF,T(a)f6 is irreducible.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

399

If o,o' E C and T is a bounded Mp(n, R) intertwining operator from IET(u)+6 to IEr('")+61 then T commutes with the projection operators Pq in (69). Hence T maps & T ( u ) + 6 to & T ( u ' ) + 6 . Since the functions in P ( V ) are analytic vectors for Mp(n,R), we conclude (as in the proof of (70)) that T intertwines the g' actions. Hence o = (T' by Theorem 9.2. The orthogonality of the decomposition also follows from Theorem 9.2. 0 12.2. Square-integrable representations of Sp(n, R)

The irreducible unitary representations of Go = Mp(n,R) that occur in Theorem 12.1 are called highest-weight representations. Some of them also appear as discrete summands in the decomposition of the left regular representation of on L2(Go) (these representations are called squareintegrable). We now apply Harish-Chandra's criterion [17] to determine which of the representations E'(")f6 are square-integrable. It is convenient to give separate statements of the result depending on the parity of k.

Theorem 12.2: (notation of Theorem 10.1) Let k = 21 uE

c.

(a) i f n > I

+ 1 then

+ 1 be odd. Let

is newer square-integrable.

I E T ( ~ ) + ~

(b) ifn = L+1 then E'(')+6 is square-integrable if and only if u = (A) -1) E G-1 and depth()\) = 1 . h

(c) If n _< 1 then IET(u)+6 is square-integrable for all o E C. Proof: The general condition on the highest weight A for squareintegrability is ( A + P , y-)< 0 ,

(71) where p is one-half the sum of the positive roots and y- is the co-root to the highest noncompact root y. For q ( n ,R) we have p = [n, n - 1,.. . , 2,1] and y = 2 ~ 1 so , y- = ~1 (see Section 11.1).We must check this condition when A = ~ ( o )b, with 6 = [ - k / 2 , . , . , -k/2]. ). (A+p, y-)= T ( o ) ~Let T ( D ) ~denote the first coordinate of ~ ( u Then k / 2 n.Since k = 21 1, the Harish-Chandra condition (71) is

+

+

+

7(0)1

< l + 1- n.

(72)

Case (a): n > Z+1. The formulas for T ( O ) in Theorem 10.1 show that ~ ( 6 is either 0 or -1 in this case. But 1 - n 1 5 -1, so (72) is never satisfied.

+

+

Case (b): n = 1 1. Now the right side of (72) is zero. The formulas for T ( O ) show that T ( O ) I < 0 if and only if o = (A, -1) E with d = 1.

c-1

)

R. Goodman

400

Case (c): n 5 1. Now the right side of ( 7 2 ) is positive. The formulas for T ( ( T ) show that T ( c ) ~I 0 for all (T E C, so ( 7 2 ) is always satisfied. 0

Theorem 12.3: (notation of Theorem 10.4) Let k = 21 be even. Let (T E C. (a)

If n > I then TE'(')+6

is never square-integrable.

(b) If n = 1 then lET(u)+6 is square-integrable if and only i f u = (A, 0 ) E and depth()\) = 1. (c) I f n

< 1 then E T ( u ) f 6is square-integrable for all

(T

Go

E C.

Proof: When k = 21 the Harish-Chandra condition ( 7 1 ) becomes

1. The formulas for T ( ( T )in Theorem 10.4 show that T ( ( T ) is~ either 0 or -1 in this case. But 1 - n 5 -1, so ( 7 3 ) is never satisfied. Case (b): n = 1. Now the right side of ( 7 3 ) is zero. The formulas for show that T ( ( T ) ~< 0 if and only if (T = (X,O) E GOand depth()\) = 1.

T((T)

Case (c): n < 1. Now the right side of ( 7 3 ) is positive. The formulas for T ( ( T ) show that T ( ( T ) 5 ~ 0 for all (T E C, so ( 7 3 ) is always satisfied. 0

Examples (1) Assume k is even. Then the oscillator representation a ( n kis) single2n then we see from the formula for the valued on Sp(n,R). If k &correspondence that every GL(n, C)-highest weight )\ that satisfies the Harish-Chandra inequality is of the form T ( ( T ) 6, for some (T E C. Thus every highest-weight discrete-series representation of Sp(n, R)occurs in the reduction of a ( n kin) this case.

>

+

( 2 ) Let (T be the trivial representation of O(Ic) (denoted by 7r(',l) in Sections 10.1 and 10.2). Then (T E C for all n 2 1 and T ( ( T ) = 0, by Theorems 10.1 and 10.4. The representation, call it 7 r + , of Mp(n,R) that corresponds to (T is square integrable if and only if 2n < k . It occurs with multiplicity one in W 2 ( M n x k ) , and has highest weight 6 = [ - k / 2 , . . . I - k / 2 ] . This weight parameterizes the one-dimensional representation g H det(g)-'/' of the maximal compact subgroup of Mp(n, R).Since consists of the constant functions, the space lHI: := lE6 of 7r+ is the completion (in the Fischer norm) of the space P ( M n x k ) G ,where G = O ( k , C ) . ( 3 ) Let (T be the representation g H det(g) of O ( k ) (denoted by 7r('>-') in Sections 10.1 and 10.2). Then (T E C if and only if n 2 Ic. In this case

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality T(0) = [ 0,.

401

. . ,o, -1,, . , , -11

n-k

k

by Theorems 10.1 and 10.4. The representation, call it T - , of Mp(n, R)that corresponds to cr is never square integrable. It occurs with multiplicity one in H 2 ( k f n x k ) when n 2 k , and it has highest weight

x = [ - l c / 2 , . . .,-Ic/2] n

+ [ O , . . . ,o, -1,. * .,-11. n-k

k

This is the highest weight of the representation (det)-k/2 @ Ak(Cn)*of the maximal compact subgroup of Mp(n, R). In this case T" = C g k , where g k ( Z ) is the determinant of the bottom k x Ic block of z E M n x k when we take the orthogonal group G = 0 ( C k , w ) as in Section 10. Thus the space MI? := Ex of T - is the completion (in the Fischer norm) of the space ( P ( h ' l n x k ) G g k ) @ A'((@.")*. Note that for fixed n , one obtains a distinguished set of n irreducible unitary highest-weight representations of Mp(n, R) this way by taking k = 1,.. . , n.

(4) By the harmonic duality theorem, (W:,n*) are the only irreducible Mp(n,R) modules that occur with multiplicity one in w 2 ( h f n x k ) . More details and other models for the representations lE7(")+6 can be found in

161. Final Remarks: In Schur-Weyl duality we took tensor powers of the representation of GL(n, C)on Cn (the representation of smallest dimension) to obtain all the irreducible finite-dimensional polynomial representations of GL(n, C). The two irreducible components r* of the oscillator representation on MI2 (C")are the smallest unitary highest-weight representations of Mp(n,R) in the sense of Gelfund-Kirillov dimension (see [31]). As we already noted in Section 11.3the representation on w 2 ( M n x k ) is the Ic-fold tensor product of this representation: k

w2(Mn,k)

=@

w2((Cn) (Hilbert-space tensor product).

(74)

Thus the action of the group O ( k ) on the right-side of (74) is another instance of a hidden symmetry.j jThe unitary representations that occur in the decomposition of this tensor product are the mathematical analog of the elementary particles, some familiar and some exotic, that physicists create by high-energy collisions of the basic particles.

R. Goodman

402

13. Brauer algebra and tensor harmonics In this final section we use duality to decompose the space of Ic-tensors under the action of the orthogonal or symplectic group G. This was first done by Brauer [4], who determined the generators and relations of the Gcentralizer algebra. The complication here is that this algebra is not a group algebra (as was the case when G = GL(n,C)). However, just as in the case of Howe duality, there is an analog of the harmonic duality of Section 9 in this situation. The centralizer algebra contains C [ 6 k ]as a subalgebra, and there is a subspace of harmonic tensors (in Weyl’s terminology completely traceless) which decomposes in a multiplicity-free way under the jointly commuting actions of G and 6:k.The full space of Ic-tensors then decomposes as the sum of spaces of partially harmonic tensors (see [16, Sec. 10.31, [9] and [lo] for details). 13.1. Centralizer algebra and Brauer diagrams Let G be the full isometry group of a nondegenerate bilinear form w on a finite-dimensional complex vector space V . We assume w to be either symmetric or skew-symmetric. For f E V * define f b E V by w(fb,

v) = ( f ,v> for all v

E

v.

The map f H f b is then a G-isomorphism between V * and V. Define a G-module isomorphism T : V*@2k ---f End(VBk) by T ( f 1€3

‘ . . €3 f2k)U

= W ( f ; €3

fi €4

”

’

€3 f k k ,

u)fi” @3 fi €4

’

’ . €4 f k k - 1

(75)

for fi E V * and u E V B k . Here we have extended w to a bilinear form on V@‘“by k

w(u1 CZJ

. . . CZJ uk,v1 CZJ . . ’ CZJ V k )

w(ui, vi) for ui,vi E

=

v.

k l

Theorem 13.1: Let Zk be the set of two-partitions of (1,.. . ,2k}. For E E sk let A, E (V*B2k)Gbe the corresponding complete contraction. Then EndG(VBk)= Span{T(Xc) :

E

E

%} .

Proof: Since T is a G-module isomorphism this is a immediate consequence of Corollary 7.5. 0 Theorem 13.1 only gives a spanning set for the centralizer algebra EndG(Vwk)as a vector space. To describe the multiplicative structure of this algebra it is convenient to introduce a graphic presentation of the set

Multiplicity-Ree Spaces and Schur- Weyl-Howe Duality

403

of two-partitions. We display the set { 1 , 2 , . . . , 2 k } as an array of two rows of lc labeled dots, with the dots in the top row labeled 1 , 3 , . . . , 2k - 1 from left to right, and the dots in the bottom row labeled 2 , 4 , . . . , 2 k . Consider the set XI, of all (unoriented) graphs whose vertices are the two rows of dots, and such that each dot is connected with exactly one other dot by an edge. (A dot in the top row can be connected either with another dot in the top row or with a dot in the bottom row.) An example with lc = 5 is shown in Figure 1. We call an element of XI, a Brauer Thus we can identify the set 3 k of two-partitions with x k ; if E E k corresponds to the Brauer diagram x E xk, we shall write A, for the complete contraction