HARMONIC ANALYSIS AND REPRESENTATIONS OF SEMISIMPLE LIE GROUPS MATHEMATICAL PHYSICS AND APPLIED MATHEMATICS Editors: M. FLATO, Universite de Dijon, Dijon, France R. R1CZKA, Institute of Nuclear Research, Warsaw, Poland with the collaboration of: M. GUENIN, Institut de Physique Theorique, Geneva, Switzerland D. STERNHElMER, College de France, Paris, France VOLUME 5 HARMONIC ANALYSIS AND REPRESENTATIONS OF SEMISIMPLE LIE GROUPS Lectures given at the NATO Advanced Study Institute on Representations of Lie Groups and Harmonic Analysis, held at Liege, Belgium, September 5-17, 1977 Edited by J. A. WOLF, M. CAHEN, AND M. DE WILDE D. REIDEL PUBLISHING COMPANY DORDRECHT : HOLLAND/BOSTON: U.S.A./LONDON: ENGLAND Library of Coagress CataJogiDg in Publkatioa Data Nato Advanced Study Institute on Representations of Lie Groups and Harmonic Analysis, Liege 1977. Harmonic analysis and representations of semi simple Lie groups. (Mathematical physics and applied mathematics; V. 5) Includes bibliographies and index. 1. Harmonic analysis- Congresses. 2. Lie groups Congresses. 3. Representations of groups Congresses. I. Wolf, Joseph Albert, 1936- 11. Cahen, Michel. III Wilde, M. De. IV. Nato Advanced Study Institute. Liege, 1977. V. Title. VI. Series. QA403.N37 1977 515'.2433 8().10768 ISBN·13: 978·94.()Q9·8963·4 D .·ISBN·13: 978·94·009·8961·0 DOl: 10.1007/978·94.()Q9·8961·0 Published by D. Reidel Publishing Company, P.O. Box 17, 3300 AA Dordrecht, Holland Sold and distributed in the U.s.A. and Canada by Kluwer Boston Inc., Lincoln Building, 160 Old Derby Street. Hingham, MA 02043, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, Holland D. Reidel Publishing Company is a member of the Kluwer Group All Rights Reserved Copyright © 1980 by D. Reidel Publishing Company, Dordrecht. Holland and copyright holders' as specified on the appropriate pages within. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any informational storage and retrieval system, without written permission from the copyright owner. TABLE OF CONTENTS PREFACE Vll R. J. BLATTNER / General Background 1 J. A. WOLF / Foundations of Representation Theory for Semi- simple Lie Groups 69 V. s. VARADARAJAN / Infinitesimal Theory of Representations of Semisimple Lie Groups 131 P. c. TROMBI / The Role of Differential Equations In the Plancherel Theorem 257 M. ATIYA~ and W. SCHMID / A Geometric Construction of the Discrete Series for Semisimple Lie Groups 317 M. ATIYAH and W. SCHMID /Erratum to the Paper: A Geo metric Construction of the Discrete Series for Semisimple Lie Groups 379 M. FLAT O and D. STERNHEIMER / Deformations of Poisson Brackets. Separate and Joint Analyticity in Group Representa tions. Non-linear Group Representations and Physical Applica- tions 385 J. SIMON / Introduction to the I-Cohomology of Lie Groups 449 H. FURSTENBERG / Random Walks on Lie Groups 467 SUBJECT INDEX 491 [A more detailed Table of Contents is given at the beginning of each paper.] PREFACE This book presents the text of the lectures which were given at the NATO Advanced Study Institute on Representations of Lie groups and Harmonic Analysis which was held in Liege from September 5 to September 17, 1977. The general aim of this Summer School was to give a coordinated intro duction to the theory of representations of semisimple Lie groups and to non-commutative harmonic analysis on these groups, together with some glance at physical applications and at the related subject of random walks. As will appear to the reader, the order of the papers - which follows relatively closely the order of the lectures which were actually given - follows a logical pattern. The two first papers are introductory: the one by R. Blattner describes in a very progressive way a path going from standard Fourier analysis on IR" to non-commutative harmonic analysis on a locally compact group; the paper by J. Wolf describes the structure of semisimple Lie groups, the finite-dimensional representations of these groups and introduces basic facts about infinite-dimensional unitary representations. Two of the editors want to thank particularly these two lecturers who were very careful to pave the way for the later lectures. Both these chapters give also very useful guidelines to the relevant literature. In the paper on the role of differential equations in the Plancherel theorem, P. Trombi studies the partial Fourier transform of continuous functions on a reductive group G with values in a complex Frechet space and gives a proof of the Plancherel theorem. V. Varadarajan gives, in a paper on the infinitesimal theory of representation of semisimple Lie groups, a construction of irreducible Harish-Chandra modules which will include in particular fundamental series of representations: the main emphasis in this paper is on the infinitesimal point of view. In the paper by M. Atiyah and W. Schmid, which is here reproduced from the Inventiones, a geometric realization of the discrete series, analogous to the Borel-W eil theorem in the compact case, is used to obtain the main properties of the discrete series of representations. An erratum to this beautiful paper is added at the end. At Liege, W. Schmid gave only two lectures on the regularity of invariant eigendistributions on vii viii PREFACE semisimple Lie groups. A detailed version of these results has appeared in the Inventiones and is not reproduced here. The next paper by M. Flato and D. Sternheimer contains essentially three parts: in the first part (Chapters 1 and 2) one finds an alternative formulation of quantum mechanics on phase space when the associative multiplication of functions is suitably deformed. The second part (Chap ters 3 and 4) deals with analytic vectors in group representation. The third part (Chapters 5 and 6) deals with non-linear representations of Lie groups in an infinite-dimensional space and the possibility of linear izing these representations. The contribution of J. Simon develops the I-cohomology of represent a tions. It is in particular of fundamental importance for the study of non linear representations. The last paper by H. Furstenberg describes various qualitative aspects of the theory of random walks on a Lie group G and in particular boundar ies of the group G and Jl-harmonic functions (Jl = probability measure on G). We are pleased to thank NATO who gave us the basic financial means to organise this Summer School and who gave us useful practical advice in the early stages of the organization. The University of Liege offered us a warm hospitality and some financial support which was very helpful. The beautiful Sart Tilmant Campus was an ideal location for the lectures. We express here our gratitude. We thank the Solvay foundation, and in particular its director, I. Prigogine, who gave us some support which was useful to invite mathe maticians from non NATO countries. The firm Faulx et Champagne offered generously to each participant a nice briefcase and we are happy to thank them. Finally, Springer Verlag gave us the permission necessary to reproduce the Atiyah-Schmid article which appeared in the Inventiones. We thank them for their cooperation. ROBERT J. BLATTNER GENERAL BACKGROUND* CONTENTS 1-. Introduction 1 2. Harmonic analysis on R/21tZ and on R" 3 3. Locally compact Abelian groups 10 4. Compact groups 19 5. General locally compact groups 28 6. Representations of Lie groups 40 7. Induced representations 47 References 65 1. INTRODUCTION These lectures will be devoted to the general subject of harmonic analysis on locally compact and Lie groups. We shall begin with the classical problem of decomposing a function of period 21t on [R into harmonics of the fundamental 'tone':!(x)'" I: a eillx Already in this simple setting we lIez ll • face two questions: (1) What sort of regularity properties should! possess for the decomposition to make any sense at all?; (2) In what sense does the series converge? These questions (or their analogues) will persist through out our investigations. After dealing with periodic functions on IR, we will take up the situation in IR ". Here harmonics of a fundamental 'tone' are replaced by plane waves, and the summation must be replaced by a multiple integral. The generaliz ation to locally compact Abelian groups is quite simple. In the process of generalizing, we lose those powerful tools (Schwartz distributions) associated with the differentiable structure of 1R", but the generality gained allows application of the theory to situations which could not be touched before, such as the additive or multiplicative groups of a locally compact non-Archimedean field. The main theorems are those of Bessel, Parseval, Plancherel, Herglotz, and Bochner, and the duality theorem of Pontrjagin. *This work was supported in part by NSF Grant MCS75-17621. 1 J. A. Wolf, M. Cahen. and M. De Wilde (eds.), Harmonic Analysis and Representations o!Semi-Simple Lie Groups. 1--67. Copyright © 1980 by D. Reidel Publishing Company. Dordrechr. Holland. 2 ROBERTJ. BLATTNER To generalize the foregoing to the non-Abelian case requires recasting the problems. One reformulation notes that the plane waves on (Rn (or the unitary characters on a general locally compact Abelian G) are eigen functions under translation: in the non-Abelian case these translation eigenspaces are replaced by translation invariant subspaces which can be given a complete locally convex topology in such a way as to be topo logically irreducible. If these spaces are given invariant Hilbert space structures and if jeL (G, Jl), where Jl is left Haar measure, then the 2 harmonic decomposition of f will result from the decomposition of L G, Jl) ( 2 under the left regular representation as a direct sum (or direct integral) of irreducible unitary representations. In case G is compact, this is accom plished by the Peter-Weyl theorem. If G is only locally compact, the situation is technically more complicated and will require us to develop direct integral theory. The other reformulation aims to decompos.e only those j invariant under the inner automorphisms of G. Plane waves are replaced by 'characters' of irreducible representations: functions such as x Trace(1t(x)), where n is an irreducible representation. Now if G is t-+ compact, 1t will be finite-dimensional and so its character will be a well defined function. But if G is not compact, n can be infinite-dimensional so that strictly speaking Trace(1t(x)) will not be well-defined. In case G is a Lie group, there is hope that Trace(n(')) can be interpreted as a dis tribution, and if G is semisimple or nilpotent that hope is realized. With either of the foregoing reformulations there is the problem of finding all irreducible unitary representations of G, or at least enough of them to perform the harmonic decomposition of a givenj. In many cases, this can be done by 'inducing' representations of closed subgroup H of G up to G. When G is discrete, induction arises naturally as the (right or left) adjoint of the functor that assigns to every representation of G its restric tion to H. When G is locally compact, this formulation of induction works in general only when H has finite index in G. For arbitrary closed H, there is a natural construction due to Mackey and Rieffel that serves well and is natural, although it no longer has a nice functorial interpretation. In case G is a Lie group, it is possible to extend this construction to situations in which one induces from representations of subalgebras of the complex ification Be of the Lie algebra 9 of G, and this extension can be used to produce important irreducible representations of solvable and semi simple groups. These notes are organized as follows: In Section 2, we go over harmonic analysis on 1R/21tl and on IRn. These results are extended to locally compact GENERAL BACKGROUND 3 Abelian G in Section 3. The compact case is covered in Section 4. Section 5 deals with harmonic analysis on general locally compact G, including direct integral decomposition theory for unitary representations. Lie groups are dealt with in Section 6, and, finally, in Section 7 we discuss induced representations and their use in producing irreducible representa tions. 2. HARMONIC ANALYSIS ON 1R/2nl AND ON IRn Let f be an 'arbitrary' function on IR with values in C and which has period 2n. Our object is to decompose in some sense f as a linear com inx bination of the harmonic functions x ...... e , nE 71., so that we will have L inx (2.1) f(x)~ ane nelL for some choice of coefficients {an}. We are immediately faced with the problem of giving sense to the sum on the right-hand side of(2.1). One way of doing this is to interpret it as a 'weak sum'; that is, for functions g of period 2n in a certain class ~ we want 2n 2n 1 f f (2.2) f(x)g(x) dx= an einXg(x) dx, o o where all the integrals are to converge absolutely and where the sum on the right-hand side is to converge unconditionally. The functions in ~ are known as test functions. For each choice of ~ we will have a class of func tions f for which (2.2) holds for some choice of {an} and for all gE~. Obviously, in order for to be a useful class of test functions, the class of ~ decomposable functions f associated to ~ must be large; moreover, the coefficients {an} must be completely determined by f Suppose f} contains all the harmonic functions x ...... einx• Then, if feL ([O, 2n]) and if (2.2) holds for all gE~, we must have 1 2fn . (2.3) an = 21n e -lnxf(x) dx for nE I, o because f~n e - inx eimx dx = 2nbmn. {an}, regarded as a function on 71., is called the Fourier Transform off and is written! :j(n)= an, nE I. We hence forth always assume ~ contains these harmonics.