Unitary Representations and Harmonic Analysis, Second Edition (North-Holland Mathematical Library)

UNITARY REPRESENTATIONS AND HARMONIC ANALYSIS An Introduction Second Edition North-Holland Mathematical Library Board...

Author: Mitsuo Sugiura

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UNITARY REPRESENTATIONS AND HARMONIC ANALYSIS An Introduction Second Edition

North-Holland Mathematical Library Board of Advisory Editors: M. Artin, H. Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H. Halberstam, L.V. Hormander, J.H.B. Kemperman, H.A. Lauwerier, W.A.J. Luxemburg, L. Nachbin, F.P. Peterson, I.M. Singer and A.C. Zaanen

VOLUME 44

NORTH-HOLLAND AMSTERDAM OXFORD NEW YORK TOKYO

Unitary Representations and Harmonic Analysis An Introduction Second Edition

Mitsuo SUGIURA Professor Emeritus The University of Tokyo

1990 NORTH-HOLLAND AMSTERDAM OXFORD NEW YORK TOKYO

KODANSHA LTD. TOKYO

Published in co-edition with Elsevier Science Publishers B.V., 1990 Distributors f o r Japan: KODANSHA LTD. 12-2 1 Otowa 2-chome, Bunkyo-ku, Tokyo 112, Japan Distributors outside Japan, U.S.A. and Canada. ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 21 1, 1000 AE Amsterdam, The Netherlands Distributors f o r the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y. 10010, U.S.A.

ISBN: 0 444 88593 5 ISBN: 4 06 203509 x (JAPAN)

Copyright 0 1975, 1990 by Kodansha Ltd. All rights reserved. No part of this book may be reproduced in any form, by photostat, microfilm, retrieval system, or any other means, without the written permission of Kodansha Ltd. (except in the case of brief quotation for criticism or review) Printed in Japan

To the memory of Edward T.Kobayashi

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Preface to the Second Edition

The second edition differs from the first in the following places. In Ch. I, Theorem 5.6 on the Fourier series expansion of piecewise C'-functions has been added. In Ch.11, the proof of Theorem 8.1 has been more fully implemented by adding two Lemmas. In Ch.111, Theorem 1.9 has been replaced by Proposition 1.5. In Ch.IV, the proof of Theorem 5.1 is rewritten using Ch.III.Prop.l.5 instead of Proposition 5.7 which has been deleted. In Ch.V, Q 3, the description of limits of discrete series has been added. In the first edition, we dealt solely with the representations of SU(1,l). In the present edition we have added Q 10 on the realization of irreducible unitary representations of the group SL(2, R). Appendix G on Fatou's theorem is added in connection with the limits of discrete series. Bibliography is tripled in length by adding some recent works and Notes have been rewritten accordingly. The author would like to express his hearty thanks to W.L. Baily Jr. who read through the manuscripts of the 2nd edition and suggested many improvements. He also wishes to thank K. Nishiyama who provided me with an Errata for the first edition.

M. SUGIURA

January 1989

vii

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Preface

The principal aim of this book is to give an introduction to harmonic analysis and the theory of unitary representations of Lie groups. Fourier series and integrals were introduced into mathematics in connection with the problems of vibrating string and heat conduction. They are techniques of representing a vibration as a superposition of harmonic oscillations e'a'. The techniques of harmonic analysis turned out to be powerful tools in mathematical analysis and physics. In the meantime, the group theoretical character of harmonic analysis was discovered. The harmonic oscillation e4=( is an irreducible unitary representation of the additive group R of real numbers. It is a representation of the torus group T = R/2nZ when x is an integer. Thus Fourier series and integrals are deeply connected with the theory of unitary representations of the groups T and R. This connection was fruitfully pursued by many mathematicians and was extended to cover a wider class of Lie groups. In the 19203, H. Weyl constructed the representation theory of compact Lie groups. He determined the irreducible representations, gave an explicit formula for the characters. Moreover he developed the Lo-theoryof Fourier series on compact (Lie) groups with F. Peter. In 1947, two important papers on infinite dimensional unitary representations appeared. One was Gelfand-Naimark's paper on SL(2, C) and the other was Bargmann's paper on SL (2, R). Since then, many interesting works have been done in the field of unitary representatiods of non compact and non commutative Lie groups and of harmonic analysis on them. In the development of the theory, certain difficulties were encountered in connection with the existence of operator algebras (von Neumann algebra) of type I1 and type 111. For example, the uniqueness of (direct integral) decomposition into irreducible unitary representations is lost for representations of type I1 or 111. Fortunately, for a Lie group whose factor representations are always of type I (a type I group), the theory of unitary representations can be constructed along similar lines to the case of compact and abelian groups, although it will generally be more complicated. All the Lie groups treated in this book are type I groups.

X

PREFACE

This book is essentially self-contained. No special knowledge on harmonic analysis, unitary representations or Lie groups is required. However, the reader is expected to have some familiarity with Lebesgue integrals and fundamental concepts of functional analysis. For the reader's convenience, basic facts on these materials are summarized in the A p pendices. The author has preferred a concrete calculation on a given matrix group rather than to develop the general theory for Lie groups or locally compact groups. However, general arguments are adopted without hesitation when they give a clearer insight. I hope that the resultant book will kindle interest for further development of the theory. The following six groups are treated in this book: the onedimensional torus group T,the special unitary group SZ7(2), the 3-dimensionalrotation group SO(3), the n-dimensional vector group R", the Euclidean motion group of the plane M(2) and the real special linear group SL(2, R). In each chapter, all the irreducible unitary representationsof these groups are concretely constructed and harmonic analysis on them is developed. In particular, the Fourier transforms on.the spaces of rapidly decreasing functions, C"-functions with compact supports and square integrable functions are discussed. Chapter I deals with the ordinary Fourier series from the viewpoint of unitary representations of the torus group T.Since the compactness of the group T is decisive, the general theory of unitary representations of compact groups (Peter-Weyl theory) is given in $3. In & the I,irreducible unitary representations of the group T are determined and the results in $3 are applied to the group T. In $5 and $6, the Fourier series of smooth functions and distributions on T are discussed. In Chapter 11, the theory of Fourier series is developed to the non commutative compact Lie groups SU(2) and SO(3). Using the theory of charactersin $2 and the calculation of the Haar measure in $3, it is proved in §4 that the representations constructed in $1 exhaust the irreducible unitary representations of SU(2). In $5, the notion of Lie algebra is introduced for a linear Lie group. $5 is the preparation to $6 which discusses the Fourier series of smooth functions on SU(2). In $7, the irreducible representations of the rotation group SO(3) and their realizations on the spaces of spherical harmonics are discussed. $8 gives the only exception of the requkment for self-containedness. In this section, E. Cartan's theory of highest weight and Weyl's theory of characters are freely used. The aim in $8 is to show that the results in $6 for SU(2) hold for general compact Lie groups. The reader may wish to skip over $8. In chapter 111, Fourier integral and unitary representations of the group R m are treated. The theory is based on L. Schwartz's theorem (Theorem 1.2) on the Fourier transforms of rapidly decreasing functions. Since R"

PREQACB

xi

is not compact, the notion of the direct integral is needed for irreducible decomposition of representations. Using Bochner's theorem on positive definite functions, an arbitrary unitary representation of Rnis decomposed as the direct integral of irreducible representations in $3. The PaleyWiener-Schwartz theorem characterizing the Fourier transforms of Cmfunctions with compact supports is proved in $4. In $5 the notion of tempered distribution is introduced and their Fourier transform is studied. Chapter IV deals with the Euclidean motion group M(2) as a simple example of non compact and non commutative Lie groups. The group M(2) has a series of S n i t e dimensionalirreducible unitary representations which are good examples of induced representations. Harmonic analysis on M(2) is discussed. In particular, the inversion formula of Fourier transform on M(2) is proved in $3 for rapidly decreasing functions and the Fourier transforms of such functions are characterized in $5. The final and longest Chapter V treats the group SL(2, R), which is a prototype of non compact semisimple Lie groups. The group SL(2,R) appears in many branches of mathematics in different forms. A realization of SL(2, R) as the special unitary group SU(1, 1) of indefinite Hermitian form z1Zl-z212is particularly useful for constructing the representations. The group SL(2, R)z SU(1, 1) has the three different series of irreducible unitary representations; the principal continuous series, the (principal) discrete series and the complementary series. After a short preparation for Iwasawa decomposition in $1, these three series of representations are constructed in $2, $3 and $4 respectively. In $6, the irreducible unitary representations of SL(2, R) are classified by the infinitesimal method. In $7, the characters of irreducible representations are computed explicitly. $8 gives the Fourier inversion formula for the C"-functions with compact supports. In $9 harmonic analysis of two-sided K-invariant functions is discussed. The Notes at the end of the book include certain historical facts and supplementary results together with references €or further study. The bibliography contains a list of papers and books more or less directly related to the subject of this book. The present volume is based on lectures given by the author over the past decade at several universities including Osaka Univ., New Mexico State Univ. and the Univ. of Tokyo. I would like to express my sincere gratitude to Professor W. L. Baily for carefully reading the manuscript and for offering many suggestions to improve the exposition. I would also like to thank Dr. T. Hirai for his advice on the content of Ch. V, and Dr. Y. Shimizu for his help in proofreading. It is also a pleasure to thank the National Science Foundation for its financial support during my stay in the U.S. Finally, I wish to thank

xii

PREFACE

the staff of Kodansha Scientific, whose endurance enabled me finally to complete this book. The book is dedicated to the late Dr. Edward T. Kobayashi who provided me an opportunity to lecture most ofits contents and made the notes for me.

May, 1975

Mitsuo SUGIURA

Contents

Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Preface . . .. . . . . . . .. . . , . . . . .. . . . . . .. . . . . . . . . . . .. . . .. . . . . . . . . . . .. i~ Conventions and Notations . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . xv

Fourier series and the torus group T . . . . . . . . . . . . 1

I.

$1. Introduction, 1 $2. Fundamental definitions, 5 $3. Unitary representations of compact groups, 14 &I. Fourier series of square integrable functions, 30 $5. Fourier series of smooth functions and distributions, 34

11.

Representations of SU(2) and SO(3)

. . . . . . . . . . . .47

$1. Construction of irreducible representations of SU(2), 47

$2. Characters of compact groups, 50 $3. Haar measures on SU(2), 54 &I. Enumeration of irreducible representations, 57 $5. Lie algebras and their representations, 59 $6. Fourier series on SU(2), 76 $7. Representations of SO(3) and spherical harmonics, 82 $8. Fourier series on compact Lie groups, 93

111. The Fourier transform and unitary representations of RM . . . . . . . . . . . . . . . . . . . . . . . .lo1 $1. Rapidly decreasing functions, 101 $2. The Plancherel theorem and the decomposition of the regular representation, 115 $3. Positive definite functions and Stone’s theorem, 122 &I. The Paley-Wiener theorem, 142 35. Tempered distributions and their Fourier transforms, 150

IV. The Euclidean motion group . . . . . . . . . . . . . . . . . .155 $1. Construction of irreducible representations, 155 §2. Classification of irreducible unitary representations, 165 93. Fourier transforms of rapidly decreasing functions, 169 &I. The Plancherel theorem, 179 95. Determination of 9 ( G ) and B ( G ) , 187

V. Unitary representation of SL(2, R) . . . . . . . . . . . .205 91. The Iwasawa decomposition, 205 92. Irreducible unitary representations, 213 I. PRINCIPALCONTINUOUS SERIES, 213 93. Irreducible unitary representations, 235 11. F?RINCIPAL DISCRETE SERIES,235 111. T H E LIMIT OF DISCRETE SERIES 246 &I. Irreducible unitary representations, 255 IV. COMPLEMENTARY SERIES, 246 95. K-finite vectors, 265 96. Classification of irreducible unitary representations, 280 97. The characters, 306 98. Inversion formula, 343 99. Harmonic analysis of zonal functions, 362 910. Irreducible unitary representations of SL (2, R),388 I. DISCRETE SERIES, 389 11. COMPLEMENTARY SERIES, 391 111. PRINCIPAL CONTINUOUS SERIES, 393

Appendix ...................................................... 395 Notes .......................................................... 411 Bibliography . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .417 Index . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .449

Conventions and Notations

Theorems and Propositions are numbered separately in each section. Therefore both Proposition 3.1 and Theorem 3.1 exist in 43. A Proposition or Theorem is referred simply as (say) Proposition 3.1 in the same chapter. A results in a different chapter is cited as (say) Ch. 11, Proposition 3.1. Notations. 2, N,R, C, denote respectively the set of integers, the set of non negative integers, the set of real numbers and the set of complex numbers with their usual algebraic and topological structures. Re and Im signify the real and imaginary parts of a complex number. A x B denotes the direct product of sets A and B. M n ( C ) denotes the algebra of all matrices of degree n with coefficients in C. Let X be a topological space. Then C ( X ) denotes the vector space of all C-valued continuous functions on X. L ( X ) denotes the subspace of C ( X ) consisting of all functions with compact supports. The uniform norm I(flJm=supIf(x)l is a norm on L(X). Z#X

Let H be a Banach space andf be an H-valued function on I=(a, + m) and g be a real valued function on I. If g(t)-'f(t) is bounded on some subinterval (b, + m) of I, then we write

f(t)= O(g(t))

( t d + m 1.

If lim g(t)-tf(t) = 0, we denote t-+-

f(O=o(g(tN (t-. + m). Similar notations are used for two sequences (fn)neN and ( g n ) n c N . The complexification of a real vector space V is denoted by V C . VC is a complex vector space whose element z is uniquely written as z = x + i y , (x, y E V). If V is a real Lie algebra, then Vc is a complex Lie algebra. The support of a function f is denoted by supp (f).

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CHAPTER I

Fourier series and the torus group T

$1. Introduction A unitary representation is a homomorphism of a given group into a group of unitary operators on a Hilbert space (cf. $2 for the precise definition). The theory of unitary representation is a natural generalization of classical Fourier analysis. Fourier analysis is a method of decomposing “any” oscillation into simple oscillations called harmonic oscillation. So Fourier analysis is also called harmonic analysis. A harmonic oscillation is a motion given by an equation x = a cos n t + b sin nt, where t is the time and x is the coordinate of a moving point. The equation for x can be expressed as follows: x = d cos(nt a) (1.1)

+

where d= .Ja’ +ba, cos a = a/d and sin a =: -b/d. The number d in (1.1) is the amplitude, n, the frequency and a, the initial phase of the oscillation. The phase a is not related to the nature of the oscillation but depends only on the choice of the origin of the time scale. Fourier analysis assures us that any function in a suitable family of functions (such as smooth functions, functions of bounded variations) with the period 2a can be developed in a Fourier series:f can be expressed as w

f(t> =3

(1.2)

2

+ C (a,

+

cos nt b, sinnt)

n-1

where the coefficients a, and b, are determined by (1.3)

an=-

x

s:’

f(t) cos n t dt,

b, =

J:’

f(t) sin nt dt.

The expansion (1.2) means that the oscillation f is a superposition of harmonic oscillations with integral frequencies n=O, 1, 2, .... By Euler’s relation e“ =cos t

+i sin r,

1

2

FOURIER SERIES AND TORUS GROUP

T

the Fourier series (1.2) can be rewritten as

-

f(t)=

(1.4)

C

C,e*nt,

n=--

where the coefficients cn are determined by

(1.5)

1

c --(an n-

2

- ibn)=- f(t)e-tncdt 2n Is’’ 0

c- ---(a,+ib,)=~;; 1 n-

2

r

(n=O, 1, 2, .-).

of(t)e*ncdt.

The function xn(t)= etnc appearing in (1.4) expresses a circular motion with a constant angular velocity n of which the projection to the real (or imaginary) axis is a harmonic oscillation. From our point of view, the “complex” form (1.4) of a Fourier series is preferable to the “real” form (1.2), because the functions xn(t) are strongly connected to the group structure. We shall call the function e‘nt a harmonic oscillation rather than (1.1). The function Xn(f)=efRthas the following four properties: 1) Ixn(t)l= 1 for all t E R. 2) Xn(t)xn(s)= Xn(t +s). 3) x,, is continuous. 4) Xn(2R)= 1 . The properties I), 2), and 3) mean that xn is a continuous homomorphism from the additive group R of real numbers to the one dimensional unitary group U(1) = { z E CI IzI = I}. That is, xn is a unitary representation of the real number n. On the contrary, the group R. This fact hol4s for an’ property (4) holds only when n is an integer. And the function x,, for an integer n is a u n i t a j representation of the torus group T = R/2nZ which is isomorphic to U(1) and to SO(2). (See the last part of this section for the definition of the topological group T). Now we study the Fourier series for a square integrable function. All complex valued functions which are square integrable on the interval [ 0 , 2 ~ form ] a Hilbert space La(T)with the inner product

Another remarkable property of the family of functions ( z J n r Zis its orthogonality : 5) (xn, zn)=6n,m The determination of the Fourier coefficients in (1.5) is explained by the orthogonality relation 5). If a functionfin La(T)can be expanded as

LNTRODUCTION

3

in La(T),then the coefficient cn must be equal to

for all n. So the first problem in the theory of Fourier series for square integrable functions is as follows: Problem A : Can any function f in La(T)be expanded in La(T)as in (1.6)? The answer is affirmative and will be proved in 54. Here we explain the meaning of the expansion in La(T). In a Hilbert space H , a family ( x . ) , , ~of elements in H is called summable with the sum x in H if for any positive number E , there exists a finite subset Bo of A such that for any finite subset B 2 Bo, the inequality Ilx- Ex,l/ < E holds. x is denoted by Ex,. In particular if A = N (the set ..B

urA

of all non-negative integers), this definition implies n

limllx n+co

C xkll= 0, k=O

and if A = 2 (the set of all integers), this implies n

A family ( x . ) . ~of~ elements in a Hilbert space H is called orthonormal if it satisfies

An orthonormal family ( x . ) . ~in~H i s called complete if it satisfies one of the equivalent conditions stated in the following proposition. Proposition 1.1. The following three conditions l), 2) and 3) for an orthonormal family (x,),,~ in a Hilbert space H are mutually equivalent: 1) every element f in H can be expanded in the form

f=

c

(f, xm)xm

9

.CA

2) the Parseval equality

Ilflla

=

I(Lx.4' USA

is valid for allfin H,

4

FOURIER SERIES AND TORUS GROUP

T

3) iff( E H) is orthogonal to xu (that is, (f,x,)=O) for all a in A, then fmust be equal to 0. Proof. 1)02). This equivalence is clear by the equality

c
n

n

(1.8)

Ilf-

I(f,

x U k ) x R t ~ i 2 = ~ ~ f ~ i 2 -

xUk)ll*

t-1

0-1

2) =>3)is trivial. 3) *l). Put (1.9)

g=

1cf, x&a urA

The right hand side of (1.9) is convergent by the BesseZ’s inequality (1.10)

c I(f,

xa)lls Ilfll’

aeA

which follows from (1.8). So g is a well defined element in H and the equality (1.9) gives the equalities (g, x.)=cf, xu) for all a E A ,

that is, ( g - f , x a ) = O for all a e A . Then the assumption 3) implies g-f=O and f= Ccf, xa)xa. RtA

q.e.d.

Using the concept of completeness of an orthonormal family, problem A can be re-expressed as follows: Problem A’: Is the orthonormal family (x,JnrZ complete? Many proofs for the completeness of the family ( x ~ ) , are , ~ ~known. We shall give a proof from the representation theoretical point of view. Our proof depends on a theorem of Peter and Weyl which asserts that the matricial elements of “all” irreducible unitary representations of a compact group G form a complete orthogonal family. By the Peter-Weyl theorem, the above Problem A’ is reduced to the following Problem B. Problem B : Does there exist an irreducible unitary representation “different” from every x,,? The precise meaning of “different from every” is “not equivalent to any” which is explained in the next section. Now we shall give the precise definition of the torus group T.A set G is called a topological group if it satisfies the following three conditions: 1) G is a group, 2) G is a topological space satisfying Hausdorff separation axiom, 3) the mapping ( x , y ) ~ x y - ’is a continuous mapping from the direct product space G x G into G.

FUNDAMENTAL DEFINITIONS

5

Examples of topological groups. The additive group R of real numbers is a topological group with its usual topology. The condition 3) is satisfied because of the inequality I(x-y)-(x'-y')l4 Ix-x'I +ly-y'l. The complex general linear group GL(n, C ) of order n is a topological group with the topology defined by

( 2 Igrj-hrjla) . Any 111

distance d(g, h) =

subgroup N of a topological

j-1

group G is a topological group with the relative topology of G. So the subgroups of GL(n, C ) , such as SL(n, C), SO(n), U(n), SU(n), are all topological groups. Let H be a closed normal subgroup of a topological group G. Then the set G/H of all cosets gH= {ghlh E H ) of G modulo H forms a group with the product defined by

(XH) 01H)=(XYIH. The group G/H is called a factor group of G. The topology on G/H is given by determining the family &'(G/H) of open sets in G/H. Let p : g w g H be the canonical projection of G onto G/H. Then d ( G / H ) is defined as follows: (1.11) &'(G/H)= {A1 A c G/H and p-l(A) E d(G)). I t is easily verified that the family @(G/H) satisfies the axioms of open sets: any unions and finite intersections of open sets are open. The factor group G/H with the topology defined by ( 1 . 1 1) is a topological group and is called a factor group of the topological group G. For example the group 2 r Z = (2mln E Z } (2 is the group of integers) is a discrete subgroup of R . So the factor group R/2nZ is defined and it is called the 1-dimensional torus group and is denoted by T.The group T is compact because it is the image of a compact interval [0,2r] by the continuous mapping p . The group T is commutative because R is commutative.

$2. Fundamental definitions Definition 1. A unitary representation of a topological group G is a strongly continuous homomorphism U of G into the group U ( H ) of unitary operators on a Hilbert space H. The Hilbert space H i s called the representation space of U and is denoted by H( U ) . The dimension d( U) of H(U) is called the degree of the representation U. More precisely, a mapping U :g H U g of a group G into the group U ( H ) is called a homomorphism if it satisfies Ugh= U,Uh for all g and h in G.

6

FOURIER SERIES AND TORUS GROUP

T

A homomorphism U of a topological group G into U ( H )is called strongly continuous if the mapping gr-.U,,x is a continuous mapping of G into H = H( U)for all x in H. Remark 1. The equality IIUgX-

Unxll =IIUg--IAX-XIl

proves that a homomorphism U of G into U ( H ) is strongly continuous if it is strongly continuous at the identity element e of G . Example 1. Let R be the additive group of real numbers with its usual topology. Each real number s defines a function x , on R by Xs(t)=eft*. Then xI is a unitary representation of R . The representation space H(xs) is a one-dimensional vector space C (the field of complex numbers). The complex number xa(t)is identified with a scalar multiplication m+XS(t)x on C which is a unitary operator on C . Example 2. Let H=M,,(C) be the linear space of all complex square matrices of degree n. H is a Hilbert space with the inner product ( A , B) =Tr(AB*) . A unitary representation A of the unitary group U(n) on H = M , ( C ) is defined by A g ( X )=gXg-1 . The representation A is called the (complex) adjoint representation of

Wn). ExampZe 3. Let H = L a ( R ) be the Hilbert space of all complex-valued square-integrable functions on R . The inner product is defined by ( f , g ) = / -a -I(f)mdt.

A unitary representation R of the group R on L2(R)is defined by

( R t f ) ( s ) = f ( ~ +t), f E L 2 ( R ) . Since the Lebesque measure on R is invariant under the translation S H s + t , the operator R, is a unitary operator on H = L 2 ( R ) . It is easily verified that R is a homomorphism. There remains to prove the strongly continuity of R. By Remark 1, it is sufficient to prove that for any f in H and for E >0, there exists a neighborhood N of 0 in R such that if t belongs to N , then the inequality IlRlf -fll< holds. Since the space L(R) of all continuous functions with compact supports

7

FUNDAMENTAL DEFINITIONS

is dense in H=La(R) (cf. Appendix DS), there exists a function L(R) satisfying

(O

in

(2.1) I V - ~ I I~ € 1 3 . Since Rt is a unitary operator, the inequality (2.2) IIR~~-R < €13 ~~II holds for all r in R. Let U be a fixed compact neighborhood of 0 and C be the support of (o. Then since C is compact, (O is uniformly continuous on R . Hence there exists a neighborhood N = -N of 0 such that N c U and (2.3) llRt~--(ollm<~/3 J;;; for all t c N , where rn is the Lebesque measure of the set C + U = {s+ t 1s E C, t E U}. Then we have Il(O11 5

6 Il(O11-

and (2.3) prove the inequality

.(2.4)

IIRt~-pII < ~ / 3 for any r in N.

The inequalities (2.1), (2.2) and (2.4) prove that

IIRtf-fll 5 IIRtf-Rtvll+ IIRtp-~ll+ l l ~ - f l l < €13 €13 €13= E

+ +

for any t in N. So we have proved the strong continuity of R. The unitary representation R is called the right regular representation of R . Similarly the right regular representation R is defined for any locally compact topological group G. Definition 2. Let G be a locally compact topological group and let H(R) be the Hilbert space of square-integrable functions on G with respect to a right Haar measure (cf. $3) on G. For any element g in G, let R, be the operator on H(R) defined by

fv)-

(R$ )(h)= f ( h g ) f e Then R is a unitary representation of G (Appendix D.6). R is called the right regular representation of G . Similarly the left regular representation L is defined by (L,f)(h)=f (g-'h). Other examples of unitary representations will be given in later chapters. Definition 3. Two unitary representations U and V of a group G are called eqzrivalent and the equivalence is indicated by U E V if there exists an isometry (a linear mapping preserving the norm) A of H ( U ) onto H( V ) satisfying

8

FOURIER SERIES AND TORUS GROUP

T

AU,= V g A for all g in G.

It is easily seen that the relation “ z ” is, in fact, an equivalence relation. In a sense, the “atoms” of unitary representations are the irreducible representations which are defined as follows. Definition 4. Let U be a unitary representation of a group G . A closed linear subspace F of the representation space H ( U ) is called invariant under U if we have U g F cF for all g in G. A unitary representation U is called irreducible if H ( U ) + {0} and if H(U) and {0} are the only invariant closed subspaces of H ( U ) under U. The unitary representations in example 1 is irreducible, but it can be proved that those in examples 2 and 3 are not irreducible (cf. Chapter 111, Proposition 2.1). Non-irreducible unitary representations are “decomposed” into irreducible representations. Here, the meaning of “decomposition” should be clarified. At this point, some difficulties arise for infinite-dimensional unitary representations and the notion of direct sum decomposition must be generalized to that of direct integral decomposition which will be explained in Chapter 111. In this section we explain the more classical notion of direct sum decomposition.

Definition 5. A Hilbert space H i s called a direct sum of a family (Ha)aEA of closed linear subspaces and one writes

H= @ H a ,X
if the following two conditions are satisfied: 1 ) the subspaces Ha are mutually orthogonal (H, IHBif cx ;f:B), 2) H is spanned by (the set of all finite linear combinations of the elements in u H, is dense in H). CrCA

If H is decomposed in the form H = @ H,, then any element x in H can arA

be expressed uniquely in the form X=

EX.,

x,EH,,

URA

and the norm of x is given by 11x112=

c

llxalla*

arA

Therefore at most countably many x,’s are different from 0.

FUNDAMENTAL DEFINlTIONS

9

Definition 6. If the representation space H of a unitary representation

U of a group G is decomposed into a direct sum of U-invariant closed subspaces then the unitary representation U is called a direct sum of unitary representations (U')meAwhere U a is the restriction of U on He. Proposition 2.1. A finite-dimensional unitary representation U of a group G can be decomposed into a direct sum of irreducible representak

tions (Ui)ldtSk: U = @ U i . i=1

Proof. We prove the proposition by induction on n = dimH(U)=degU. Any representation of degree 1 is irreducible. So the proposition is trivial if n = 1. Assume that Proposition 2.1 is valid for every representation with degree less than n. Take a non-zero U-invariant subspace V of H(U) having smallest dimension. Then V is an irreducible subspace of H(U). The orthogonal complement VL of V is also invariant under U.This can be seen from the equalities (U,V*, V)=(V', U,-,V)=(V', V)= {O}. Since the space V' has dimension smaller than n, V* is a direct sum of irreducible subspaces. So H itself is a direct sum of irreducible subspaces. q.e.d.

Next we give a criterion for the irreducibility of unitary representations which is well known for the finite-dimensional case under the name of Schur's lemma. Definition 7. Let U and V be two unitary representations of a group G. A linear operator T of H ( U ) into H(V> satisfying TU,= V,T for all gE G ,

is called an intertwining operator between U and V. Proposition 2.2. Let U and V be two finite-dimensional irreducible representations of G and T be an intertwining operator between U and V. Then either T= 0 or T is a linear isomorphism of H( U)onto H( V). Proof. By the intertwining property of T, the image Im T= T(H(U)) and the kernel Ker T= T-l(O) of T are invariant under V and U respectively. By the irreducibility of U and V, we have ImT= {O) or H(V) and KerT=H(U) or {O}. If ImT= {O) or Ker T=H(U), then we have T=O. Otherwise Im T= H(v> and Ker T= {O}, i.e., T is an isomorphism of H(U) onto H(V). q.e.d.

10

FOURIER SERIES AND TORUS GROUP

T

Remark 1. If U and V are not equivalent, then the only possible intertwining operator T between U and V is 0. This can be seen from Proposition 2.2 and Proposition 2.6 which will be proved later. Proposition 2.3. Let U be a unitary representation of a group G. Then a closed subspace V of H( U ) is invariant under U if and only if the orthogonal projection P on V commutes with U, for all g in G. If V is invariant under U,then the orthogonal complement V l is also invariant under U. Proof. H = H( U ) is decomposed as a direct sum

(2.5)

H = V @ V'

where V' = [ z E H ( ( y ,z ) =0 for all y E V ) is the orthogonal complement of V. If V is invariant under U, then V' is also invariant under U.This fact was proved in the proof of Proposition 2.1. Let x=yi-z,

YE

v,2 6 V'

be the decomposition of an element x in H according to (2.5). Then the orthogonal projection P is defined by Px =y and we have u g x = u , y + u g z , UoyE v, U,Z€ VL,

and PU,x= u,y= UgPX.

We have proved (2.6) PUg=UgP for all g E G . Conversely if (2.6) is valid, then the equalities U,V=U,PH=PU,H=PH= V show that V is invariant under U.

q.e.d.

Let B ( H ) be the algebra of bounded linear operators on a Hilbert space H and M be a subset of B(H). The commutant M' of M is defined by M ' = [ A E B ( H ) I A U = U A forany U E M } . We define M", M"', ..., by M" = (W)',M"'= (M")', ... .

Proposition 2.4. (Schur's Lemma) Let U be a finite-dimensional unitary representation of a group G and M = { UglgE G}. Then U is irreducible if and only if the commutant M' of M is equal to C1. Proof. Let V be a U-invariant subspace of H( U ) and P be the orthogonal projection on V. By Proposition 2.3, P belongs to M'. So if M ' = C I , P = a l , a E C . Since P a = P , a = O or 1, i.e. P=O or 1.

This implies that V = [O} or H ( U ) and U is irreducible. Conversely assume

FUNDAMENTAL DEFINITIONS

11

that U is irreducible. Then Proposition 2.2 proves that M’ consists of 0 and invertible elements. Let A be an element in M’ and a be an eigenvalue of A (the root of the equation, det(x1 -A)=O). Then a1 - A belongs to M’ and is not invertible. So a1 - A must be equal to 0. Since A can be q.e.d. an arbitrary element in M’, we have proved M’=Cl. Proposition 2.4 can be proved without the assumption that U is finitedimensional. Before proving this fact, we give some preliminaries. Let M an N be subsets of B ( H ) . Then the following properties of commutant are obvious from the definition of M’. (2.7) and (2.8)

MC N

3

M ’ 2 N’

=> M”c

N”.

M c M”.

(2.7) and (2.8) imply M ‘ 2 M ” ‘ . On the other hand (2.8) applied to M’ gives M’cM”’. So we have M’ = M”’. (2.9) A subalgebra M of B ( H ) is called a von Neumann algebra (or W*-algebra) if it is equal to M“. It can be proved that a subalgebra M containing 1 is a von Neumann algebra if and only if M is self-adjoint ( M = M * ) and M is weakly closed (cf. Dixmier [l]). Let M be a subset of B ( H ) . Then (2.8) and (2.9) prove that M” is a von Neumann algebra containing M . If A is a von Neumann algebra containing M , then (2.7) proves that M ” c A ” = A . So M“ is the smallest von Neumann algebra containing M . Theorem 2.1. (Schur’s Lemma) Let U be a unitary representation of a group G and M = {U,lg E G } . Then U is irreducible if and only if the commutant M’ is equal to the set C1 of scalar operators. Proof. The proof of the “if part” of the theorem is same as the corresponding part in the proof of Proposition 2.4. We assume that U is irreducible and prove that M ’ = C l . Any operator A E B ( H ) can be written uniquely in the form A=H1+iHa, H,*=Hj ( j = l , 2)

by putting Hl = 2-1(A +A*), H , = (2i)-l(A-A*). The operator A belongs to M’ if and only if Hl and Ha belong to M ’ . So it is sufficient to prove that any self-adjoint operator A in M‘ is a scalar operator. Let (2.10)

A=

1-

ItdE,

-w

12

FOURIER SERIES AND TORUS GROUP

T

be the spectral representation of a bounded self-adjoint operator A belonging to M’. Then we have El E {A}” (2.1 1) (cf. Appendix E.4). Since {A} c M‘, we have {A] ” c &f“’ = M‘ (2.12) by (2.7) and (2.9). (2.11) and (2.12) prove that E,EM’ forall R E R . Therefore by Proposition 2.2, E,H is invariant under U. E,H is equal to [O} or H, because U is irreducible. Since E, is monotone increasing and right continuous, we have 1 ROSA (2.13) E,= 0 I
(

COROLLARY to Theorem 2.1. Any irreducible unitary representation of a commutative group G is one-dimensional. Proof. By the commutativity we have M c M’. By Theorem 2.1, the irreducibility of U implies M’ = C 1. All the operators U,, are scalar operators and every linear subspace of H( U ) is invariant under U. If dimH(U) > 1, then there exists a closed U-invariant subspace V different from {0} and H(V). For example take any one dimensional subspace. Therefore dimH( U ) must be equal to 1. Remark 2. The proof of Theorem 2.1 depends solely on Proposition 2.3 and the fact that any W*-algebra is generated by the projections. Definition 8. Let U be a unitary representation of a group G. An element x in the representation space H ( U ) is called a cyclic vector if the set [ U,xlg E G} (topologically) spans H(U). That is, the space of all finite linear combinations of the elements in { U,,xlgE G} is dense in H(U). A

unitary representation U which has a cyclic vector in H ( U ) is called a cyclic representation. Proposition 2.5. Every unitary representation U is a direct sum of cyclic representations. Proof. Let r be the set of all families r = (Ha)aeA of mutually orthogonal closed U-invariant subspaces Ha such that each H , has a cyclic vector with respect to the restriction of U to Ha. Then r is an inductively ordered set. The order is given by the inclusion. In fact, if 0 is a totally ordered

FUNDAMENTAL DEFINITIONS

13

r, then TO= u 7 is the upper limit of # in r. By Zorn’s Lemma, there exists a maximal element r = (Ha)aeA in r. Then we have subset of

14.

H=@Ha. .Ted

Because if we assume that HZ @ H a , then there exists a non-zero element .
x in ( @ Ha)’. Let Hobe the closed subspace spanned by { U,xlg E G } .Then a cA 7 u [Ho}is larger than r. This contradicts the maximality of 7. q.e.d.

Proposition 2.6. Two unitary representations U and V of a group G are equivalent if and only if there exists an intertwining operator T between U and V which is a bijective bounded linear operator of H ( U ) onto WV). Proof. The adjoint operator T* of T is a bounded linear operator from H ( V ) onto H ( U ) defined by (x, T*Y)=(Tx,V> for all x E H( U)and y E H( V ) . Since T satisfies TU,-I = V,-lT for all g E G,

T* satisfies U,T*=T*V,

for all g E G .

The bounded operator T*T on H(U)commutes with U, for all g e G . Let M = { U,lg E G} ;then T*T belongs to M’. Since (T*Tx, x)=IITxll280 and T is injective, T*T is a bounded strictly positive definite self-adjoint operator on H(U). There exists a bounded strictly positive definite selfadjoint operator S such that S’= T*T. S is bijective.

T*T=J AdEl 0

be the spectral representation of T*T. Then

Since S is a weak limit of a linear combinations of Ez’s and El belongs

14

FOURIER SERIES AND TORUS GROUP

T

to {T*T}" (cf. Appendix E), S belongs to [ T * T ) " c M " ' = M ' Let W = TS-1. Then W is an injective, bounded, linear operator from H ( U ) onto H(V). Since W*W=S-lT*TS-l=l, IIWxll2=(W*Wx,x)=[lxl[*.W is an isometry of H(U) onto H(V). W is an intertwining operator between U and V, because WU,= TS-l U,=TU,S-l= V,TS-l = V, W for all g E G.

$3. Unitary representations of compact groups First we define the Haar integral on a locally compact group G . Let' L =L(G, R ) be the vector space of real valued continuous functions on G with compact supports. A Radon integral m on G is a positive linear form on L (cf. Appendix D.3). The linear form m on L can be extended to a linear form on the space L1=L1(G,m) of all complex valued m-integrable functions on G (cf. Appendix D). The extended linear form is also called a Radon integral and is denoted by

A right Haar integral I on a locally compact group G is a Radon integral on G which satisfies

(3.1) Z# 0 and Z ( R , f ) = I ( f ) for all g e G and f E L , (3.2) where the right translation R, is defined by (R,f)(h)=f(hg). A left Haar integral on G is similarly defined by replacing (3.2) by (3.2)'

Z ( L , f ) = I ( f ) for all g E G and f~ L ,

where (L,f)(h)=f(g-lh). A locally compact group G is called unimodular if a right Haar integral of G is also a left Haar integral. In a unimodular group G , the value of a Haar integral I for a function f is denoted by (3.3) and L1(G,dt) is denoted simply by L1(G). All the groups discussed in this book are unimodular. Proposition 3.1. For any locally compact group G , there exists a right Haar integral. Let Il and labe two right (or left) Haar integrals on G. Then there exists a positive constant c > 0 such that Zl = c12.

UNITARY REPRESENTATIONS OF COMPACT GROUPS

I5

We omit the proof of Proposition 3.1. Instead of relying on this general existence theorem, we shall construct explicitly a Haar integral on each Lie group discussed in this book as a Riemann (and Lebesgue) integral with respect to suitable coordinates of the group. If G is a compact group, then the constant 1 belongs to L and Z(I)>O. Dividing Z by the constant Z(1), we obtain a Haar integral satisfying

(3.4) 1(1)= 1. A Haar integral satisfying (3.4) is called normalized. There exists a unique normalized right Haar integral on G. f ( t ) d t is normalized

Example 1.

Haar integral on

T = R/2nZ. Proposition 3.2. The normalized right Haar integral Z on a compact group G is invariant under the right (left) translation R,:i w t s (L, : t ~ s - l t and ) the inversion S : t w t - 1 :

(3.5)

/$t)dt

= / p ) d f = / Q f ( s - l r ) d t = / Q f ( t-')dt

for all S E G and f E L 1 ( G ) . Proof. Since the process of extending the integral from a linear form on L to one on L1(G') isinvariant under these transformations, it is sufficient to prove (3.5) for any function f in L. Then the right invariance is contained in the definition itself. Let s be a fixed element in G and put J ( f )=Z(L,f) for any YE L. Then J is a normalized right Haar integral on G , because J ( R t f )=Z(L,R$) =Z(RtL,f)=Z(L,fl = J ( f ) and J( 1) =I( 1) = 1 . Hence, the uniqueness of Z implies J = Z, namely Z(L, f)=Z ( f ) . To prove the invariance under S, let (Sf)(t)=f(t-l) and

K(f)=&Yf). Since S(Rtf ) =L4S.f).

K is a normalized right Haar integral on G . Therefore, K=Z and Z is invariant under S.

q.e.d.

In the following we always denote the values of the normalized Haar integral on G as in (3.3). Now we return to our original subject, unitary representations of a compact group G . All the results in this section are obtained from Theorem 3.1 on the following page.

16

FOURIER SERIES AND TORUS GROUP

T

Theorem 3.1. Any unitary representation U of a compact group G is a (Hilbert space) direct sum of finitedimensional irreducible unitary representations. In particular, any irreducible unitary representation of a compact group is finitedimensional. Proof. The last part of the theorem is obtained from the first half by assuming that the representation U is irreducible. TO prove the first half of the theorem, we can assume that the representation U is, cyclic by Proposition 2.5. Let z be a cyclic vector in H ( U ) = H with norm llzll= 1. A Hermitian form (x, y ) on H i s defined by <x, Y>

=/ (

~

x,8 z) (2, Ury)h.

The Schwarz inequality implies the inequality

I<x,V > l S llxll llvll Thus ( x , y ) is a bounded Hermitian form on H and there exists a bounded Hermitian operator K on H satisfying (x, y ) =(Kx, y). From the definition of K, we have (3.6)

Jff

(Kx,v)=

(Utx, Z) (z, utY)dt

for all x, y E H . The Hermitian operator K is strictly positive definite, i.e. (3.7) Since

(Kx,x)>O if xfO.

(Kx,x)=

s

j(Usx, z)ladtzO,

K is positive semi-definite. If (Kx, x) =0, then (Ut,x,z ) = (x, Ut-1z)=0 for all t E G, and since z is a cyclic vector, the last equality implies x=O. Thus (3.7) is proved. As a corollary to (3.7), any eigenvalue R of K is positive. That is,

if K x = l x and xfO, By the defining formula (3.6), we have

(3.8)

(KUsx, y)=

then A>O.

(ucrx, z ) (z, uty)dt

=IG

(U'X,Z) (z,Urr-ly)dt

=(Kx, U r l y )=(U*KX,y ) for all x , y E H. We have proved that

17

UNITARY REPRESENTATIONS OF COMPACT GROUPS

13.9)

KU,=U,K Let I be a real number and put

forall s e G .

H(I) = { x E HlKx =2x1 . Then (3.9) proves that 13.10) U,H(I)c H(I) for all s E G. The most important property of the operator K is its compactness. A bounded linear operator K on a Hilbert space H is called compact if any weakly convergent sequence (Xn)nrN in H is mapped by K to a strongly convergent sequence (KXn)nrN (cf. Appendix B). That is, K is compact if the condition lim (X-Xn, y)=O for all y E H implies lim IlKx- Kxn(l=O. n-a,

n--

Since lim(Kx, Kxn) = IIKxlla and n--

+

IIKx-KXnlI’= IIfill’ -(Kx, Kxn)- (KXn, fi) llfinll’ it is sufficient to show that (3.11)

3

lim llKXnll= ilKxli n+-

in order to prove the compactness of K. By Banach-Steinhaus theorem {cf. Appendix A), a weakly convergent sequence (Xn)nrN is b ~ u ~ d e d . There exists a positive number M such that I l x n l lM~ for all n 6 N. The last inequality and the Schwarz inequality prove that I( UtXn, Z) (2, UsXn) (Uit-a, z)lS M’ for all s and t in G. Then by Lebesgue’s bounded convergence theorem (cf. Appendix C.6),

s

lim IIKxnlla=lim(Kxn,Kxn)=lim n--

It--

=lim/DJ n*-

n--

(UtXn,

(UtXn, Z) (z,

UtKxn)dt

D

z) (z, usxn) (Ust-lz, z)hdt

(t

=n Kxll’ . Thus (3.1 1) and the compactness of K are proved. Let E be the set of eigenvalues of K. Then the Hilbert-Schmidt theorem (Appendix B.5) shows that (3.12) H - @H(4, Ar E

18

FOURIER SERIES AND TORUS GROUP

T

and dimH(2)c + m for all 2 E E (remark each R E E is strictly positive: A>0). On the other hand each eigenspace H(2) is invariant under U, by (3.10). If the restriction of U to H(R) is denoted by UA,(3.12) implies

u= @ u.'. It E

Since each U.'is a finite-dimensional unitary representation of G, UA is a direct sum of irreducible (necessarily finite-dimensional) unitary representations (Proposition 2.1). Theorem 3.1 is proved. Another fact basic for the representation theory of compact groups is the orthogonality relations between matricial elements of irreducible representations which were found and effectively used for finite groups by I. Schur. A continuous homomorphism U of a topological group G into the unitary group U(n) of order n is a unitary representation and is called a matricial (unitary) representation. The number n is called the degree of U and is denoted by d ( U ) .

Theorem 3.2. (Orthogonality relations). Let U and V be two irreducible matricial unitary representations of a compact group G and let urr(g)and vrl(g)be the (i,j)-element and (k,&element of the matrices U, and V , respectively. Then the inner products in La(G)between matricial elements are given by

Proof. Let n = d ( U ) and m = d ( V ) and A be a matrix with n rows and m columns. Then the matrix B=/

UtAVt-ldt 0

satisfies U,BV,-l= B i.e. U,B= BV, for all s E G . The matrix B is a linear operator of Cm= H( V ) into Cn=H( U ) and is an intertwining operator (Definition 7 in 92) between V and U. Since both V and U are irreducible, Proposition 2.2 and Remark 1 after it prove that B = 0 if U is not equivalent to V. If U = V , Proposition 2.4 proves that B=bl. The value of the scalar b is obtained by taking the traces of both sides of the last equality:

UNITARY REPRESENTATIONSOF COMPACT GROUPS

19

1 b=-TrB. n If we take Ejl as A, where Ejl is a matrix whose (p, &element is d p j d l q , then the (i, k)-element of the matrix B is equal to n

m

n

on the one hand and on the other hand is equal to 0 if U&V and equal to

if U = V. The theorem is proved. Remark 1. Without the assumption that U and V are matricial, the orthogonality relations can be expressed more generally as if U&V (3.13) (U,x, y ) (Vtz, w)dt= d(U)-l(x, z) (y, w), if U = V.

1

r'

(3.13) is easily obtained from Theorem 3.2, because any unitary representation U is equivalent to a matricial unitary representation U'. The equivalence is realized by taking an orthonormal basis (et) of H( U)and defining U,' as the matrix of the linear transformation U, with respect to the basis ( e t ) . To state the next theorem (Peter-Weyl theorem)' neatly, we introduce the following notation.

Definition 1. The direct sum of m copies of a unitary representation

U is denoted by mU : mU= U @ ... @ U. m times

Definition 2. The set of all equivalence classes of irreducible unitary representations of a topological group G is denoted by 6 and is called the dual of G. Theorem 3.3 (Peter-Weyl). Let G be a compact group and 6 the dual of G. Choose a matricial unitary representation UA =(u$,l) for each class 1 in 8 once for all and denote the degree d(U2) of U Aby d(2). Then the family

20

FOURIER SERIES AND TORUS GROUP

T

9={ q u t j ’ I A E e , 16i,j6d(R)l is a complete orthonormal family in La(@. Let d(l)

H f a = z C u r j Afor

2 ~ 8and i € { 1 , 2,..., d(2)}.

j-1

Then the subspace Hi’ of La(G) is invariant under the right regular representation R of G. The representation induced by R on the subspace Hia is equivalent to U’.The Hilbert space L1(G)and the regular representation R are decomposed as follows: d(l)

L’(G)= @ @Hi’ 2.8

(-1

Proof. The family 9 is orthonormal by the orthogonality relations (Theorem 3.2). Since

z

d(2)

(Rtutj’)(s)=urj’(st)=

utk’(s)uxj’(t)

k-1

the subspace Hi2 is invariant under R and the restriction RtlHf2of the operator Rt to Hii is represented by the matrix Uta. It remains to prove that {Ht212E 6, 1SiSd(2)) spans the Hilbert space La(G). d(l)

Suppose that the closed subspace W = @ @Hf2is not equal to L%(G). ace f-1

Then the orthogonal complement W of W is a non-zero closed subspace of La(@ which is invariant under R. Let U be the unitary representation of G induced on the space WL.Then U is a direct sum of irreducibleunitary representations (Theorem 3.1). There exists a class 2 E 8 and an invariant subspace E+ {0} of W Lon which the regular representation R induces a representation in the class 2. Let n be the degree d(2) of U’.Then there exists an orthonormal basis (xi)lrirnof E such that n

(3.14)

Rtxj=Cutj’(t)xc

-

(1 5j5n)

1-1

The equalities (3.14) imply (3.15)

xjE&Hf’ i-1

( l s j s n ) and E c & H i a c W. f-1

The last inclusion contradicts the fact that (0)# E c WL.Hence (3.15) proves that W=L’(G). The proof of (3.15). The equality (3.14) can be written as

UNITARY REPRESENTATIONS OF COMPACT GROUPS

21

c n

x,(sO = ut,W&).

(3.16)

i-1

Putting s= e (the identity element of G), we get

This “proof” is incomplete because (3.14) is a system of equations in La(G)and the two sides of (3.16) are equal only when s does not belong to a null set N(t) (of measure 0) for all f E G. The identity element e may be contained in the exceptional set N(t). The exact proof is as follows. Let

[

M(s)= t E GI x&)

ud(t)xt(s)}

# t-1

and

and let xar be the defining function of the set M and m the normalized Haar measure on G. Then by Fubini’s theorem

=SslS.

1

x ~ ( st)ds , dt =

SG

so

m(N(t))dt=0

.

m(M(s))=0 for almost all s E G There exists thus a null set N such that m(M(s))=O for all S E G-N. Since m(G)= 1>0, there exists an element s in G such that m(M(s))=0. For this element s, the equality (3.16) is valid for almost every t in G. Fix such an element s and put st=a. Then (3.16) can be written as

c n

XAU) =

(3.17)

ut,2(s-lu)xi(s)

(-1 n

=

u*l’(a)Ck k=1

9

22

FOURIER SERIES AND TORUS GROUP

T

where c k = ~ U ~ ~ ~ ( Sis- a~ constant ) X ~ ( S(s)is a fixed element). The equality i=1

(3.17) is valid for almost all a in G. So we have proved x j E &Ti2. (-1

Theorem

3.3 is proved. By the definition of completenessof an orthonormal family (Proposition 1.1), the content of Theorem 3.3, 1) can be expressed as follows. COROLLARY 1 to Theorem 3.3. panded in the Fourier series:

Any function f in Lz(G) can be ex-

(3.18) where the series converges in the norm of L2(G). The Parseval equality (3.19) holds for any f in L2(G). Now we shall rewrite (3.18) and (3.19) in another way. Definition 3. The Fourier trandormf of a function f in L1(G)is the matrix-valued function on the dual 6 of the compact group G defined by

(3.20)

sn

f(n) =

f ( t ) u i , - l d t , for R E 6 .

The value f(;O off at R E d is a matrix of degree d(2). In the following, we denote the Hilbert-Schmidt norm of a matrix A=(au) of degree n by IlAll: n

llAlla=Tr(AA*)=

~aij~a.

6, j = 1

COROLLARY 2 to Theorem 3.3. Any functionf in L2(G)can be expanded as (3.21) where the series converges to the function f in the norm of L2(G).The Parseval equality for the function f E L2(G)can be written as (3.22)

UNITARY REPRESENTATIONS OF COMPACT GROUPS

23

Proof. By the definition of the Fourier transformf, the (i,j)-element

f(& of the matrix {(I) is given by

Therefore (3.21) and (3.22) are easily obtained from (3.18) and (3.19) respectiGely. In fact, we have

c c

d(l)

T r ( f ( 4 VQ2> =

f(&*utm

i, j = l d(l)

=

(f,uiJ2)utj2(g)

f, j=l

and q.e.d. In order to see clearly the meaning of the Fourier transform on L2(G), we introduce on the dual 6 of a compact group G a function space ~ “ 6given ) as follows: Definition 4. The space L2(6)is the set of all fucctions p on the dual

6

of a compact group G with values in the set 6Mn(C)of matrices n=l satisfying 1) p(l) E M ~ ( ~ ) ( cfor ) all I E c3 2) Cd(~)llp(J)ll2 <+a . 2c8

The space La((e)is a Hilbert space with the inner product

(%$I =

c

4 a w 4 a m * )*

2fA

Theorem 3.4. The Fourier transform fl: f ~on f a compact group G induces an isometry of the Hilbert space La(G)onto L2((e). Proof. The Parseval equality (3.22) proves that .Finduces an isometry of La(G)into La((e).Moreover f l is surjective. In fact, let p be a function in La(@.Then the series

24

FOURIER SERIES AND TORUS GROUP

T

converges to an element f€’L%(G)in the norm of La(G), because

Since the family B = { Jd(Du(/} is orthonormal,

(5 u ~ , ^ ) = ~ ( &for~ all

1E d

and i, j .

The last equality and (3.23) prove that f(~)=(p(~) for all R E € , and c p = 9 f .

q.e.d.

The most essential part of harmonic analysis on a general compact group is contained in Theorem 3.3. Theorem 3.3 assures us that the irreducible unitary representations play the same r61e for a compact group as the harmonic oscillators etncdo for the torus group. Now we draw some useful consequences of the Peter-Weyl theorem.

Definition 5. For any two integrable functions f and g on a compact group G, their convolution f *g is defined by r

(3.24) We can prove that the above integral exists and has a finite value for almost all s in G.

Proposition 3.3. Let f, g and h be integrable functions on G. Then we have : 1) The integral in (3.24) converges and has a finite value for almost every s in G. The convolutionf*g belongs to L1(G)and satisfies Ilf*gll15 IIf

1 1 1

llgll1 *

2) The convolutionf *g is also expressed by the following integrals: (3.25) and

3)

ProoJ 1) Since and since

f*(g*h)=(f*g)*h. p(s, t)=f(t-I)g(ts) is a measurable function on G x G

25

UNITARY REPRESENTATIONS OF COMPACT GROUPS

=

llflll

l M 1 = llflll llgll1<

+

Q) f

where f(t)=f(t-l), p is integrable on G x G. By Fubini's theorem (cf. Appendix C), (f * g) (s) has a finite value for almost every s in G and is an integrable function on G. Integrating the inequality

I(f*g)(s)l SJ I f ( W l Ig(ts)ldt 0

-

with respect to s, we get IIf * gll15 Ilf 111 llglll 2) (3.25) is obtained by replacing t with ts-l in (3.24). (3.26) is obtained by replacing t with r-l in (3.24) and (3.25). 3) As in the proof of l), we can apply Fubini's theorem and change the order of integration in the following integral:

On the other hand, we have (f*(g

* h))(u)=j-Gf(t-l){J

g(s-"wds)dr

*

Thus 3) follows from the associative law (sr)u=s(tu) in the group G. Remark 2. Proposition 3.3 and its proof are valid for any locally compact unimodular group G. Proposition 3.4. Let f and g be two square integrable functions on a compact group G. 1) Then the convolution f * g is a bounded continuous function on G and satisfies (3.27) Ilf * gll-5 Ilf 111llgllz * dU)

2) Let x2= C uii2 be the character of U2.Then the Fourier series of a i-1

function f E La(G) can be written as

26

FOURIER SERIES AND TORUS GROUP

T

3) The Fourier series

d(R)Tr(&)U89 irl!

of the convolution h =f * g converges to h uniformly on G. Proof. 1) From the definition of convolution and the Schwarz inequality it follows that (*)

I(f* g)(s)ls[

(t

lf(t-1)l Ig(ts)latr llflla llRtglla llglla for almost all s E G

=

.

(3.27) is proved. Similarly, we have

-

ICf*g>(s)-Cfdfg)(s')l$IIf IIa IIR8g-RJ%Ila The strong continuity of the right regular representation R (Appendix D.6) then entails the continuity off * g. Hence (*) is valid for all s E G. 2) The Fourier series (3.18) off E L2(G)is equal to r

=

144(f*X J 0 )

*

2.e

3) The inequality (3.27) implies that the convolution operator T, :g H f * g is a bounded linear mapping from La(G) into C(G).Since the Fourier series of g converges in the norm of La(G),the series f*g=p(w*(g*XJ) 2,e

=

c 4 2 ) (f* (

g>

*XA

If@

converges uniformly on G. The right hand side of the last equality is the Fourier series of the function h =f * g by 2). Hence the Fourier series of h =f * g converges uniformly to h =f * g. q.e.d. Remark 3. Proposition 3.4, 1) and its proof are valid for any locally compact unimodular group G. Definition 6. Let b be the orthonormal family in Theorem 3.3. A finite linear combination of the elements in b is called a representativefunction

UNITARY REPRESENTATIONS OF COMPACT GROUPS

27

on G. The set R(G) of all representative functions on a group G is a complex vector space with the usual addition and scalar multiplication of functions. The vector space R(G) is an associative algebra under pointwise multiplication: (uv) (g)=u(g)v(g); for, utj2ukpis a matrix element of the tensor product representation U Q U p and is a linear combination of matricial elements of the irreducible components of U2@Up. R(G) is called the representative algebra of G. Remark 4. Among continuous functions on a compact group G, representative functions are characterized as follows. A continuous function f on a compact group G is a representative function if and only if dim[Rgf )gE GI is finite where [Rpf(gE is the linear space generated by the set { R gf Ig E G}

.

Theorem 3.5. (Uniform approximationtheorem). Any continuousfunction on a compact group G may be uniformly approximated by representative functions. Namely R(G)is dense in C(G). Proof. Let % be the set of compact neighborhoods of the identity element e in G and let V be an element in %. Then there exists a continuous function hv :G+[O, 13 satisfying hv(e)= 1 and hv =O on the complement V c of V, because the space of a (Hausdorff) topological group is a comhV(t)dt. Then gV is a continuous

pletely regular space. Let gv=hv/ S V

function on G satisfying

S.

gV(t)dt= 1 and gv 2 0. Since a continuous

function f on a compact group G is uniformly continuous, for any E > O there exists a compact neighborhood V of e such that (3.28) By the inequality

llR,-1f -f 11- c E

for any s in V.

I-.

K f *gv)(t)-f(f)l=I

51.

f(ts-')gv(s)ds-/ If@-1)

f(t)gv(s>4

-f(r)lgv(s)&

and (3.28), we get

llf* gv -f II- 5 E * The last inequality shows that a continuous functionf may be uniformly . the other hand, approximated by the functions in the family (f * g v ) Y t l POn since the Fourier series off *gV is uniformly convergent to f *gv (Proposition 3.4), f * g , is uniformly approximated by the finite partial sums of its Fourier series and any finite partial sum of the Fourier series is a representative function. q.e.d.

28

FOURIER SERIES AND TORUS GROUP

T

Remark 5 . For any continuous functionf with compact support on a locally compact group G, and for any E>O, there exists a compact neighborhood V of e in G and a continuous function g, whose support is contained in V such that

-

(3.29) Ilf*gv-fll-SE Let R be the directed set of all compact neighborhoods of e with the order of inclusion. Then the net (gv)rsm is called the approximate j&ntity of L. It satisfies limllf* gv-fll..=O by (3.29). P

The Parseval equality (3.22) proves that the Fourier transformf of a functionfin L2(G)is equal to zero if and only iffis equal to 0 in L'(G) (namely IIf 112 =O or f=O almost everywhere). This relationship holds for any functionfin L1(G).

Theorem 3.6. (Uniqueness of Fourier transform). The Fourier transform fl:fnf is injective on L1(G).Namely the Fourier transform f of a function f in L1(G)is equal to 0 if and only if f=O almost everywhere. Proof. If f=O almost everywhere, then f=O. Conversely let f be an element in L1(G) satisfying f=O. This assumption implies that (3.30)

for all a, i and j

Jfff (r )GV )d t =0

by (3.23). Since TF :g H is an irreducible unitary representation (the contragredient representation of Ul)of G, it is equivalent to a representation W for a certain class Y in 6. The mapping I n v is a bijection of 6 onto itself. (3.30) is thus equivalent to (3.31)

I Gf ( t) u( t) dt=o for all u E R(G) .

(3.31) implies (3.32)

/ / ( t ) h ( t ) d t = ~ for all h E C(G)

because there exists a sequence ( u , ) , , ~in R(G) satisfying lim Ilh- unllm=O n-+-

by the uniform approximation theorem (Theorem 3.5) and we have then P

J.

P

f(t)h(t)dt=lim n--

J

f(t)u,(t)dt=O

0

by (3.31). Replacing h(t) in (3.32) with h(t-ls), we get (3.33) f * h = O forall h € C ( G ) .

UNITARY REPRESENTATIONS OF COMPACT GROUPS

29

Let E be a positive number and V be a compact neighborhood of e satisfying IIR,-if-flll
(3.34)

If*

gv -fh 51.

1

0

If(f~-')-f(t>lsv(s)~~~

s llR'-lf-flll

(3.33) and (3.34) prove Ilflll
Therefore

llflll=

for any t > O .

1.

Jf(g)ldg=O and f=O almost everywhere.

q.e.d.

Theorems 3.4 and 3.6 give a characterization of square integrable functions among the integrable functions on a compact group. Theorem 3.7. Let f be a function in L1(G). Then f belongs to L'(G) if and only if the Fourier transformf offbelongs to the space La(&)defined in Definition 4. Proof. Iff belongs to L3(G),thenf belongs to La(&)by Theorem 3.4. Conversely letfbe a function in L1(G)whose Fourier transformf belongs to La(&).Then by Thorem 3.4, there exists a function$ in L2(G)such thatfl=f. Since La(G)cL1(G),the uniqueness property of the Fourier transform on L1(G) (Theorem 3.6) proves that fi=f almost everywhere. q.e.d. Sof belongs to L2(G)and f=fl in L2(G).

At the end of this section, we prove the uniqueness of irreducible decomposition whose existence was proved in Theorem 3.1. Theorem 3.8. Let U be a unitary representation of a compact group

G and (3.35)

u= @ u= arA

be a decomposition of U into a direct sum of irreducible representations (Uu)..A and (3.36) H= @Ha IlrA

be the corresponding decomposition of the representation space H = H(U). Then we have the following:

30

FOURIER SERIES AND TORUS GROUP

1) Let

T

C? be the dual of G and put H(;c)=Uae2 @ P for each

1E C? and

L = { I E C ? ~ H ( R ) # { O Thenwehave }}. H=@H(I). If&

2) Let V be a closed subspace of H invariant under U and assume that the representation U vinduced on V by U is an irreducible one belonging to a class A E 6. Then V is contained in H(;c). Hence, H(1) is the closed subspace of H generated by closed subspaces of H on which U induces irreducible representations of class 1. In particular the space H(1) is independent of the decomposition (3.35). 3) If U = @ Va is another irreducible decomposition of U, then there BcB

exists a bijectionf from A onto B such that Ua is equivalent to VfCa)for every a E A. Proof. 1) is a direct consequence of the assumption (3.36). 2). Let P be the orthogonal projection onto V. Then P commutes with every U,(Proposition 2.3). The closed subspace PHa is a subspace of V # {O} and is invariant under U. Since U v is irreducible, PHa is equal to either V or {O} If PHa= V, then P I P is a non-zero intertwining operator between Ua and U pand hence is an isomorphism of Ha onto V(Proposition 2.2). In this case, Ua is equivalent to UP(Proposition2.6). Therefore if a # A, then PHa= {0} and V c H a * . Hence VcH(p)L-for all p # and ~ Vc n &$=H(R).

.

P*l

3) By 2), the subspace H(1) is uniquely determined and is independent of the decomposition (3.35). Therefore in order to prove 3), it suffices to show that if H(1)= @ H a = @ Ha a-rA(2)

BcB(l)

be the two irreducible decomposition of H(R), then the cardinal number m(1) of A(A) is equal to the cardinal number rn'(1) of B(1). But this is easily seen from the relations d(R)m(A)=dimH(A) =d(;c)rn'(,I) and 0 c

d(1)< cu .

Definition 7. The cardinal number m(1) introduced in the proof of Theorem 3.8 is called the muZtipZicity of the representation (class) 1 in U. We denote m(A)=(U: A ) .

&Fourier i. series of square integrable functions Now we shall return to the Problems A and A' posed in $1. The results in $2 and $3 enable us to answer these problems. The Peter-Weyl theorem

FOURIER SERIES OF SQUARE INTEGRABLE FUNCTIONS

31

(Theorem 3.3) shows that Problem A' is equivalent to the following Problem A". Problem A": Is each irreducible unitary representation of the torus " ~ some integer n? group T equivalent to a character ~ , , ( t ) = e
Theorem 4.1. Let x be an irreducible unitary representation of the group R of real numbers. Then x is equivalent to Xf(t)=etEt

for some real number c. xf is equivalent to x,, if and only if f = 7. Proof. Since the group R is commutative, an irreducible unitary representation x is one-dimensional (Corollary to Theorem 2.1). By replacing x with an equivalent one, we can assume that the representation space of x is C and that x is a continuous homomorphism of R into the multiplicative group U(1) of complex numbers with absolute value 1. First we show that x is a differentiable function. The homomorphic property of x is expressed by

d s + t ) = x(s)x(t> for all s and t in R. Integrating (4.1)with respect to t from 0 to h, we get

(4.1)

(44.2)

s;X(s+t)dt=X(s)

s:

x(t)dr.

Since x(O)= 1 and x is continuous

s'

(4.3)

x(t)dt# 0

for all h#O with suEciently small absolute value. On the other hand

(4.4)

Sh

x(s+

0

x+h

=J

t)df

x(u)du.

t

Since x is continuous, the right hand side of the last equality (4.4) is a differentiable function of s. So by (4.2), (4.3) and (4.4), we have proved that x is differentiable. Let ~'(0)= c. Then (4.1)yields

(4.5) The function x also satisfies

(4.5)'

x(O)= 1

.

The function xc(t)= ect also satisfies the differential equation (4.5)and the initial condition (4.5)'. So x coincides with zc, because the function p(s)=

32

FOURIER SERIES A N D TORUS GROUP

T

-

ce-cLx(s)f ce-ctx(s) =0 for all s E R and is thus equal to the constant p(O)=l. Moreover since x is a unitary representation ~ ( s ) s )= 1 for all s E R. Differentiating the last equation at s=O, we get c+E=O. Thus c is a purely imaginary number, and E =i-L is a real number. Hence we get ~ ( s )=eic* = x&). Since GL(1, C)=C* is commutative, x- is equivalent to xI,if and only if XC = x,. If xc = x?, then we get 6=71 by differentiating x&) = xq(t) at t =0.

e-"x(s) has the derivative p'(s) =

In the proof of Theorem 4.1, we have proved the following Corollary. COROLLARY to Theorem 4.1. If x is a continuous homomorphism of R into the multiplicative group C* = GL(1, C), then there exists a complex number c such that ~(s)= ec* for all s E R .

Theorem 4.2. Any irreducible unitary representation x of the torus group T =R/2nZ is equivalent to x,(t) =efnt for some integer n. x,, G x,,, if and only if n=m. Proof: As usual, we identify a function on T with a periodic function on R with the period 2a. Then x is identified with an irreducible unitary representation of the group R . By Theorem 4.1, x is equivalent to xc for 7 ~ )= 1 some real number E. Moreover since x F has the period 2a, ~ ~ ( 2=el"" and 6 must be equal to an integer n. The last part of Theorem is clear from Theorem 4.1. q.e.d Once the irreducible unitary representations of the group T are determined, the results in 53 can be automatically applied to the group T.By Theorem 4.2, the dual f of the group T is identified with the set Z of integers. Theorem 4.3.

1) The right regular representation R of the group

T = R/2nZ is decomposed thus: 2) The family b = { Xn In E Z } is a complete orthonormal family in Ls(T). 3) Any function f in La(T)can be expanded in its Fourier series which converges in the norm of La(T).Namely,

(4.6)

f = Cf(n)xn , nc2

4) The Parseval equality

where

FOURIER SERIES OF SQUARE INTEGRABLE FUNCTIONS

33

1

II~= I I ~ If(n>12 ~

(4.7)

neZ

holds for any functionfin La(T). Proof. 1) and 2) are direct consequence of Theorem 4.2 and Theorem 3.3 (Peter-Weyl theorem). 3) and 4) are equivalent to 2) (Corollary 1 of Theorem 3.3, cf. Proposition 1.1). Example.

1- n’1 = -.6 -

a2

n-1

This series whose sum was found by Euler is, in its essence, the Parseval equality for the periodic function f defined by f(t) = i when 0s t <2 ~ In . fact we have

and

SimilaiX;, the Parseval equality for the function f(t) = t a 4 J d s

Remark. The convergence of Fourier series (4.6) is the convergence in the norm of L2(T).Namely (4.6) means

However L. Carleson [l] proved in 1966 that the Fourier series of any function in La(T)converges almost everywhere. The proof of Carleson’s theorem is too complicated to be given here. In the next section, we shall see that Fourier series of C1-functions converge absolutely and uniformly. In general, the pointwise convergence of Fourier series is a delicate problem. For example, in 1926, A. Kolmogoroff showed that there exists an integrable function on T whose Fourier series diverges everywhere (cf. Zygmund [l] pp. 310-314). Let L 2(Z )be the Hilbert space of all complex-valued functions a on 2 satisfying

1la(n)la< + m . U.d

34

FOURIER SERIES AND TORUS GROUP

T

The inner product on L 2 ( Z )is defined by (a, b) =

c

a(n)b(n) .

ntZ

Theorem 4.4. The Fourier transform 9:f w f induces an isometry of La(T) onto La(Z).A function f in L1(T)belongs to L'(T) if and only if the Fourier transformf belongs to La(Z). Proof. Theorem 4.4 combines particular cases of Theorem 3.4 and q.e.d. Theorem 3.7.

By the Parseval equality (Theorem 4.3, 4)). Iim f ( n )=o (4.8) Inl-+-

for all functions f in La(T).The equality (4.8) is valid for any function f in L1(T).

Theorem 4.5 (Riemann-Lebesgue). we have 1)

If(n)l

s llflll

Let f be a function in L1(T).Then

for all n E 2 and lim f ( n ) =o .

1nI-m

Proof.

1) Since l e t n t l = l ,

we have

2) For any fcL1(T) and E > 0, th g e exists a continuous function g such that Ilf-glll<s (cf. Appendix D). Then there exists a trigonometrical polynomial P such that ((g- P ( ( , < E by the uniform approximation ~ llg-Plll<E. l ~ ~ ~Letgr be ~ theorem (Theorem 3.5). Since ~ ~ g ~we get the degree of P. Then P can be written as P ( t ) = Z, c,etmt. ImlSr

Hence, if In1 > r, then we have I'(n) = (P,x.) =0. Therefore by the result of l), we have

I f W l = ICf-P)^(4lS Ilf-Pll1S Ilf-gll1+ Ilg-PII1<

26

if

I4 >r.

We have proved that lim f ( n )=O. l.l-+-

95. Fourier series of smooth functions and distributions The theory of Fourier series of square integrable functions discussed in

~

~

,

FOURIER SERIES OF SMOOTH FUNCTIONS AND DISTRIBUnONS

35

$4is a particular case of the general theory for compact groups. But some of the results for the ordinary Fourier series can not be generalized to an arbitrary compact group. In this section, we discuss the properties related to the differentiable structure of the torus group T . The results of this section can be generalized to every compact connected Lie group (cf. Ch. 11, especially $8). Let p be the canonical projection of R onto T = R/2nZ which maps a real number t to the coset t + 2 r Z containing t. A function f on T is called dtyerentiable if the function fop on R is differentiable in the ordinary sense. Let k be an integer >0 or m . Then a function f on T is called of class C kor a Ck-functionif the function f o p has continuous derivatives of degree up to k. The set of Ck-functions on T is denoted by C k ( T ) . As before, we identify a function f on T with the function f op on R . The derivative and k-th derivative off is denoted by f ’ and f A function f on T is called a piecewise Cl-function if there exists a finite set A = {al, ..., an}in T such that f is a C1-function on T - A and there exist finite limits lim f(x) and lim f ’ ( x ) for each point akcA.

+

o*ak*0

Z-akfO

Theorem 5.1. Iff is a continuous piecewise C1-function on T , then the Fourier coefficients off andf’ satisfy (5.1) (f’)”(n)=infl(n) for all n E Z and (5.2)

f (n)=o( $)

The Fourier series off converges to f absolutely and uniformly on T . The function f is uniquely determined by the Fourier transform f ’ . ProoJ By integration by parts, we get

=inf

(n)

because f(O)=f(2r). Since f’E LZ(T), the Parseval equality (4.7) (or Theorem 4.5) implies lim (f’)^(n)=O. Inl-+-

The last equality and (5.1) prove (5.2). By the relation (5.1), the Schwarz and I, the Parseval equality (4.7), inequality in L 2 ( Z ) ,the Example in & we get

36

FOURIER SERIES AND TORUS GROUP

~

~

45

~

~

+f

00

T

for ’ all ~ t~E T,.

<

Hence the Fourier series off converges absolutely. Since the right hand side is independent of t, the series converges uniformly on T. Therefore the sum fi of the series is a continuous function on T. Since Ilplla d Ilpl.. for all Q E C(T), the Fourier series off converges tofi in L’(T). On the other hand it converges to f in La(T) by Theorem 4.3. Therefore f ( t ) = fi(t) for almost every t E T. Since f and fi are continuous,f(t)=fi(t) for all r E T, because if the open set E= { t a Tlf(t)#;fi(t)} is not empty, its measure must be positive. Iff=O, then f=fi=O. Thus f is uniquely determined .by! q.e.d.

COROLLARY to Theorem 5.1. Let k be an integer >O and f be a Ckfunction on T. Then we have (5.3) ~f‘k”)*(n) =(inyif(n) for au n E and

z

(5.4)

(I++

i(n)=o(&)

a).

Proof. Applying Theorem 5.1 k-times, we get (5.3). (5.4) is a direct consequence of (5.3) and the Parseval equality (or Riemann-Lebesgue theorem).

is called a rapidly decreasing function on Z . Theorem 5.2. Letf be a continuous function on T.Thenf is a C‘-function on T if and only if the Fourier transformf is a rapidly decreasing function on Z. Proof. Iff is a C”-function on T,thenf is a rapidly decreasing function on 2 by the Corollary to Theorem 5.1. Conversely let f be a continuous function on T whose Fourier transformf is rapidly decreasing. Then the Fourier series (5.5) nrZ

off converges absolutely and uniformly on T,becausef satisfies

FOURIER SERIES OF SMOOTH FUNCTIONS AND DISTRIBUTIONS

37

(cf. Example in $4). The sum f 1 of (5.5) is continuous and equal to f almost everywhere and hence everywhere (cf. the proof of Theorem 5.1). The series obtained by differentiating (5.5) term by term is

1in{(n)e(nc

(5.6)

.

ne.3

Since f satisfies {(n) =o (:,a)

,

(5.6) converges absolutely and uniformly

on T . Therefore f is differentiable and f ' ( t ) is given by (5.6). Since f is rapidly decreasing, this process can be repeated an arbitrary number of times. We have proved that f is a Ck-function for any k e N,namely, f is a C"-function. COROLLARY to Theorem 5.2. Let f be a function in L1(T).Then the Fourier transformf off is rapidly decreasing if and only iff coincides with a C"-function5 almost everywhere. Proof. Iff coincides a C-functionfi almost everywhere{={; is rapidly decreasing. Conversely iff is a function in L1(T)of which Fourier transform{ is rapidly decreasing, then the Fourier series (5.5) off converges uniformly to a C"-functionfi, as the proof of Theorem 5.2 shows. Then fcoincides withfi almost everywhere (cf. Proof of Theorem 5.1). q.e.d. Definition 2. For each natural number k E N = {O,1,2, ...) , a seminorm on C"(T) is defined by

Px(S)=SuP If'"(t)l tcT

*

The family of seminorms {pklkE N ) defines a topology-on C"(T). C"(T) is a topological vector space when supplied with this topology. Any finite set { k l , ..., k,) of natural numbers and any finite set { E X , . ., E,,) of positive numbers define a set U(kl, ..., kn;El, ..., en)= { f E C " ( T ) IpPl(f)dEt ( l s i s n ) } The family of all such sets U(kl, ..., k,; el, ..., en) is a basis of neighborhood of 0 in C"(T). Then C"(T) is a complete, metrizable, locally convex topological vector space. In what follows, we denote this topological vector space by S ( T ) .

.

Definition 3. For each natural number k E N,a seminorm qk on the space 9 ( Z ) is defined by

4&)

= SUP Inkdn)l nc.3

-

By the countable family of seminorms (q& E N ), the structure of a com-

3%

FOURIER SERIFS AND TORUS GROUP

T

plete, metrizable, locally convex topological vector space is defined on ~ ( 2 From ) . now on, the space Y ( Z ) is always assumed to be given this topology. Theorem 5.3. The Fourier transform F :f wfis an isomorphism of topological linear spaces from g ( T ) onto Y ( Z ) . is a linear mapping from g ( T ) onto Proof. The Fourier transform 9~ ( 2(Theorem ) 5.2). The mapping 9- is injective by the uniqueness of Fourier series (Theorem 5.1). By Theorem 4.5 and the Corollary to Theorem 5.1, the inequality

1= SUP I nLP(n)I = SUP I (f'")"(n) I s llf(k)llls llf(k)ll=Pk(f) holds for any f E g ( T ) and k E N. So the mapping F 44f

is a continuous mapping of g ( T ) onto 9'(Z). On the other hand, the Corollary to Theorem 5.1 shows that

for all f e g ( T ) and k > 0, because (f(lC))"(O)= 0. If k =0, we have

Therefore r-lis continuous.

q.e.d.

Definition 4. A continuous linear form on d ( T ) is called a distribution on T. The set of distributions on T is denoted by g ' ( T ) . g ' ( T ) is a vector space with the ordinary operations. The value Tdf) of a distribution T a t an element f in g ( T ) is denoted by (f, T). The mapping (f, T ) H (f, T) is a bilinear form on s ( T ) x s ' ( T ) . Example 1. An integrable function 'p E L'(T) is identified with a distribution by putting

FOURIER SERIES OF SMOOTH FUNCTIONS AND DISTRIBUTIONS

39

Example 2. Let a be a point in T. Then the linear form 6, on 9 ( T ) defined by

=f(a) is a distribution called the Dirac measure at a. Example 3. Let T be a distribution on T. Then the linear form T’ on d ( T ) defined by (f, T’)= - (f’, T> is a distribution called the derivative of T. If Tis a C1-function, the distributional derivative T’ is equal to the ordinary derivative of the function T as is easily seen by integration by parts. The derivative 60’ of a Dirac measure is equal to a linear form (f, 6,)

f - (f’,6,)

= -f’(a).

Now we shall introduce the notion of Fourier transform of a distribution and give a characterization of distributions by their Fourier transforms. Since g(T)ngP(Z), there exists a natural isomorphism between @(T) and gP’(2) (the space of continuous linear forms on 9 ( Z ) ) . So we give first a characterization of . y ’ ( Z ) . Let ‘pn be the element of 9 ( Z ) defined by

.

for each m E Z Then any function ID in Y ( Z )may be expanded in terms of {pn}nsz: ’pn(m)=,6

(5.7)

The series in the last equality converges to space 9 ( Z ) . For,

‘p

in the topological vector

and thus

for any k E N , because ‘p is rapidly decreasing. Let S be an element in .-V’(Z) and put S(n)= ((on,S ) for any n E 2. Then by (5.7) and the continuity of S, we have (‘pY

S> =

c

‘p(n>S(n>*

r.rZ

An element S in .-V’(Z)is uniquely determined by and identified with the

40

FOURIER SERIFS AND TORUS GROUP

T

function on 2 defined by n ~ S ( n )As . a function on 2, an element in 9 ’ ( Z )is characterized as follows. Definition 5. A complex-valued function S on 2 is said to be tempered or slowly increasing if there exists a constant ChO and a natural number k2O such that the inequality IS(4lS Clnlk

(5.8)

holds for all n E 2. (Both C and k depend on S.) Theorem 5.4. A complex-valued function S on 2 belongs to the dual space q ’ ( Z )of g ( Z ) if and only if S is tempered. Let En be the element in 9 ’ ( 2 )defined by (5.9) (9,En)= ( D ( I Z ) for all n E 2 and p E 9 ( Z ) . Then any element S in 9 ’ ( Z )can be expanded in the form (5.10)

S=

c

S(n)En

n.2

in the weak*-topology (cf. Appendix F) on 9 ’ ( 2 ) . Proof. Let S be an element in y’(2).Since the mapping (5.11) is a continuous linear form on 9 ( Z ) ,there exists a finite number of positive constants cl, ..., cz and natural numbers kl, ..., kl such that the inequality 1

I(p,

1ciqr,(p)

S>ls

$=1

holds for all Y E g(2).Let k be Max{kl, ..., k z } .Then we have @i(Pn)= In ktI = < In Ik for all n~ 2 and all iE {1,2, ..., I} and 1

1

ci >0.

for all n E 2, where c=

We have proved that S is tempered.

t=1

Conversely, let S be a tempered function on 2. Then S defines a continuous linear form by (5.11). For, we have

I<$% s>lr

c

nrZ

Ip(n)S(n>lScCIn“W nrZ

FOURIER SERIES OF SMOOTH FUNCTIONS AND DISTRIBUTIONS

dCqo(P0)+

41

IT% p + 2 ( ( 0 )

*

(We can omit the &st term of the last inequality if kL1.) The series Cp(n)S(n) is absolutely convergent by the latter inequalities. For any E>O, there exists an integer m such that

This inequality implies that I
1 S(n)En)I
lnlsm

We have proved (5.9).

q.e.d.

Definition 6. The Fourier transform f' of a distribution T o n T is the element in .9"(Z) defined by (5.12)

(9, f')= (9,T ) for all 9 E g ( T ) .

Theorem 5.5. The Fourier transform fl': T+f' of distributions on T is a topological isomorphism of g ' ( T ) onto .9"(Z) with respect to the weak*-topologies. A function S on Z is the Fourier transform f' of a distribution T on T if and only if S is tempered. Every distribution T can be expanded in a Fourier series

(5.13)

T=

1f'(-n)xn, nrz

where XR(t)=efnc.The series in (5.13) converges in the weak*-topology of g ' ( T ) . Proof. Since the Fourier transform 5 : (0-9 of C"-functions is a topological isomorphism of g ( T ) onto g ( Z ) , the Fourier transform fl': Twf' defined by (5.12) is an isomorphism of topological vector spaces from g ' ( T ) onto y ' ( T ) . g ' ( T ) and g ' ( T )are given their weak*topologies as dual spaces. As a function on 2, f' is defmed by

f'(n)= ((on, f) for all n E z . Since the Fourier transform f l z n = f n of the C"-function fn=pn forall ~ E Z ,

(5.14)

(5.14) can be written as (5.15)

f'(n)=

( ~ n 7

T)

Xn

is equal to

FOURIER SERIES AND TORUS GROUP

42

T

On the other hand, the Fourier transform F'x-,, of a distribution x-" is equal to En (defined by (5.9)) because

(4,f-n) The element (5.16)

=(

~ 3X - n )

= @(n) = (4, En)*

f in Y ( Z ) can be expressed as f = f(n)E.

C

ncz

in the weak*-topology of Y ( Z ) (Theorem 5.4). Applying the inverse mapping F ' - l of the Fourier transform Y' to both sides of (5.16), we get

The series converges in the weak*-topology of B'(T). Example 4. Let T = 6, (the Dirac measure at 0). Then the Fourier expansion (5.13) of &, is given by,

because 6, (-n) = (x-,,, 6,) = x-. (0) = 1 for all n e Z . In Theorem 5.1. we assumed that the function f is continuous as well as piecewise of class C' on T.Now we shall drop the continuity assumption and prove that the Fourier series of any piecewise C1-functionf is everywhere convergent and that its sum at x is equal to 2-'lf(x 0) f(x - O)] (Theorem 5.6). First we give an example of a piecewise C'-function which together with its Fourier series, will be used in the proof of Theorem 5.6. Examples. We start from the principal value Log z of the natural logarithm of a complex variable z. Log z is defined by

+ +

L o g z = l o g lzl +iargz, - - a < a r g z < a The function w = Log z is a branch of the inverse function of the exponential function z = ew. Hence Log z is holomorphic on the domain E = C - { x E R l x S 0)and has the derivative

The following equality is easily verified by differentiating both sides : (5.17) Log(1

- (+ Z) = 1 n-1

,>.-I

z " . f o r a l l z E D = { z E C I IzI < l}.

FOURIER SERIES OF SMOOTH FUNCTIONS AND DISTRIBUTIONS

The real and imaginary parts of (5.17) for z following two series: 1 -log(l 2

(5.18)

- (+ 2rcos8 + r2) = 1 "-1

4=

(5.19)

4

5 r < 1) give the

I).-1 rncos no, and n

(- I).-I r"sin n8, n-1

where

= reie (0

43

n

denotes the angle indicated in Fig. 1.

~~

-%

U

Fig. 1

On the circle U = ( z E CI I z I = 1}, the series (5.19) turns out to be < n) and integrate both sides of the equality

of interest. Put z = de (I 8 I

from 0 to z = eie along the radius C = reie (0 5 r 5 1, 8 = fixed). Then we have Log (1

+ Z) = z - 2 + . . . + (22

l)"-'>-'

+ R&),

dC. By Fig.1 and the law of sine, we have

where R,(z) =

+ C l = IsinOI- Isin41 '1 2 rlsine1. In particular, we get 11 + clz 1 if 181 4 2 . Hence we have 11

The above inequalities show that the remainder term R, (z) tends to zero when n tends to +m. Therefore the equality (5.17) holds for every point z on the unit circle U except at the point z = - 1. If z belongs to U then

44

FOURIER SERIES AND TORUS GROUP

T

r = 1 and the angle 4 in Fig.1 is equal to 0/2, and so the equality (5.19) becomes (5.20)

When 0 = fa, the right side of (5.20) is equal to 0. Put (5.21)

Then g(0) is a periodic function with the period 2a and (5.22)

g (e) = g (e

+ 2nn) =

I?

--Ic<~
if if

e=fa

for any integer n. The graph of the function g is shown in Fig. 2 by solid lines. A

__

2

Fig. 2

At the point of discontinuity x = nx, g(na) = 0 and we get

(5.23)

+ 0) + g(nz - O)},

1 g(nx) = -{g(nx

2

n

E 2.

Putting 8 = a - t in (5.21), we get the function h (t) = g (x - t). h (t) is expressed as (5.24)

and (5.25)

LO, At the point of discontinuity t (5.26)

t = 2nx for n E Z. = 0,

2, x h ( + 0) = -

h satisfies the equality h(-O)=--

A

2'

FOURIER SERIES OF SMOOTH FUNCTIONS AND DISTRIBUTIONS

45

The graph of the function h is shown in Fig. 2 by dotted lines. Put 1

k (0 = $ g ( t )

+ h (t>l.

Then we have

(5.27) The period of k (t) is equal to n and by (5.22) and (5.25) we get

i" 4'

(5.28)

k(r)=/O,

O
t=nn(nEZ)

I-:

-n
Using the series (5.24)' we can prove the following Theorem 5.6. Let f be a piecewise C'-function on the torus group

T = R / 2 n Z . Then the Fourier series o f f converges everywhere and

+ +

its sum at t is equal to L{f (t 0) f (t - 0)). In particular the sum is 2 equal to f(t) iff is continuous at t . Proof. Let t l , . . . ,t, be the points of discontinuity off on T. Put Ck=f(fk+o)-f(fk-o),

1s k d m ,

and

where h(t) =

1n-I sin nt. Then 4 is a piecewise C1-functionand continun-1

ous on T.Clearly 4 is continuous at t # tk and by (5.26),

46

FOURIER SERIES AND TORUS GROUP

T

Since 4 satisfies the conditions of Theorem 5.1, the Fourier series of 4 converges absolutely and uniformly on T,and its sum at t is equal to f i t ) . 1 Since the Fourier series of h converges at t to -{h(t

2

the Fourier series off converges at t to $-{f(t

+ 0) + h(t - 0)},

0) + f ( t

- o)}.

CHAPTER II

Representations of SU(2) and SO(3)

In this chapter, we shall construct the irreducible representations of two non-commutative compact Lie groups SU(2) and SO(3) and study the Fourier series of smooth functions on them.

$1. Construction of irreducible representations of SU(2) The group SU(2) (the special unitary group of degree 2) consists of all matrices g of degree 2 satisfying g*g=l and d e t g = l .

As a matrix group of degree 2, SU(2) acts on the vector space Ca= ((zl,Z a ) l Z i , Za E C } of all row vectors. SU(2) acts on the right. Namely, the action of an element g = (1.1)

(zi)

E SU(2) is given by

z = ( z ~za)Hzg=(azi , +cza, bzi +dz&

and the action satisfies (1.2) 4giga) =(zgilga (1.3) 21 = z for all z E Ca and g1, g, E SU(2). The action of SU(2) on Ca may be naturally transferred to suitable function spaces. Let E be a vector space of functions on Cahaving the property that if a function f belongs to E, the function z~ f ( z g ) belongs to E for every g E SU(2). Then we can define a representation T of SU(2) on E by

V,f) (4= f W . Proposition 1.1. Let V,, be the subspace of homogeneous polynomials of degree n of the polynomial algebra C[zl,za]. Then the dimension of V, is n+ 1. A continuous representation U" of SU(2) on V,, is defined by (1.4) (Ug"P)(Z>= P W .

47

48

REPRESENTATIONS OF

su(2)AND

Proof. Since any polynomialfin V, can be written uniquely as a linear combination of n + 1 monomials

(05 k 5 n), dimV,, is equal to n + l . If (O belongs to Vn, then Ug"p belongs to V,, because the action (1.1) of g E SU(2) is a homogeneous linear transformation of the variables z1 and za. Un is a homomorphism of SU(2) into GL(Vn)by (1.2) and (1.3). So (1.4) defines a representation U" of SU(2) on V,. The representation Un is continuous because the matrix elements of U," with respect to the basis ( $ 7 k ) O s k s n are polynomials in the matrix elements a, b, c and d of g. Remark 1. Actually, the formula (1.4) defines a continuous representation on V, of the group GL(2, C). (1.5)

Proposition 1.2.

yDk(z1,zq)

=Z1kZan-k

Let (9,$) be the inner product on Vn defined by

Then the representation Un of SU(2) is a unitary representation on V,,. Proof. The space Caof all column vectors

is the dual space of Ca under the pairing given by the bilinear form ( z , a ) =za, for z E C a and a E C'. For any column vector U E Ca, let (pa be the polynomial in Vn defined by ya (z)= (ZU)".

By the definition (1.6) of the inner product, we have ((pa,( O a ) = f k(n-k)!( !

a

~ ) a , L a a n-- % -~ k b ~ n - i

k=O

k-0

=n ! ( ~ 1 3 1 +aaFa). =IZ ! (a, b)"

where (a, b) is the canonical hermitian inner product in Ca. Since g E SU(2) is a unitary matrix it satisfies (ga, gb)=(a, b) for any a and b in Ca. (1.8) On the other hand,

CONSTRUCTION OF IRREDUCIBLE REPRESENTATIONS OF

(1.9) (Upnpa) ( ~ > = p a ( z g ) = ( ( z g ) a > ~ The equalities (1.7), (1.8) and (1.9) prove that

su(2)

49

=pga(z).

(1.10) (Ugn9a7 ugnpJ= (cpa@b) for any g E SU(2) and a, b E Ca. It remains to prove that the set A={pa

I

aECa} ¶xi

contains a base of the space V,. Let o=e”be unity. Then the following n + 1 polynomials

a primitive n-th root of

+

(1.11) (zl okz2)n, (05 k 5 n - 1) and Za* in the set A are linearly independent. To prove the last fact, it is sufficient to show that the determinant D whose k-th row consists of the coefficients of the k-th element in (1.11) is not equal to zero. If k d n , the (k+ 1)-th row of D is

The last row of D is 0,o ,..., 0,l. Expanding D by the last row and dividing the k-th column by get the Vandermonde determinant of the quantities (0, oa,..., 0 - l ) . Thus we have proved

D=hl(Z) k=l

n

(ot-ok)#O

OSk
because wk# w1 for any k and 1 satisfying 0 5 k < 15 n - 1.

q.e.d.

Theorem 1.1. For any non-negative integer n E N, the unitary representation Un of SU(2) constructed in Proposition 1.1 is irreducible. Proof. By Schur’s Lemma (Ch. I, Proposition 2.4 or Theorem 2.1), it is sufficient to prove that if a linear transformation A on the space V, commutes with Ugnfor every g E SU(2), then A is a scalar operator cl. Let a E U(1) be a complex number with the absolute value 1. Then

is an element in SU(2). If ( ( o k ) o s k s n is the basis of V,,definedin(1.5), then (1.12)

U;ca, p k = ask-"

IDk

50

REPRESENTATIONS OF

Su(2) AND SO(3)

and (1.13)

UE,,,A ya=AUt(,, yk=Uak-fAyr

for any a E U(1). On the other hand it is easily seen that an element y in V,, is proportional to (Dk if and only if U;(,)~ = C L ~ ~ - for ~ ( Devery U E U(1). The equality (1.13) proves that Avk=Ck(Dk, ck E c, for any kE [O, 1,2,

..., n}. Put cost

-sin t

)

fortER.

Then we have (1.14)

AU&

(on

=A(zl cos t +zs sin t)" =

i(l) i();

coskt sinn-ktApr

k=o

=

COSktSinn-kt c k v k

t=O

for all t E R. Choosing t such that cos t sin t # O and comparing (1.14) with (1.15), we get Cx=C,,, O S k S n . q.e.d. We have proved that A is a scalar operator. Remark 2. It will be proved in $4 that the irreducible unitary representations of SU(2) are exhausted by the family { U"ln E N}. Namely any irreducible unitary representation of SU(2) is equivalent to U" for some n E N.

$2. Characters of compact groups To prove a theorem announced at the end of the last section, we shall use the theory of characters of compact groups which is developed in this section.

CHARACTERS OF COMPACT GROUPS

51

Definition 1. Let U be a finite-dimensional representation of a group G.Then the function XU on G defined by xuk) =Tr

uil

is called the character of U.

Proposition 2.1. The characters of representations of G have the following properties: 1) If U s V, then x u = x v ; 2) xu(ghg-9 = xu(h); 3) X U ~ ~ = X U + Xand ~ ; 4) xu(e)=degU where e is the identity in G. Proof. These properties are direct consequences of the corresponding properties of the trace: Tr(AI3A-l) =TrB, Tr(A@B) =TrA +TrB, Tr( 1n) =n.

Proposition 2.2 (Orthogonality relations of characters). Let U and V be two irreducible unitary representations of a compact group G. Then the inner product of their characters in L2(G)is given by 1 if U Z V (xu, xv) = 0 if U g V .

{

Proof. By Proposition 2.1, l), we can assume that Uand Vare unitary matricial representations. By the orthogonality relations for the matrix elements (Ch. I, Theorem 3.2), we have (xu, x v ) = O if U g V. If U z V, then we have xu=xv, by Proposition 2.1, 1). Again by the orthogonality relations, we get

Proposition 2.3. Let G be a compact group. 1) A finite dimensional unitary representation U of G is irreducible if and only if (xu, xu) = 1.

2) Two finite dimensional unitary representations U and V of G are equivalent if and only if X U = ~ V . Proof. The "only if" parts of both assertions have been proved in Propositions 2.2 and 2.1. Conversely any finite-dimensional unitary repre-

52

REPRESENTATIONS OF

Su(2) AND s0(3)

sentation U of G may be decomposed as a direct sum of irreducible representations (Ch. I, Proposition 2.1) : Let n

U=@ miUr i-1

be the irreducible decomposition of U,where we can assume that

U&U,

if i+j

and

mizl

for all i.

Then by Proposition 2.1, 3), we have n

The orthogonality relations for characters (Proposition 2.2) show that (2.2) and

mr =(xu,

xu3 n

(2.3)

(xu, xu) = Em? (-1

If xu satisfies (xu, xu)= 1, then the equality (2.3) proves that n = 1, m l = 1 and U = Ul is irreducible. If xU=xV, then we have (XU, XUi) = (xv, for any irreducible representation Ui. The equality (2.2) proves that the multiplicity mi of Usin U is equal to the multiplicity of Ui in V. We have proved that n

U z @ rntUtE V . i-1

q.e.d. Now we compute the character of the irreducible representation U" constructed in $1. First we observe the following facts. Proposition 2.4.

1) Any element g in SU(2) is conjugate to an element

2) A functionfon SU(2) satisfying f(uhu-l)=f(h) for all u and h in SU(2) is completely determined by its restriction to the subgroup H = (h(B)(BE

53

CHARACTERS OF COMPACT GROUPS

[O, 2all. The restrictionf(0) =f(h(e)) is an even function of e. Pro05 1) A unitary matrix g is transformed to a diagonal matrix h by a unitary matrix u :g=uhu-l. If g belongs to SU(2), h also belongs to SU(2) and is equal to h(0) for some 0 E [0,2x].The unitary matrix u can be taken from SU(2), because we can replace u by i u (2 E U(1)). 2) The first half of 2) is a direct consequence of 1). The last half is based on the equality 1 ere 0 0 -1 0) -o)(o e-ib)(l

(Y

=(:"

i).

q.e.d. Remark 1. The subgroup H= {h(O) I e E [0,27c]] is isomorphic to the torus group T = R/2aZ by the isomorphism Owh(0). The centralizer C of H in SU(2) is equal to H itself, namely if an element g E SU(2) commutes with every element of H, then g belongs to H. For, the equality

(" w' ) ( a0 2

O

a-'

)=("0 O )(" w') 4-1

2

'

implies that ay =u-'y, az =a-l',

so that y=z=O if a2#l.

Therefore the subgroup H is a maximal abelian subgroup of SU(2). In particular, if H' is a subgroup of SU(2) satisfying H c H and H s T n for some n r 1 , then H = H . H is, for this reason, called a maximal torus in SU(2). Another maximal torus of SU(2) is given by

Two maximal tori H and K are conjugate. More generally, any maximal torus A of SU(2) is a maximal commutative family of unitary matrices which can be transformed to diagonal matrices simultaneously. This means that the group A is conjugate to H in SU(2).

Proposition 2.5. Let n be a non-negative integer and zn be the character of the representation Un of SU(2) constructed in $1. Then we have

54

SU(2) AND SO(3)

REPRESENTATIONS OF

for every integer my and xn(l)= n + 1, Xn( - 1)= ((2.5)

l)n(n+

1).

If the eigen values of g E SU(2) are e r g and e-f@y then xn(g) is given by (2.4) or (2.5) according to whether e+ma for every integer m or 8=mx for some integer m. Proof. Putting a=eieYfrom the equation (1.12), we get

if aa# 1. We get (2.4). The first equality in (2.5) is immediate by Proposition 2.1, 4) and Proposition 1.1. The second equality is clear from (1.12) for the case a = - 1. The last part of Proposition 2.5 follows from Proposition 2.4. Remark 2. The values of xn at & l are the limits of zn(h(0)) when 0 tends to 0 and 7c. x n ( ~ ) = ~ n ( h ( 0 )is) a continuous function on the torus T=RI27cZ=H.

$3. Haar measures on SU(2) In order to apply the results in $2 to our group SU(2), we must know the normalized Haar measure of G = S U ( 2 ) for computing the inner product in La(G).For this purpose we use the fact that SU(2) is homeomorphic to the three-dimensional sphere S3. Proposition 3.1. Let S3 be the unit sphere in R4:

Then the mapping

is a homeomorphism of S8 onto SU(2). Proof. It is easily verified that the inverse matrix of a unimodular matrix g=(:

j)

ad-bc=1,

is given by d g-l=(-c

-b

a).

HAAR MEASURES ON

su(2)

55

Therefore we have the equivalences,

g=(:

f;)

E SU(2)=g*=g-l,

det g = l

-

e d = Z , b=-cand I a l a + I c l a = l . We have proved that @(x)belongs to SU(2) for any x E S3 and that any element g in SU(2) may be written uniquely as g = @ ( x ) ( X E S3). 0 is a bijection of S3 onto SU(2). It is clear that @ is continuous. Since the sphere S3 is compact and SU(2) is a Hausdorff space, the inverse mapping 0 - 1 is also continuous. q.e.d. Remark. Identifying R4 with the field of quaternions H by the mapping ( x ~ , x ~ , x ~ , x+~xai+ ) H xx~j + x4k, the above mapping @ is an isomorphism from the multiplicative group Sp(1) of quaternions with norm 1 onto SU(2). @(x) is the matrix of right translation R,:q~-+qxwith respect to the C-basis {l,j} of H . From now on we identify SU(2) with S8 by the mapping 0-l.

Proposition 3.2. The right translation R, : xHxg (x, g E SU(2)) of SU(2) is a rotation (an element of SO(4)) of S 3 (identified with SU(2)). Pro05 The mapping 0 of S 3 onto SU(2) can be extended to a linear isomorphism of R4 onto the vector space

1

xl+ixa, -x3+ix4> X I - ixa V = ( x 3+ ixr, The extended mapping is defined again by (3.1) and is also denoted by 0.Identifying x E R4 with @(x)E V, we get (3.2) [Ix [la=detx. The right translation R, on SU(2) is extended to a linear transformation on the space V = R' by defining R,x= xg. Then we have ]I xg IJa= det(xg) =det x = [I x [Ia

{

for all x E R4 and g E SU(2). Therefore the right translation R, is an orthogonal transformation of R' :R, E O(4). Since SU(2), which is homeomorphic to S3, is connected, the orthogonal transformation R, q.e.d. has the determinant 1 :R,E SO(4). As a hypersurface in R 4 , S3 has a Riemannian metric induced by the Euclidean metric of R 4 . In order to express the metric explicitly, we introduce 3-dimensional polar coordinates on S3.

56

REPRESENTATIONS OF

su(2)AND s 0 ( 3 )

Let x=(xl, xl , x8, x 3 be.a point on S8: XI' xS' xa' ~4~ = 1.

+ + +

Since

- 1g x l g 1, there exists a unique real number 8 such that

x1=cos e, (05e 5 n). (3.3) Then (xl, Xa, x4) satisfies Xaa + xsa x 2 = sinze. Namely (x,, XI, A) is a point on a 2-dimensional sphere with radius sin 8. So (x,, x8, x,) can be written as x, = sin 0 cos p, xs = sin @ sin p cos G, ( O g p ~ n 0, 5 4 5 2 n ) (3.4) x4 = sin e sin p sin 50. (3.3) and (3.4) give the polar coordinates of a point X E Ss.The polar coordinates are local coordinates on Sa. They are not defined for all points of S8. Note that 9 and (I cannot be determined if e =0 or 7r and qj can not be determined if p=O or R. However these exceptional points constitute a 1-dimensionalsubmanifold {(cos17, &sin 8, O,O)lOr@ 4 x 3 and its measure is equal to 0. So we can neglect the exceptional points in the calculation of the Haar measure.

+

Proposition 3.3. Identify SU(2) with S8 and supply the latter with the polar coordinates given by (3.3) and (3.4). Then the normalized Haar integral of G = S U ( 2 ) is given by

(3.5)

1

f(x)dx=&/l/:[>(e,

( ~ , ( ~ ) s isinpdedpdqj. n~e

Proof. Since the Riemannian metric of S8 is induced by the Euclidean metric of R4,its line element is given by

+sinzed$+ sina@sin'p

=dea

d#'. Therefore the volume element dx= (det(g<,))*d8dPd+ is given by dx= sinat?sin (D dB dp d$. (3.6) The volume element dx is invariant under rotations, hence dx is a Haar measure of G. Since the total volume of G with respect to (3.6) is equal to

1:1:1:

sinV sin p de dp d@=2na,

ENUMERATION OF IRREDUCIBLE REPRESENTATIONS

the normalized Haar integral is given by (3.5).

57

q.e.d.

Definition 1. A function f on a group G is called a class function or a centralfunction if it satisfies f(ghg-') =f(h) for any g and h in G. For the class functions f on SU(2), the Haar integral (3.5) may be simplified as follows.

Proposition 3.4. I f f is an integrable class function on SU(2), then the normalized Haar integral off is given by (3.7) wheref(0) =f(h(O)). Proof: Let x be an element in SU(2). Then the characteristic equation of the matrix x is ta-(Tr x)t+ 1 =0, because det x = 1. If (e, p, 4) are the polar coordinates of x, then the above equation becomes ~ - 2 cos t e+l=o. The eigen-values of x are ete and e-ra.By Proposition 2.4, there exists an element u E SU(2) such that x = uh(e)u-'. Therefore a class functionf satisfies f(x)=f(h(e))=f@, o,o), and is independent of the variables p and (I.Applying the general formula (3.5) to a class functionfwe get (3.7). Remark. As an application of Proposition 3.4, we give another proof of the irreducibility of Un (Theorem 1.1). By Propositions 2.5 and 3.4, we get

's:

(zn,Xn)= ;sina(n+1) e d e =1

n

S n ( 1 -cos 2(n+ 1)e)de = 1. o

By Proposition 2.3, we have proved the irreducibility of U". $4. Enumeration of irreducible representations After the preparations in 992 and 3, we now give a proof of the theorem announced at the end of $1 :

58

REPRESENTATIONSOF

SU(2) AND SO(3)

Theorem 4.1. Any irreducible unitary representation U of SU(2) is equivalent to one of the representations Un constructed in $1. Proof. An irreducible unitary representation U of a compact group SU(2) is finite-dimensional (Ch. I, Theorem 3.1). Let x be the character of U and put ~ ( 8=) x(e) sin 8,

where x(e)= x(h(0)). Since x is a class function, ~ ( 8 is) an even function and .$ is an odd function of 8. To distinguish the inner product in La(H)from the inner product in La(G),we denote the former by cf, g ) H . Then we have

=“I n

x(e) Xn-l

(8) sina@ d8

o

=z(x, xn-l) 2

for all nh 1

by Propositions 2.5 and 3.4. Since E is an odd function, we have (E, l)x=O. (4.2) By (4.1) and (4.2), the Parseval equality for the group H z T (Ch. I, Theorem 4.3) yields (4.3) On the other hand, the Haar integral formula for class functions on SU(2) (Proposition 3.4) shows that (4.4)

= 2 11 E IIHa* By (4.3) and (4.4), we get the Parseval equality

LIE ALGEBRAS AND THEIR REPRESENTATIONS

59

(4.5)

Suppose that none of the representations Un is equivalent to U,then for all n E N, by the orthogonality relations (Proposition 2.2). (4.5) and (4.6) imply

(4.6)

( x , Xn)=O

II x Ilaa=O. The last equality contradicts the fact that U is irreducible and 11 x 1 (Proposition 2.2). Therefore our assumption is false and U is equivalent q.e.d. to some U". Remark. For each compact group G, the set of characters {xu/U E 61 is an orthonormal base of the space of central functions in La(G).

55. Lie algebras and their representations In this section, we introduce the notion of the Lie algebra g of a linear Lie group G. The Lie algebra g is first defined as a vector space of matrices which is closed under the bracket operation [X, y l = X Y - YX. Then the Lie algebra g is interpreted as the set of left invariant vector fields, namely, first order differential operators commuting with left translations on G. To reduce the preliminaries to a minimum, we restrict ourselves to linear Lie groups and establish the notion of Lie algebra on the basis of concrete calculation of exponential and logarithmic functions of matric variables. Let M,(C) be the associative algebra of complex matrices of degree n. Then, as a vector space, Mn(C) is isomorphic to C"' and has an inner product ,'

(5.1)

( A ,B)=

1 arj6,=Tr(AB*). t , j=1

The norm [IAII=(A, A)l'a defined by this inner product is the HilbertSchmidt norm of A . It satisfies (5.2)

1I AB I1d II A II II B 11.

The algebra Mn(C) is a Banach algebra with the norm ] ] A11. For any matrix A E M,(C), the series

(5.3)

60

REPRESENTATIONS OF

su(2)AND s0(3)

is convergent because the majorant series

fa is convergent. n!

n-0

Definition 1. The sum of the series (5.3) is called the exponential of A and is denoted by exp A or ea. From the definition of exp A, the following properties are easily obtained; (5.4)

(5.5)

exp('A) = '(expA), exp(x) =%, exp(A*)=(expA)*, P(exp A)P1=exp(PAP1).

By choosing P suitably, P A P ' becomes a triangular matrix and its diagonal elements are eigenvalues of A. So we get by (5.5): (5.6) If A1, ..., An are the eigenvaiues of A, then eal, ..., eaS are the eigenvalues of exp A. n

By (5.6) and the relation TrA =

1At, we get (-1

det exp A = eTrA. The most important relation for the exponential function is the following. If AB= BA, then exp A exp B= exp(A +B). (5.8) Proof. For any m E N,we get (5.7)

(5.9)

where A k B1

R, = Max(b,l)>m k+l52m

Let C=Max(llAII, 11B11). Then, by (5.2), the remainder term R, which consists of m(m + 1) terms is estimated as

So the limit of (5.9) when m tends to + m gives (5.8). Since (5.10) exp O= 1 (identity matrix of degree n), by putting B= -A in (5.8) we get (5.11) expA is a regular matrix and (exp A)-l= exp( -A). Since for any complex number z satisfying [z- 1I < 1, the series expansion

LIE ALGEBRAS AND "HEIR REPRESENTATIONS

z00

logz=

61

(- 1p-1

(2-1).

n-1

is valid and the series converges absolutely, the series 00

(5.12)

(A- 1)" n-1

is convergent for any A E M.(C) satisfying IIA- 111< 1. Definition 2. The sum of (5.12) is called the logarithm of A and denoted by log A.

Proposition 5.1. (5.13) explogA=A (5.14) log expA =A Proof. Put

ifllA-111< 1 if llA II
and

(5.16) Then it is sufficient to prove (5.17)

lirn lim A(k, I)= k-+-

&+-

lirn lim B(k, I)=O k-+-

I-+-

Since the powers of A- 1 and of A are mutually commutative, the coefficients of (A - 1)' and of A a in the expansions of (5.15) and of (5.16) are equal, respectively, to the coefficients of x a in the polynomials of x obtained from (5.15) and from (5.16) by replacing the matrices A- 1 and A with a variable x. If k, I ~ s ,then the terms of degree 5 s are same in these polynomials and in the power series for the functions (5.18) log exp x -x. exp log (1 +x)-(1 +x), Since the power series expansions of the two functions appearing in (5.18) represent the constant 0 (for small real values of x), we see that the coefficients of (A- 1)' and of A' in (5.15) and (5.16), respectively, are equal to 0 if k,1 2 s . Now we estimate the non-zero coefficients. The coefficients of (A- 1)' and of A' in (5.15) and in (5.16), respectively, are smaller than or equal to the coefficients of x* in the ordinary power series

62 (5.19)

REPRESENTATIONS OF

c a(c 7 - 1

m-0

- 1

X P ) ~ +

su(2)

AND

1 (1 +x) =exp log--l-x

s0(3)

+ 1+ x =

1 1 +x+ l-x

p=1

and in

(5.20)

l-(ez-l) =x

+x

1 +log- 2-e".

The series in (5.19) and (5.20) are convergent if Ixl
(5.21)

I I 4 k 011, llB(k, 1)llSZI c 4 s=r+1

The right hand side of (5.21) tends to 0 if r tends to there exists SO in N such th& if k,l r s othen

(5.22) Let 1 tend to

+ co . For any

E

>0,

I I A k 0114~ and IIB(k, 1)Ilse. + co in (5.22). Then we have lllim A(k, 1)114E and lllim B(k, 1)114s 1*+-

t-+-

if k 2 s o . We have proved (5.17).

q.e.d.

Definition 3. The Lie product (or bracket) [X, Y] of two matrices X , YE M,, (C) is defined by

[X, yl=XY-YX The commutator (g, h} of two non-singular matrices g, h E GL(n, C) is defined by {g, h} =ghg-'h-'.

Proposition 5.2. Let X and Y be two matrices in Mn(C).When the real variable t tends to 0, we have the following asymptotic expansions:

(5.23)

ta exp tXexp tY=exp t(X+ Y)+$X,

Y]+U(ts)],

a

~

)

63

LIE ALGEBRAS AND THEIR REPRESENTATIONS

(5.24)

exp t X exp tY exp(-?X)=exp {t Y+ta[ X,Y ] + O ( t 3 ) } ,

(5.25)

{exp tX, exp c Y } =exp { t a [ X ,Y ] + O ( t S ) } , Proof. Replacing 2 in 1 log z = (2 1) --(Z 2

-

- l)a

+ O(12- 118)

by ta t' exp t X exp t Y = (1 + t X + T X ~ O(tS))( 1 t Y -Y a O(t3)) 2

+

+ +

+

ta

=l+?(X+ Y)+-i-(Xa+2XY+ Y')+O(?3). we get

1

t2

log (exp tx exp t ~= )t(X+ Y ) ---(t(X+ 1 2

+ ~ X +YY') +

0(t3)}

Y ) + tpa a + 2 X Y + Y')+O(t3)}a +O(t3)

+

ta ++x,

= t(x+ Y )

Y ] o(t3).

(5.23) is proved. Since ta (5.26) exp tXexp tYexp( - t X ) = ( 1 + t ( X + Y ) + T ( X a + 2 X Y + o(t3))(1-t~+~x1+0(f3))=1 ta +tY++[X,

Ya)+

Y ] + ya)+o(t3),

we have log(exp t X exp t Y exp ( - t X ) ) =

?a r Y + $2[X,

r]+ Ya)+ O(t3)--T{ tY + O(P)} 1

+ O(P)

= t Y + t a [ X , Y ] + O(tS).

(5.24) is proved. By (5.26), we have

k p

tx, exp t~

(

I(

t'

= I+~Y+ t' ~ ( ~ [Y]+YS)+O(P)) x,

) +t"X, Y ]+ O(t

x 1- t Y + y Y'+ O(P) = 1

0)

and

+

log {exp tX, exp t Y } =tl [X, r] O(t8).

Proposition 5.3. For any two matrices X, Y EM,(C), we have

q.e.d.

64

REPRESENTATIONS OF

SU(2) AND SO(3)

yY

lim exp - exp=exp (X+ Y), ( n n X nz lim exp -, exp L n } =exp [x,Y]. n*{ n

(5.27)

I--

(5.28)

Proof. By Proposition 5.2, we have

(exp

x exp Y ” n ) = {exp (+ ( X + Y)+O($))}’ =exp(x+ Y+o(+)).

The last equality is verified by using (5.8) n times. (5.27) is proved. Similarly we have

q.e.d.

Definition 4. Let K be a field. A vector space g over K is called a Lie algebra over K if a bilinear mapping ( X , Y)t+[X, u] of g x g into g is given which satisfies the following two identities 1) and 2 ) : 1) [ X , X ] = O 2) [X, [YY Zll + [Y, [Z, XI1 + [Z, [X,Yl1 =o for any elements X, Y and Z in g . 2) is called the Jacobi identity. If a vector subspace fj of g is closed under the operation [X, u] (i.e. when X , Y belong to Jj, then [X,r] belongs to q), then lj is Lie algebra and is called a Lie subalgebra of g. Example 1. The space M n ( K )of all matrices of degree n with elements in a field K is a Lie algebra over K if the Lie product [X, Y ] is defined by [X, Y] = X Y - Y X The identities 1) and 2) are easily verified. This Lie algebra is denoted by gxn, K ) . Example 2. A vector space A over a field K is called an algebra over K if a bilinear mapping of A x A into A is given. We denote the bilinear mapping by (a, b)Ha.b and call a - b the “product” of a and b. A linear mapping D on an algebra A is called a derivation if it satisfies D(a-b)=Da.b+a-Db for all a, bE A. The set D(A) of all derivations of an algebra A is a Lie algebra and is called the derivation algebra of A .

LIE ALGEBRAS AND THEIR REPRESENTATIONS

65

Example 3. Let M be a C"-manifold and fl be the associative algebra of all real valued C"-functions on M.Then a derivation of fl is called a C"-vector field. Proposition 5.4. If G is a closed subgroup of GL(n, C), then the set (5.29)

g= {XEgI(n, C)lexp tXE G

for all t E R}

is a Lie subalgebra of gl(n, C). Prooj If X belongs to g and a is a real number, then a x belongs to g by the definition of g. If X, Y are two elements of g, then X + Y and [X, Y ] belong to g by Proposition 5.3. Delinition 5. A closed subgroup G of GL(n, C ) (for some integer n) is called a linear Lie group. The real Lie algebra defined in Proposition 5.4 is called the Lie algebra of the linear Lie group G. Proposition 5.5. Denote the Lie algebras of GL(n, C), GL(n, R),SL(n, C), U(n), O(n, C ) and Sp(n, C )by the corresponding small German letters.

These algebras are characterized as follows: aNn, C ) = M?l (C), gl(n, R) = {XEgI(n, C ) I r = X ) , GCn, C ) = {XEgl(n, C ) I TrX=O], n(n) = { X € gqn, C)I X*+X=O), o(n, C ) = { X E gI(n, C ) I ' X + X = 0) ,and Bp(n, C) = {XEgf(2n, C) I 'XJ+ JX = O}, where J is a matrix of the form

Proof. Let g be the Lie algebra of GL(n, C). Then g is a subalgebra of gl(n, C) by the definition (5.29) of g. Conversely any element-- XE gl(n, C) belongs to g by (5.1 1). If X E gl(n, C ) satisfies X=X, then exp tX=exp tX for all t E R and X belongs to gI(n, R).Conversely if exp tXE GL(n, R)= {gE GL(n, C)lg=g] for all t E R,then differentiating the equality exp tX=exp t X at t = O , we get x = X by the second equality in (5.4). Similarly by (5.7) we can prove det exp tX=l for all t if and only if TrX=O. By (5.4) (exp tX)*(exp tx> = 1 for all t E R if and only if X* X = 0. In fact if X* +X= 0, then exp t X * =exp (- tx) = (exp tX)-l and if (exp tX*) (exp tX) = 1 for all t E R,we get X* + X=O by differentiating the equality at t = O . Similarly we get the description for ~ ( nC, ) and @(a, C).

+

Proposition 5.6. Let G and H be two closed subgroups of GL(n, C )and let g and be their Lie algebras.

66

REPRESENTATIONS OF

Su(2) AND SO(3)

1) If G I H , then g 2 G . 2) The Lie algebra a of G n H coincides with g n 8. 3) The Lie algebra go of the connected component Go of G coincides with g. Proof. 1) is clear from the definition. 2) The inclusions G 1G nH and HIG n H implies g a , 2 a and g n Z, 2 a. Conversely let X be an element in g n 8. Then exp ~ X G En H for all t E R and X belongs to a. 3) By l), we have g I g o . Conversely let X be any element in g. Since R is connected and the mapping Xwexp tX is continuous, exp t X belongs to Go for all t E R and X belongs to go. q.e.d.

=I

Proposition 5.7. The Lie algebras of SL(n, R), SU(n), O(n), SO(n), Sp(n) and Sp(n, R) are described as follows: I?K(n,R ) = (XEgK(n, C) I X E X and TrX=O}, I?u(n) = (XE gK(n, C) I X*+X=O and TrX=O}, o(n) = {XEgK(n, C) I X Z X and "x+X=O},

I?o(n)

=o(n),

2p(n) = {XEgK(n, C) Gp(n, R ) = {XEgl(n, C )

I X*+X=O and LXJ+JX=O}, I X = x and tXJ+JX=O}.

Proof. Since SL(n, R ) =SL(n, C )n GL(n, R), SU(n)=SL(n, c )n U(n), = O(n, C ) n GL(n, R), SO(n)= O(n, C )n SL(n, R), O(n) Sp(n) =SP(n, C ) n W ) , Sp(n, R ) = Sp(n, C )n GL(n, R), Proposition 5.7 follows from Proposition 5.5 and Proposition 5.6.

Proposition 5.8. Let x be a continuous homomorphism of R into GL(n, C). Then x is differentiable and x(t)=exp tX for all t E R where X = ~'(0). Proof. The proof of Proposition 5.8 is same as that of Ch. I. Corollary to Theorem 4.1. Proposition 5.9. Let G be a linear Lie group, g, the Lie algebra of G, V , a finite-dimensional vector space over C,and let U be a continuous homomorphism of G into GL( V). Then a representation U' of g on V is defined by

and we have (5.30)

UexpLg = exp(t U'(X)) for all

E R.

67

LIE ALGEBRAS AND THEIR REPRESENTATIONS

Proof. By proposition 5.8, the continuous homomorphism x:twUexPtx of R into GL( V) is differentiable and U ' ( X ) = (due,, $=/dt) is defined and satisfies (5.31). We next prove that U' is a Lie algebra homLmorphism of g into gI(V). By the definition of U', U'(aX)=aU'(X) for all a € R and for all X E g. By (5.27), we have exp tU'(X = lim n--

+ Y) = U,,,

r(X+Y)

t n

= lim U exp-X

(

exp-Y n

t V ( X > exp ;u~(Y)) t (uexp?ueXp+ ) =lim exp n

>") n

n--(

=exp t(U'(X)+ U'(Y)) for all t E R. Hence we have U'(X+ Y) = V'(X)+ U'( Y) Similarly by (5.28), we have U'([X, Y]) = [ U'(X), U'( Y)]

for all X and Y in g. for all X and Y in g.

Definition 6. The representation U' of g is called the direrentialrepresentation of U and is denoted by the same symbol U if there exists no possibility of confusion. Example 4. Let g be an element of a linear Lie group G. Then the linear transformation Adg on the Lie algebra g of G is defined by (Adg)X=gXg-' for X E g.

Since exp t(gXg-'>=g(exp tX)g-l for all t E R, gXg-' belongs to g and Adg is actually a linear transformation on g. And the mapping Ad: g H Adg is a continuous real representation of G on g. The representation is called the adjoint representation of G. The differential representation ad of Ad is defined by (exp t x ) Y(exp( - tm)] t = O =X Y - YX= [x, Y]. The representation ad of a Lie algebra g is called the adjoint representation of g. Now we come back to our original subject, i.e., the representations of SU(2). First we give a basis of the Lie algebra Su(2) of SU(2). Proposition 5.10. Put

Then they form a basis of the Lie algebra Bu(2). Their Lie products are given by [Xi, Xa ]=x3. (5.31) [X,, X3 ]=XI, [X3, XI ]=x,,

68

REPRESENTATIONS OF

Proof.

sU(2) AND s0(3)

Since en(2)=

{(:+cyb+2) - I

a, b, C E R)

by Proposition 5.7., ( X + ) l s i s 8is a basis of Bn(2). The Lie products are easily calculated. Proposition 5.11. Let U" be the irreducible representation of SU(2) constructed in $1. and (X,)lst., be the basis of Su(2) given in Proposition 5.10. Then the differential representation of U" is given by i

(k +( n-k )yt+l

U"(X1)pt =1

(ot-l

(5.32) we have by Proposition 5.8

1

sin-

2

sin - cos 2

cos-

-

and exp tX3= Then we have by (5.32) and (5.33) c u n ~ i ) (21, p ~za)=

i =

jk

t-0

y

-

({;[ 2

-)

+

t t z1cos +iza sin 2t t ( i z1 sin z) 2 (n-W z1 za"-t-l 2

~ (n+ -k ) (or+1] (21, za)

C U ~ ( X p, )t ] (z1,zs) =[${(zl cos

and t -+ 2

za sin

t

2

2

LIE ALGEBRAS A N D THEIR REPRESENTATIONS

69

Similarly we have

q.e.d. Remark. The eigenvalues o f the operator iU"(X8) (or iUn(Xl) or iU"(X2)) are called the weights of the representation Un. It should be remarked that the weights determine the irreducible representations of SU(2). In fact, the class of U" is the only class of irreduciblerepresentations of SU(2) whose maximal (or highest) weight is equal to n/2. With the inner product defined in Proposition 1.2., the representation space Vn of Un is a Hilbert space and U" is a unitary representation. The Hilbert-Schmidt norm \\All of a linear transformation A of Vn is defined in terms of this inner product. More explicitly, since (5.34)

is an orthonormal base of V,, the Hilbert-Schmidt norm l]A]lis defined by

c n

llAlla=

11Aji11'.

k=O

Proposition 5.12. Let X be any element in Bu(2), and U" be an irreducible unitary representation of SU(2). Then the Hilbert-Schmidt norm IIUn(X)ll of U"(X) satisfies the following equality.

-

IIU" (X)llSn dn+l IIXllS IIXII. Proof. For any element X E lu(2),there exists an element g E SU(2) and

70

REPRESENTATIONS OF

=2-1na(n

su(2)AND s0(3)

+ 1) 11 XIIagna(n+ 1) 11 Xl12~ ( +n1)‘ )I

q.e.d.

An associative algebra A over a field K has the structure of Lie algebra if the Lie product is defined by [x, YI =X Y -Y& where the products on the right hand side are the original products in A. This Lie algebra is denoted by L(A). In particular, the associative algebra p( V ) of all linear transformations on a vector space Y has the structure of a Lie algebra. The Lie algebra L ( y (V)) is denoted by g1(V). The Lie algebra g of a linear Lie group G is a real Lie subalgebra of an associative algebra Mn(C). Now for each Lie algebra g, we construct an associative algebra U(g) in which g is imbedded.

Definition 7. Let g be a Lie algebra over a field K, and T be the tensor algebra of g. Then T, as a vector space, is the direct sum of the spaces Bmg and the product in T is defined from the tensor product. Let I be the two-sided ideal of T generated by the set

[x@ Y -

Y @ X x - [ X , Y ] I x,YE g}. Then the factor algebra U(g) = T/Iis called the universal enveloping algebra of g. Let (i, be the canonical projection of T onto U(g) and (O be the restriction of $ to g. Now is a linear mapping of g into U(g) satisfying

d[X Yl) =lo@-) 94y>- $4Y ) Cp(J-7 =[&n cP(Y)l. is called the canonical mapping of g into U(g). The universality of U(g) is described in the following Proposition.

‘p

Proposition 5.13. Let g be a Lie algebra over a field K, A be an associative algebra over K and u be a homomorphism of g into L(A). Then there exists a homomorphism U’ of U(g) into A such that I

u yo=u.

LIE ALGEBRAS AND THEIR REPRESENTATIONS

71

Proof. The linear mapping u of g into A is extended to a homomorphism uo of T into A by defining U O ( X I @ ... @X,) =u(X1) ... u(X,). The extended homomorphism u0 sends the ideal I to 0 because

u o ( X @ Y - Y @ X - [ X , y1) = u ( X ) . ( Y ) - o ( Y ) o ( X ) - o ( [ X , Y])=O.

Therefore, the image U O ( ? ) of an element ? E T depends only on the coset t + Z and hence there exists a homomorphism U' of U(g) into A satisfying U

/

oy=o.

COROLLARY to Proposition 5.13. Let g be the Lie algebra of a linear Lie group G . Then the canonical mapping y of g into U(g) is injective. Proof. Let G be a closed subgroup of GL(n, C). Proposition 5.4 shows that g is a Lie subalgebra of gl(n, C ) = L ( M ,( C ) ) .Let u be the inclusion mapping of g into gI(n, C ) : o ( X )=X. Then there exists a homomorphism 7 of U(g) into M,,(C) satisfying ~ O J D = U . Since u is injective, (O must be injective. q.e.d.

By this Corollary, we shall always identify XE g with y ( X ) E U(g) and g is regarded as a Lie subalgebra of L (U(g)). Any representation u of g on Y may be uniquely extended to a representation U' of U(g) by Proposition 5.12. For the sake of simplicity, u' is also denoted by U . Remark. It can be proved that any finite-dimensional Lie algebra g has an injective representation (Theorem of Ado-Iwasawa cf. Bourbaki [4] Ch. I $7. THEOREM 3 and Stminaire Sophus Lie [ l ] ,expost 8). Thus y is injective for any Lie algebra g.

Definition 8. Let g be a Lie algebra and X be an element of g. Then a linear transformation ad X on g is defined by (ad X ) Y = [X, Y ] The mapping ad : Xwad X is a representation of g on g (i.e. a homomorphism of g into gl(g)). This is easily verified by the Jacobi identity. The bilinear form B on g x g defined by B(X, Y )=Tr(ad X ad Y )

is called the Killing form of g. B satisfies the following identity (5.35)

B([X, Y ] ,Z ) = B ( X , [ Y, 21) for any X , Y, Z E g, because Tr([a, b] c ) =Tr(abc)-Tr(bac)=Tr(abc)-Tr(acb)=Tr(a[b, c]).

72

REPRESENTATIONS OF

SU(2) AND SO(3)

A finite-dimensional Lie algebra g over a field of characteristic 0 is called semisimple if its Killing form is non-degenerate. A connected linear Lie group is called semisimple if its Lie algebra is semisimple. Example 5 . The Lie algebra lu(2) is semisimple. By (5.31), we have

Therefore we have (5.36)

B(X#,X,)=

-at, (lSi,jS3),

and det(B(Xc,Xi))zO.

Definition 9. Let (Xt)la/rn be a basis of a semisimple Lie algebra and gt, =B ( X , X’) and

W’)=(gt,)-’. Then the element

c n

Q=

g<’XcX,

t, 5-1

of the universal enveloping algebra U(g) of g is called the Casimir element. Proposition 5.14. The Casimir element B of U(g) is independent of the choice of the basis ( X ) .Ja belongs to the center of U(g). Proof. Let ( Y j ) l r j + n be another basis of g. Then Y, can be written as Y$=

Cacjxl j

Let Then we have

So we have

LIE ALGEBRAS AND THEIR REPRESENTATIONS

73

1

D = Xd?.

and

f

Let

[X, XI =

1ctjXjand [X, Xf]= c d i j P j

Then by (5.35) and (5.37) we have (5.38) dr, =B([X, X'],X,) = -B(X', [X, X,1) = cjc. In any associative algebra A, the identity

-

(5.39) [a,bcl = [a,b]c+b[a,cl holds for any a, b and c in A. By (5.38) and (5.39), we have

rx, QI =

c

[X, X,]X'

t

+

cX U ,

XI]

j .

for any X in g. Since g and 1 generate the algebra U(g), we have proved that the Casimir element D belongs to the center of U(g). Proposition 5.15.

(5.39)

The Casimir element D for Su(2) is given by

D= -

1

+x2 +X2).

In the representation U",U"(D) is a scalar transformation given by (5.40)

1

V"(Q)= s n ( n + 2 ) 1

Proof. By (5.36), the Casimir element D for Gu(2) is given by (5.39). By Proposition 5.1 l., we have for any k E [O, ..., n) un(D)pk = $[U"(~I) 1iT(kpx-1 i -k (n--k)$Dk+l))

.

74

REPRESENTATIONS OF

1 --n(n

-8

SU(2) AND SO(3)

+2)pk.

q.e.d.

It can be proved that every closed subgroup G of GL(n, C ) is a Lie group, namely, there exists a real analytic manifold structure on G and the group operation ( g , h)ngh-l is an analytic mapping. The analytic structure on G is defined as follows. There exists an open neighborhood Uo of 0 in the Lie algebra B of G and an open neighborhood U of 1 in G such that the exponential mapping exp is a homeomorphism of Uoonto U. Let A(1) be the set of real-valued functionsf on U for whichfoexp is a real analytic function on UO.Moreover, for any element g E G, let A ( g ) be the set of all functions f on Ug such that foR,-l belong to A(1). Then the assignment g n A ( g ) defines an analytic manifold structure on G . Since (X, Y)nlog(expXexp(- Y)) is a real analytic mapping on Uo x Uo if Uo is taken sufficiently small, the group operation ( g , h)ngh-l is analytic at (1, 1). It is then analytic everywhere by the definition of the analytic structure on G (cf. Chevalley [l]. Ch.V, gXIV, Proposition 2). In particular every linear Lie group G has a differentiable structure which is invariant under right and left translations. The differentiable structure of the group SU(2) defined above coincides with the natural differentiable structure of S3 when the latter is identified with SU(2). Definition 10. A Lie group G is said to act on a C"-manifold M dsfferentiably from the right if a C"-mapping f : M x G n M is defined and satisfies 1) m(g1gd =(mgl)g%, 2) me=m where we let f(m, g) =mg and e is the identity element of G. G is said to act on M from the left, iff satisfies 1') (g1ga)m=g1(gam) instead of 1) (where we write f(m, g)=gm). Assume that a Lie group G acts on a C"-manifold M differentiably from the right. Then a representation T of the group G on the space -F of all complex-(or real-) valued C"-functions on M is defined as follows: (T9.f) (4 =fW.

Although f l is infinite-dimensional, the differential representation T'

LIE ALGEBRAS AND THEIR REPRESENTATIONS

75

of T is defined as follows. Let X be an element of the Lie algebra g of G. Then T'(X) is defined by

Thus T'(X) It is easily seen that T ' Q is a derivation of the algebra 9. is a vector field on M and a differential operator of the first order. The mapping T' is a Lie algebra homomorphism of g into g l ( 9 ) . In a special case (G= S0(3), M = R3),this fact will be proved in Proposition 7.4. The representation T' of g has a unique extension to a representation of the universal enveloping algebra U(g). The extended representation is also denoted by T'.T'(a) (a E U(g)) is a differential operator on M. A Lie group G is said to act on M effectively, if the transformation T, : x w x g is not the identity transformation for any g + e . If the action of G on M is effective, the representation T'of the Lie algebra g is injective. In particular a linear Lie group G acts on itself effectively by ( h , g ) w hg. Then the Lie algebra g of G is isomorphic to the image T'(g)which is a Lie subalgebra of the derivation algebra D ( 9 ) . The elements 8= T ' ( X ) of T ' ( g ) are characterized among the derivations of .-Fby their left invariance. Namely an element X of D(#) belongs to T'(g) if and only if XoLp=L,oX for all g E G where ( L J ) (h)=f(g-%). Similarly T'(LI(g)) is the algebra of all left-invariant linear differential operators on G. In the following we shall always identify g with T'(g) and U(g) with T ' ( U ( g ) ) by the mapping T'.Namely, an element X of the Lie algebra g is regarded as a left-invariant vector field on G. Proposition 5.16. Let 9 be the Casimir element of sn(2) and urj" be a matrix element of the representation U".Then uijn is an eigenfunction of the differential operator 9. Specifically, we have 1 9utjn =-n(n 4-2)Utj" . 8 Proof. Let X be an element of Su(2). Then the differential representation of U nwhich is also denoted by Un is defined by

Therefore, as a differential operator on SU(2) the element X operates on U nby (5.41) = U"(g)U"(X).

76

REPRESENTATIONS OF

SU(2) AND SO(3)

So we have by Proposition 5.15, n(n +2) (nu")(g) = U"(g) U"(Q)=-U"(g). 8

$6. Fourier series on SV(2) By Theorem 4.1, the dual

6 of

G=SU(2) is naturally identified with

N- {O, 1,2, ...).The Fourier transform S : f ~on f SU(2) is defined by (6.1)

m=/

f(gW" (g-'Mg

SWO)

for anyfin L1(G).The Fourier series of a functionfin La is equal to (6.2)

c

(n+ 1) Tr(f(n)U"(g))

n-o

which converges tof(g) in L3(G). In this section we study the pointwise convergence of smooth functions on SU(2). Proposition 6.1. Iff is a C'-function on G=SU(2). Then its Fourier coefficients satisfy

Proof. Let p and (/I be any C1-function on G. Then for any element X in bn(2), we have

So we have for 9 = -2-'(XIa +X2+ X2)and C2-functionsp and cr,

(6.3) (Q(P,(I) = (9,Q4). From Proposition 5.16, and (6.3), we obtain ( Q f > d ( n )= (Q-6 W) =(-6 Q u d =8-'n(n 2) (f,ujtn)

+ = 8-'n(n + 2) (f)&).

FOURIER SERIES ON SU(2)

77

COROLLARY to Proposition 6.1. Iff belongs to Csk(G),then we have

Theorem 6.1. The Fourier series (6.2) of any Cs-functionfon SU(2) converges absolutely and uniformly to f. Proof. The Hilbert-Schmidt norm IIUgnllof a unitary matrix Ugn of degree n+ 1 is equal to J n 1. Using the Schwarz inequality for the inner product (A, B)=Tr(AB*) and (3.19) of Chapter I, we get

+

-

co

C I(n+ 1) Tr(f(Wgn)I

3

= C(n+ l)"IIP(n)ll n-1

n=1

- (n+ +

=8C

1y'a

n(n 2) n=1

II(W)Wll

Therefore the Fourier series (6.2) of a Ca-functionf converges absolutely and uniformly. The sum of the series coincides withfalmost everywhere, hence everywhere, because the sum is continuous. Remark 1. Iffis a Ca-functionon SU(2), then the Fourier series

converges absolutely and uniformly tof. This can be proved in the same way as Theorem 6.1. Namely, we have

78

REPRESENTATIONS OF

Su(2) AND s0(3)

Now we give a characterization of Cm-functionson SU(2) by their Fourier transforms. First we need a lemma. LEMMA 1. Let G be a Lie group and g the Lie algebra of G. Moreover letfbe a complex-valued function on G and k be a positive integer. Then: 1) f is a C'-function on G if and only if f(g exp rX) is differentiable at t=Ofor anygEGand

is a Ck-'-function on G for any element X in g. 2) A function f belongs to Ck(G)if and only if XkX k - 1 ... Xlfis defined and is continuous for any k elements XI, ..., X k in g. Proof. 1) Iff is a Ck-function, then the function p(g, t)=f(g exp t X ) is a CL-function on G x R. So p is differentiable with respect to t and (Xf)(g) = ap/at(g, 0) belongs to Ck-l(G). Conversely suppose that p(g, t) is differentiable at t =0 for any g E G and (Xf)(g) = ap/at(g, 0) belongs to Ck-l(G).Then p(g, t) is differentiable at any t E R, because p(g, f +s) = p(g exp tx, s), and we have (6.4)

Moreover, $(g, t, h) =f(g exp t X h) is differentiable with respect to t because d d -f(g exp t X h) =-f(gh exp (t(Adh-l)X)) dt dt =((A&-' X)f)(g exp t X h). Let XI, ...)X,, be a basis of g, and put p ( t ) = p(tl, ..., tn) =exp tlXl. ... exp t,X, Then p is an analytic diffeomorphism of an open neighborhood W of 0 in R" onto an open neighborhood V of the identity e in G. Let n

(6.5)

Ad(exp tlXl

... exp t,X,)-lX< =

aji(t)Xj. j=l

Clearly a,c(t)=a$rl, ..., t,) is an analytic function on R". Let g be a fixed element in G. The mapping gp(t)Ht =(tl, ...)t,) defines a system of local coordinates on gV, the canonical coordinates of the second kind. Now by (6.4) and (6.5), af(gp(t))/atr exists and is equal to

FOURIERSERIESON SU(2)

"

=

Cafr(O, ...,0,

ti+l,

79

... tn) (Xjf) (gy(t)).

j-1

By assumption, the right-hand side of the last equality, regarded as a function of t, is a Ck-'-function on W.Therefore f is a Ck-function on gV. Since g is arbitrary, we have proved that f is a Ck-function on G. 2) is easily proved by the induction with respect to k using 1).

Definition 1. Let G = SU(2). A function p : & = N-t

M,,(C) is called n=1

rapidly decreasing, if v(n)E Mn+1(C) and if for any k E N there exists a constant CkZO such that Ilnkp(n)ll 4ck for all n E N . The space of all rapidly decreasing functions on & is denoted by 9(&

Theorem 6.2. Let f be a continuous function on SU(2). Then f is a C"-function if and only if the Fourier transform f off is rapidly decreasing. Proof. The condition is necessary. Iff is a C"-function on G, then n(n+2)

(QY)Yn)=( 7 f(n) )

(6.6)

for every k, n E N (Corollary to Proposition 6.1). Since Qkf€ C(G)c La(G),

.

II(Q"f)^(n)ll--o

(6.7)

@-+ +

),

by the Parseval equality. (6.6) and (6.7) prove that Ilnakf(n)ll-tO + a), for any k E N. Hence, the Fourier transform f o f f belongs to 9(&). Conversely suppose thatf belongs to 9(& Then ). the Fourier series (6.2) of f converges absolutely and uniformly. In fact since Ilf(n)ll= O(n-l) the right hand side of "

"

n-0

n-0

C( n + l ) I T r ( f ( n ) ~ n ( g ) ) IC~ ( ~ + I ) ~ ' ~ I I ~ ( ~ ) I I converges. Its sum fl(g) is equal to f(g) almost everywhere by Peter-Weyl's theorem. Sincef a n d 5 are continuous,fi is equal to f everywhere. In particular, we have "

(6.8)

f(g exp t-U=

1(n+ 1) Tr(f(n)Ungexpt4 n-0

80

REPRESENTATIONS OF

SU(2) AND SO(3)

for every X E Su(2) and t E R. The series is obtained by differentiating termwise the right hand side of (6.8) with respect to t is equal to a

(n+ 1) Tr(f(n)Un,..,txUn(X))

(6.9) n=O

by (5.41). Since

II~ n ( X ) l 1 5 n J n + 1 ~ l l by Proposition 5.12., the series (6.9) converges normally, In fact, since Ilf(n)ll =0(r6), we have

-

a

1(n+ l)ITr(f(n)Un,

erp

t~Un(X))I d

1n(n+

l)sllf(n)ll llXll< + 00

I

n-0

n=O

Therefore the series (6.8) is a differentiable function of t , and the derivative ( X f ) (g exp t X ) is equal to (6.9). Since (6.9) is uniformly convergent, X f is a continuous function on G. So f is a C1-function on G by Lemma 1. Sincefis rapidly decreasing, this process can be repeated k times for arbitrary integer k>O. Thus, for any k elements Xl, ..., Xk in Su(2), XdL1 ... Xlf is defined and is equal to a

(6.10)

(X,

... Xlf

) (g)=

1(n+ l)Tr(f(n)U,"Un(Xk) ... Un(Xl)). n-0

The series converges absolutely and unifofmly. So Xr ... Xlf is continuous, and f is a CK-functionon G by Lemma 1. Since k is arbitrary integer >O, f is proved to be a C"-function on G. COROLLARY to Theorem 6.2. Let f be an integrable function on G = SU(2). Then the Fourier transformf is rapidly decreasing if and only if f coincides with a C"-function almost everywhere.

Definition 2. is defined by

For any element D E U(Bu(2)), a seminorm p D on C"(G)

pD(f )= !@fIl = where D is regarded as a differential operator on G=SU(2). The family of seminorms {pDID E U(Bu(2))I gives the structure of a complete locally convex topological vector space to C"(G). The topological vector space so obtained is denoted by d ( G ) . For any s 2 0, a seminorm q, on 9 ( d )is defined by 4 r ( d = SUP Ilngdn)ll ncN

~ ( 6is )regarded as the topological vector space defined by the family {q,lszO} of seminorms.

FOURIER SERIES ON SU(2)

*

81

Theorem 6.3. Let G=SU(2). Then the Fourier transform :fwfis a topological isomorphism of g ( G ) onto ~ ( 6 ) . Proof. 9is surjective by Theorem 6.2. Assume that two functionsj and A, have the same Fourier transform f=fl, then by the uniqueness property of the Fourier transform (Ch. I. Theorem 3.6) f coincides with f1 almost everywhere. Sincef andS, are continuous, they coincide everyis injective. where. So By the Corollary to Proposition 6.1, we get

*

(+))*llf(n)Il

=If

= ll(QW(~)Il

(QY)k)U,"_l&ll s/ol(QY)(dl llU;-lll&

a

s JnflIIJaXfll-. So we have n2k--"111f(n)lls8*llQYll-, i.e., qw-i/a(f) 4 8'

since q&) d qt(p) if s5 t, 1 if s 5 2k --. 2 Since k can be taken arbitrarily large, the last inequality proves the continuity of The inverse Fourier transform S - l is also continuous. In fact, by Proposition 5.12., Theorem 6.2. and (6.10), we get q , ( f )s 8* p & f )

(6.1 1)

*.

Il&

-

Xlf 11-5 C ( n + 1)~'~llf(n)ll IIvn(mll-..IIUn(xl)ll n-0

Since the series C (n+ l)ak+s~2/na converges if s > 2k + 512, the above in-

82

REPRESENTATIONS OF

SU(2) AND SO(3)

equality proves the continuity of X-l.(Note that every element D of U(Q) is a linear combination of “monomials” Xr ... X I . )

COROLLARY to Theorem 6.3. the family of seminorms,

The Topology of g ( G ) is defined by

(6.12) (Pdlk E N l . Proof. Let g o ( G ) be the topological vector space whose underlying vector space is C”(G) and whose topology is given by the seminorms (6.12). The identity mapping is a continuous mapping of S(G‘) onto s o ( G )because Qn is an element of U(g). Conversely the Fourier transform .9- is a continuous mapping of g o ( G ) onto 9(&) by (6.11) and X-l is a continuous mapping of g ( 6 ) onto g ( G ) . Therefore the identity mapping Id=..F-lo..F is q.e.d. a continuous mapping of go(@ onto g ( G ) .

$7. Representation of SO(3) and spherical harmonics The representation theory of the three-dimensional rotation group SO(3) is derived from that of SU(2) described in the preceeding sections, because SU(2) is a covering group of SO(3). The irreducible unitary representations of SO(3) are realized on the space of harmonic polynomials or spherical harmonics. First we give the relation between SU(2) and SO(3). Proposition 7.1. Let (et)lStg3 be an orthonormal base of three dimensional Euclidean space R3, and Hi be the subgroup of SO(3) consisting of the rotations around the axis e,(l Si53). Then the group SO(3) is generated by two of the three subgroups Hi(l S i 5 3 ) , e.g., by H, and Hs. More precisely any element g of SU(3) can be written as g=rst, r, t E H3, s E Ha Proof. We can rotate the vector ge3 into the (el, es)-plane by a rotation rl in H3. Then we can send rlge3 to e3 by a rotation slE Ha : slrlg e3= e3. Finally slrlg = t belongs Hs and g =rst where r =rl-l E H3 and s =sl-lE H z. Let (Xt)l~i63 be the basis of the Lie algebra 2 4 2 ) defined in Proposition 5.1 O., i.e., Xl=i(p X2=1(1 1 0 -1o) and

-9>,

i),

and put 1

(X,Y ) = --B(X, 2

Y)

REPRESENTATIONSOF

SO(3) AND

83

SPHERICAL HARMONICS

where B is the Killing form of Su(2). Then by (5.36), we have

Therefore the symmetric bilinear form (X, Y) is positive definite and

(Xi)lstss is an orthonormal base of Su(2)with respect,to this inner product. Hereafter we identify the vector space Su(2) with RSby the mapping tfX,-(rl,

fa,

tJ).

f-1

Proposition 7.2. The adjoint representation Ad of SU(2) (cf. 55, Example 4) is a continuous homomorphism of SU(2) onto SO(3). The kernel of the adjoint representation is [ & 1) and SU(2)/ { 2 1) G SO(3). (7.1) Proof. Since ad(gXg-l) = Adg adXo Adg-l, we have (AdgX, AdgY) =(gXg-l, gYg-') 0

= -I T , ( Adgo adXo Adg-10 Adgo ad Yo Adg-I)

2

1

= --Tr(adXadY)=(X,

Y) 2 for any g in SU(2) and any X, Y in Su(2). Therefore Ad is a continuous homomorphism of SU(2) into O(3). Since SU(2),which is homeomorphic to Ss, is connected, det Ad g = 1 for all g e SU(2); hence, Ad S U ( 2 ) c SO(3). By (5.32) and (5.33), we have 1 cos t/2 -sin "2) cos t/2 sin t/2 ( ~ exp d tXa)X1 = T(sin t/2 cos t/2 -sin t / 2 cos t/2

(p

); (

1 -isint icost icost i s i n t

=T(

=(cos t)Xl

- (sin t)&.

Similarly, we have and (Ad exp tXa)Xa =Xa (Ad exp tXs)Xs=(sin t)Xl +(cos t)Xs. Therefore cost 0 sint (7.2) 0 cost -sin r

)

84

REPRESENTATIONS OF

SU(2) AND SO(3)

and [Ad exp tX,lt E R} =H2(cf. Proposition 7.1). Similarly, -sin t (7.3) Adexp t X 8 = E : cost 0 1

"0)

and {Ad exp tX81tE R}=Ha. Since the subgroups H2and H8 generate S0(3), we have proved that Ad SU(2)=SO(3). Now we determine the kernel N of Ad. Let

be any element in N:Ad g = 1. Since (Ad g)X8=Xs,we have gX8 =Xa g, i.e.,

-ib -id)=(-: This imples b=c=O and g=

;(

-3). Moreover we have

(Ad g)X2=Xr, gX, =Xr g, i.e.,

-;)=(: SO

a9 = 1,

-f>.

a= & 1 and g = -t- 1.

q.e.d.

Theorem 7.1. For any non negative integer 1, there exists an irreducible unitary representation Do1of SO(3) which is given by (7.4) Do' Ad= U a l , ( U a l is the representation of SU(2) defined in Proposition 1.1.) Any irreducible unitary representation D of SO(3) is equivalent to Dozfor some non negative integer 1. If l#l', then Do1is not equivalent to Do1'. Proof. Let (ox (0sk g n ) be the basis of the representation space V,, of Undefined by (1.5). Then we have Q

(Un(- 1)pr) (21,z~)=(pk(-~i,

-4

=(- 1)" (or(z1,zd

and (7.5.)

U"(- 1) =(- 1)" If n =21 ( I E N),then the kernel of U" contains N = { f1) and there exists a well-defined homomorphism Do' of SO(3) into U(Vn)satisfying (7.4). DO'is an irreducible unitary representation of SO(3) as is U a L .

REPRESENTATIONS OF

SO(3) AND SPHERICAL HARMONICS

85

Let D be an irreducible unitary representation of SO(3). Then U=Dn Ad is an irreducible unitary representation of SU(2), and so is equivalent to Unfor some n E N (Theorem 4.1). Since Un(- 1 ) =D(Ad( - 1 ) ) =D( 1) = 1, n must be even and equal to 21 for some I € N by (7.5). Therefore D is equivalent to Do1. If I#I', then deg Do'f-deg Do1'and DoLis not equivalent to Do1'. q.e.d. We construct a realization of the representation Do1which has a close connection with the nature of SO(3). If we pursue the analogy to the representation Un of SU(2), we may take the representation T1on the space WI of homomogeneous polynomials of degree I in three variables x, y, z defined by

(TPtf)(4=f (%I, where x = ( x , y , z). However it turns out that T' is not irreducible if 122. (7.6)

+

For example, W2contains the invariant subspace spanned by xa y a+z'. So we must take an irreducible subspace of Wl in order to obtain an irreducible representation. Such a subspace may be obtained as an eigenspace of a differential operator, the Laplacian. Definitionl. Let

be the Laplacian on Ra and let Hi= { f W ~ Ld~f = O } . An element of H Lis called a harmonic polynomial or a solid harmonic of degree 1.

Proposition 7.3. dim W z =(z+1)(1+2) and dim HL=21+1. 2 Proof. The monomials xpyqz' ; p , q, rE N , p + q + r = l form a basis of the space Wl of homogeneous polynomials of x, y , z of degree 1. Since the dimension of the space of homogeneous polynomials of x and y with degree I-r is I-r+ 1, dim Wl is equal to I

(1+2) ( I + 1 )

dim w , = C +-0( I - r + l ) =

2

Any polynomial F i n WLcan be written uniquely as

86

REPRESENTATIONS OF

SU(2) AND SO(3)

(7.7)

Where Fn(y, z ) is a homogeneous polynomial of y and z of degree I-n. Since

We have (7.8)

So an element F in Hl is uniquely determined by giving FOand Fl and it is denoted by F(Fo,F l ) ; moreover, Fo and Fl can be taken arbitrarily. Let Wl' be the space of homogeneous polynomials in y and z of degree 1. Then the mappings (DO :FOwF(Fo,0) and p1 :B w F ( 0 , Fl) are linear Since F=O if and only if F, = O mappings from W,' and Wl'-l into Hi. ( O s n ~ l )y,o and pl are injective. The vector space Hi is the direct sum of yo( Wl') and yl( Wt-l'), because F(Fo,Fl)=F(F0,O) +F ( O , 4 ) . Since dimWl'=Z+l, we have dim Hi=(Z+1)+Z=2Z+1. q.e.6.

Proposition 7.4. Let 3-be the space of complex valued C"-functions on R 3 . Then a representation T of SO(3) on ..Fis defined by ( T , f ) (4=f (xg) for g E S0(3),f E .Fand x E R3. The Laplacian

Commutes with Tofor any g E SO(3): doT,,=T,,od.

Proof. Since SO(3) acts on R 3 from the right as a group of linear transformations, Tgf belongs to .-Fwith f and T is a homomorphism of SO(3) into G L ( x ) . For any function f in S,put Tpf=cp and y=xg and write x=(xl, xS, xs), Y = (yl, y,,ys). Then we have y j =Ex' gtj and i

Thus we get

REPRESENTATIONS OF

SO(3) A N D SPHERICAL

HARMONICS

= [(Ta0 4fl (x).

87

q.e.d.

Theorem 7.2. We preserve the notation in Propositionb7.4. 1) The space HL of harmonic polynomials of degree 1 is invariant under the representation T. Put TgIHl=Dzg.Then D1 :gHD', is a continuous, irreducible unitary representation of SO(3). 2) D' is equivalent to Doz. 3) Any irreducible unitary representation D of SO(3) is equivalent to D1 for some 1E N . Proof. 1) Iff belongs to H E ,then T,f belongs to Hc for all g E SO(3), because AT,f = TgAf =0 by Proposition 7.4. Since the matrix elements of DL(g)are polynomials in those of g , D1 is continuous. Dz(g) is a unitary operator on a Hilbert space HZ whose inner product is given by

where dx is the normalized measure on the unit sphere Ss.(In polar coordinates, dx = (4~)-1sin 8 d8dp) 2) Let Dz Ad=Bz. Then Bz is a unitary representation of SU(2). We recall that an irreducible representation of SU(2) is determined by its highest weight (cf. Remark after Proposition 5.11). Let 0

fr(x, y , z) =( x

+iyy.

Thenfz belongs to H z . From the relation (7.3), we have (B'(exp iXslfi) (x, y, z) =( x cos t +y sin t -ix sin t iy cos t)'

+

=(xe-ft

and (7.9)

+ jye-L')Z

=e-Lzt

h(X7

Y7

4

88

REPRESENTATIONS OF

SU(2) AND SO(3)

The finite-dimensional unitary representation BL is decomposed into a direct sum of irreducible representations. Let (7.10) be the decomposition into such a sum. By (7.9), one of the irreducible components, say Unci)has the weight 1. Then by Proposition 5.11, 15 n(i)/2 namely. (7.1 1) 21+ 1s n ( i ) + 1. On the other hand, by (7.10) we have (7.12) 21+ 1=deg Bz2 deg Un(O=n(i)+ 1. From (7.11) and (7.12), we see that 21+1=n(i)+l and that B L = U n ( { ) is irreducible. Since D' Ad = UsL=DLo Ad, DL is equivalent to DLo. 3) follows from 2) and Theorem 7.1. q.e.d. 0

0

Now we express the elements of Ht more explicitly in terms of known functions. Proposition 7.5. Put 0 0 0

Then ( Z , ) l s t s 3is a basis of the Lie algebra 4 3 ) of SO(3).The Lie products between them are given by

[za,Zs]=Zi, [Za, Zi]=Za, [ZI,Za]=Za. Proof. By Proposition 5.7, the Lie algebra ~ ( 3 of ) SO(3) consists of all real skew-symmetric matrics of degree 3. It is clear that (Zi)lscra is a basis of o(3). The Lie products [Z', Z,] can be calculated easily. q.e.d. Remark 1. By comparison between Proposition 5.10 and Proposition 7.5, the two Lie algebras h ( 2 ) and 4 3 ) are mutually isomorphic. This fact also follows from Proposition 7.2. The isomorphism is given by the adjoint representation ad of m(2) which is the differential representation of the adjoint representation Ad of the group SU(2). More precisely, we have seen in Example 5 in $5 that

ad(X,)=Zr

(1 riS3).

We have defined the differential representation for a finite-dimensional

REPRESENTATIONS OF

SO(3) AND

SPHERICAL HARMONICS

89

representation of a linear Lie group. Although the representation T defined by (7.6) is not finite-dimensional, the differential representation T' of o(3) is defined similarly. proposition 7.6. Let X be an element of o(3). A linear transformation T ' ( X ) is defined on the space S of C-fun0 tions on Ra by

Then T' is a representation of the Lie algebra o(3) on F. In particular we have T'(Z1)=z-

a - y-, a az

ay

a

a

T'(Z2)=x- -zaz ax

and

a

a

T'(Z*)=y--x-. ax ay

Proof. Let Z=(zr,) be an element of o(3) and f E fl be a C"-function on RS.Then

So we have for two elements Z = ( z t j ) , and Y=&)

in 0(3),

T'(aX)=aT'(X),

=T'(Y+Z)

and

= T'([Y,Z l )

Therefore T' is a representation of the Lie algebra o(3) on pression for T'(Z$)is easily obtained using (7.13).

F. The ex-

90

REPRESENTATIONS OF

s u ( 2 ) AND s0(3)

Proposition 7.7. In the polar coordinates (r, 8, p), the vector fields T'(Zi)(1 5 is 3) are represented as follows: T'(Z1)= sin p-

a

ae

ap

a q-+cot ae

T'(Z)=-cos T'(Z3)=

e cos p- a

+cot

e sin p- a

ap

--.aPa

Proof. Since x = r sin e cos p, y =r sin e sin p, z =r cos 8, we have

/a ar

ax

i a --

r

=

ae

i a ~-

a

cosecosp

cosesinp

-sine

-

- sin p

cos (0

0

a -

aY

/

\

iazf

Since the coefficient matrix A in the above equality is orthogonal, we have A-l= ' A and sinecosp

cosecosp

-sinp

sine sin p

cos e sin p

cos p

cose

- sin e

0

la

I

\

ar l a -r ae l a -, r s i n e ap,

So we have T'(Z1)=z-

a -y- a az

ay

a

cos e sin p

= r cos B (sin e sin r+ ar

a

sin e r

-r sin e sin p (cos e----)

ar

a ae

=sin p-+cot

a -+ae

cos a ) r sine -ap

a ae

e cos p-. a

ap

The expressions for TI(&) and T'(Zs)are obtained similarly.

Proposition 7.8.

Put

H = iT'(Zs),

E=2-'(iT'(Z1)

- T'(Z2))

REPRESENTATIONS OF

s0(3) AND

SPHERICAL m O N I C S

91

and F= iT'(Z1) + T'(Z2). Then in the polar coordinates, these have the expressions

(7.14) F=e-tP

(-6:

+ i cot 6 ) .a aP

Their Lie products are given by (7.15) [H, E ] = E , [H, F]=-F, [E, F]=H. Proof. The expressions in polar coordinates are easily obtained using Proposition 7.5. The Lie products of H, E and Fare obtained from those of Zl, Za and Zs and from the fact that T' is a Lie algebra homomorphism (Propositions 7.3 and 7.4). q.e.d. Since any element f in Hi is a homogeneous polynomial of degree I, it satisfies (7.16) f(rx, ry, rz) =rff(x, Y, 4 for any r 5 0 . Let D c f ) be the restriction offto Saand Hi' = { D ( f ) l f ~Hi] . Then d, is a linear isomorphism of the vector space Hi onto Hi'. HI is recovered from Hi' by (7.16). The elements of Hi' are called spherical harmonics of degree 1.

Theorem 7.3. Let Hi' be the space of spherical harmonics of degree 1. Then Hi' has a basis (Yim)-lsmL1 defined by Yim(B,v) =e*mrPi-m(cos6), where PIm(x)is the associated Legendre function defined by

Proof. As was shown in the proof of Theorem 7.2, ( ~ + i y belongs )~ to H l . Hence (7.17) Yi1(6,y)=cetl~sini8 (c=const.) belongs to Hi'. Let YLm =F1-m YIi, -15m $1. (7.18) Then YIm satisfies FYim= YLn'-l and (7.19)

92

REPRESENTATIONS OF

(7.20)

HYcm=m YZm

su(2)AND sO(3) - I s m 41.

(7.19) is obvious. (7.20) is proved by downward induction on m starting from 1. (7.20) is valid for m = l by (7.'17) and (7.14). If (7.20) is valid for m, then by (7.15) and (7.19) we get HY2m-l=HFYzm=(FH-F)YZm= (m- 1)YZm-l. Since H = -ia/ap, (7.20) implies that YZmhas the form Ycm(8,p) =etm(Pf,(0). (7.21) By (7.19), the functionsf m and fm-l satisfy

Putting s=COse andfa(@)=pm(s), we get, using (7.21), (7.22) Again putting (7.23) pm(s)=(l -s')-m'aum(s)9 (7.22) can be written as (7.24)

*

durn --&-l.

and d2-m

pm(s)=c(l- - s y - - - - - ( l - - s ' ) 2 . ds2-m Putting c = (- 1)z/l!2c,we get pm(s)=P2-m(s), fm@) =P I - m (e)~ and ~~

Ycm(e,y)=etmPPz-m(cos8). These 21+ 1 functions Yzm(- 1 S m s l ) are mutually orthogonal in Ls(Sa), because (etmp)m.Eare orthogonal in La([O,24). Therefore ( Y t n ) - z j m Lisz a linearly independent family in the (21+1)-dimensional space Hz', i.e., q.e.d. it is a basis of Hz'. Remark 2. Since the identity (7.25)

(1-m)! Pz-"(s) =(- l)"-----P2"(s) (l+m)!

FOURIER SERIES ON COMPACT LIE GROUPS

93

holds, the functions

(-1Sm51) form a basis of Hc'.The identity (7.25) can be proved group theoretically. Let UgCbe the representation of SU(2) constructed in $1. US1is regarded as a matricial representation using the orthonormal basis z ~ ~ + ~ z ~ ~ - ~ / J(l+k)! (l-k)!, (-ZSkSl). Put U g f C = ( ~ i(rg )n) -cc s k , r n + r and dk,,'(Adg)= uk,,'(g). Then the function do.rn'( - Z s m S l ) is invariant under the left translation of the elements in the subgroup K = {exp fZ& E R)of G = SO(3). Hence do,,( is a function on the sphere Sa=K\G. It can be proved that if the coset Kg corresponds to a point on S%with the spherical coordinates (8, p), then eim~Plm(cos 8)

The proof of (7.26) is similar to that of Theorem 7.3. Using (7.26) and the relations exp( - 8Za)expxZI=exp&exp( -BZ~),

we get (7.25).

$8. Fourier series on compact Lie groups The theory of Fourier series on SU(2)developed in $6 can be generalized to any connected compact Lie group G. In this section we give an outline of the theory by using freely the results of representation theory of compact Lie groups (cf. J. P. Serre [l], M. Sugiura 121). For the details of the proof, see Sugiura [4]. Throughout this section we use the following notation. G : a compact connected Lie group; Go:the commutator subgroup of G; T: a maximal toral subgroup of G ; l=dim T=the rank of G ; p : the rank of Go;n=1+2m=the dimension of G (where m is the number of positive roots of Go);g: the Lie algebra of G; 8": the complexification of g; t: the Lie algebra of T; dg: the normalized Haar measure of G,La(G)=LB(G, dg); Ck(G):the set of all k-times continuously differentiable complexvalued functions on G ; ~~A~~=Tr(AA*)l'a: the Hilbert-Schmidt norm of a matrix A; 6 :the dual of G = the set of all equivalence classes of irreducible unitary representation of G. In this section, a finite-dimensional,matricial, unitary representation of G is called, simply, a representation of G.

94

REPRESENTATIONS OF

SU(2) AND SO(3)

The difl'erential representation U' of a representation U of G is a representation of the Lie algebra g. Instead of writing U ' ( X ) , we write U(X). Since U(t) is a commutative set of skew-Hermitian matrices, there exist a unitary matrix Q and pure imaginary-valued linear forms 21, ..., r7r on t such that

RdH)

QUW)P=[

-,.,.*

0

1

-...._____

RdH) for all H in t. The linear forms R1, ..., I r on t are called the weights of U. In particular, a non-zero weight of the adjoint representation Ad of G is called a root of G with respect to T. The set of roots with respect to T is denoted by R:We fix once for all a positive definite inner product (X, Y) on g which is invariant under AdG. Put 1 x1=(X, X)lfa.A pure imaginaryvalued linear form (in particular a weight of a representation)I is identified with an element h, in t which satisfies R(H)=i(h2, H) for all H in t. We put R(H)=i(R, H) and (1, p)=(h2, hJ. Let r(G)be the kernel of the homomorphism expa: t+T. Then r(G) is a discrete subgroup of t of rank I (i.e. r(G)zZ l ) . Let Z be the set defined by Z= [ R E t l ( R , H ) E 2nZ for all H E r(G)}. Then an element R in Z is called a G-integral form on t. The set Z of all G-integral forms coincides with the set of all weights of the representations of G. We choose once for all a lexicographic order @ in t with respect to a basis. Let P be the set of all positive roots with respect to @. The elements of D = {A E ZI(2, a)LO for all a E P) are called dominant G-integral forms. The maximal element with respect to @ in the set of all weights of a representation U is called the highest weight of U. A linear form on t is a highest weight of a representation of G if and only if R is a dominant G-integral form, i.e., R E D. Since two irreducible representations of G are equivalent if and only if their highest weights coincide (cf. Serre [l] Ch. VII Th. 1, Sugiura 121 Th. 2), there exists a natural bijection from D onto the dual 6 of G which maps a dominant G-integral form R E D to a class of irreducible representations (U2) having highest weight R . We identify 6 with D by this bijection. If G is a direct product G1 x Ga of two compact Lie groups G1 and Gs, then the dual G of G is identified with 6, x 6,. The identification is realized as follows. If (A, p) is an element of x 6 , and U ' and Up are representations in the classes R and p respec-

el

FOURIER SERIES ON COMPACT LIE GROUPS

95

tively, then the representation UQU.: (gl,ga)wUg,'@Ug2pis an irreducible representation of G. Conversely, every irreducible representation U of G = GIx Ga is equivalent to a certain representation of the form U2@Up(RE P E 6%). By the mapping ( A , p ) w ( U L @ U . ) , 6, x 6, is identified with 6. In particular, if to is the center of g and To be the connected Lie subgroup of G with the Lie algebra to, then Gox To is afinite covering of G and G is isomorphic to (Gox To)/F,where F is a finite normal subgroup of Gox To,8 is the subset of 8, x foconsisting of representation classes whose kernels contain the subgroup F. Since To is commutative, any irreducible representation of TOis one dimensional (Ch. I. Corollary to Theorem 2.1.). So the degree of the ,a E f0) is equal to deg irreducible representation UA+r= UL@Up ( A B 60, UA =d(n). Hence, the degree of the irreducible representation U2(2E 6) is given by Weyl's dimension formula,

el,

where 6 =2-lC a (cf. Serre [13, Sugiura [2n. .€P

Since d(2) is a polynomial of degree m, there exists a positive constant C such that (8.2) d(~)sCl,p for a n y ~ ~ D ~ = D - { 0 ) . The Casimir element L? of the universal enveloping algebra U(g) is defined by n

(8.3)

-sa=

1g''X#X, 6.'-1

is a basis of g and (Xc,X,) = g t j , (g") =(g',)-l. where (X')l If g is simple (i.e. if the only ideals of g are {0)and g), then the operator L? defined by (8.3) differs from the Casimir element in Definition 9 of $5 by a scalar factor. The Casimir element Q belongs to the center of U(g). (Proposition 5.14). Let 2 be an element of 6. Since the representation U' and its differential representation are irreducible, V ( Qis) equal to a scalar operator cl by Schur's Lemma (Ch. I. Proposition 2.4). (Remark Schur's Lemma for finite-dimensional representations is valid not only for groups but also for Lie algebras (or for any set of operations on a vector space).) The value of the scalar c can be calculated easily. The result is as follows. Proposition 8.1.

1) For any compact group G, we have ($2) =w(R)1

96

REPRESENTATIONS OF

SU(2) AND SO(3)

where w ( R ) = ( R ,n+26). 2) The matricial element uA<,of UAis an eigenfunction of Q, regarded as a differential operator on G: sau,,I=

o(R)Ur,".

Proof. 1) There exists a basis {E.(a E R),Hi(l 5 is l ) } of gc satisfying (EuyE-,) = 1, (Hi, Hj) = at,, E, E-, E g , i(E, -E-,) E g, Hr E t and [H, E.] =a(H)E. ( a E R). Then we have

+

where Ha=[E,, E-.I is equal to ih.. Let x z 0 be a weight vector corresponding to the highest weight R. Then x is a vector in the representation space of UA which satisfies U'(E,)x=O for all a E P and U2(H)x=R(H)xfor all H E tc. Since U'(H,)x =R(H,)x = -(A, a)x, we have

+

= {(R, 26) ( R , R ) I x. 2) is proved in the same way as Proposition 5.16.

Proposition 8.2.

Iff belongs to Czx(G),then

= o(l)Xf^(~) for any R E 6, where w(R)=(R,R +26). Proof. Using Proposition 8.1, Proposition 8.2 is proved in the same way as Proposition 6.1.

LEMMA1. Let n = (nl, . . . , nI) be an element of L = 2'- {O}. Then Eisenstein's zeta function

converges if and only if 2s > 1. Proof. Let llxll be the uniform norm of x in R'i.e. llxll = max ( Ixl I , . . . , IxIl } for x = ( x l , .

. . ,xI).

97

FOURIER SERIES ON COMPAm LIE GROUPS

Then for each natural number k, the number of elements in the set L(k) = {neZ'I llnll 5 k } is equal to (2k 1)'. Hence the number T(k) 01 elements of the set S(k) = {~EZ'Illnll = k } is equal to

+

"23

T ( k ) = (2k

+ 1)' - (2- 1)' = 2 1

(2k)'-"-'.

Hence T ( k ) can be expressed as (8.4)

T (k) = ulkr-'

+ a2kr-2+ . . . + ul,

a, 5 0,u1 > 0.

Put (8.5)

Then we have

(8.6) Hence we have (8.7)

"

1

"

1

k- I

k- I

As is well known, Riemann's zeta function

n-* converges if and only "-1

cl

if s > 1. Hence (8.7) shows that (s) converges if 2s > 1. Conversely if 2s g 1, then (8.4) and (8.6) implies

Hence 5; (s) diverges if 2s S 1.

q.e.d.

LEMMA2. Let D be the set of all dominant G-integral forms and put Do = D - { 0).Then the series 11I -2r converges if and only if 2s > 1. kD0

Proof. Let I be the set of all G-integral forms and put Zo = Z - { 0).It 11[-2J converges if and only if is sufEcient to prove that the series

1

&I0

2s > 1. Let A1, . . . . ,Itl be a basis of the lattice Iand < x, y > be the inner product on t defined by

98

RJ3PRBEPJTATIONSOF

Su(2) AND SO(3)

Then we have

-’ converges if and only if 2s

Hence by Lemma 1, the series kI0

> 1. There exists a positive definite symmetric transformation A such that (x, y ) = < Ax, y > . Let a and b be the largest and the smallest eigenvalues of A. Then we have the inequalities a

< A, A > 2 (A,A) 2 b and a 2 b > 0.

1 (A, A)-’ converges if and only if 1 -* converges. Therefore 1 (A, A)-’ converges if and only if 2s > 1. Hence the series

LlO

LCIO

kI0

”hearem 8.1. Iff belongs to C2”(G)and 2k>2-’n = 2-ldim G, then the Fourier series off

c

d(wr(f(lz)u,”)

IrD

converges to f(g) absolutely and uniformly on G. Proof. The proof is essentially same as that of Theorem 6.1. In fact, since

ITflf(W,.‘>ISllf(A)llIl~,rll=~(~)”211f~~)11 =4A)1/2CU(IZ)-k II(.caWA(411 5 4 A ) 1 ’ z I nl -2kll(.cakf)A(A)ll,

FOURlER SERIES ON COMPACT LIE GROUPS

Theorem 8.2.

99

Let f be a continuous function on G. Then f belongs to

C"(G) if and only iff is rapidly decreasing i.e.

lim 121" llf(A)ll

ILI-+-

=0

for any k e N .

Proof is the same as that of Theorem 6.2. Definition 2. The space C"(G') = H(G) is topologized by the set of seminorms { p , ( f ) = IIWIL I D E U ( ~ ) . Mn(C)J ~ ( A ) E M (C) ~ ( ~for ) all A € & The space 9 ( 6 )= { Q, : 6

-

n-1

and

is rapidly decreasing} is topologized by the set of seminorms { 4 , ( f ) =P;; In)' llQ,(A)ll 1s h 01. Q,

Theorem 8.3. The Fourier transform 9 7 f -f is a topological isomorphism of g ( G ) onto 9(6). Proof is the same as that of Theorem 6.3.

This Page Intentionally Left Blank

The Fourier transform and unitary representations of Rn

$1. Rapidly decreasing functions Let Rn= R x ... x R (n factors). Then Rn is a vector space over R. The canonical inner product in R" is defined by n

(x,Y )= LX*Y' i-I

for x = ( x l , ..., x,) and y = ( y l , ...,yn). The norm 1x1 of x E Rn is defined as 1x1 =(x, x)*. _ Theorem 1.1. 1) For any 6E Rn, the mapping xc : xwei(".o)is an irreducible unitary representation of the additive group Rn. 2) Any irreducible unitary representation x of Rn is equivalent to xo for some E E Rn. 3) xE is equivalent to xc. if and only if ProoJ The proof of 1) is trivial. 2) Since Rn is commutative, an irreducible unitary representation x of Rn is one-dimensional (Ch. I, Corollary to Theorem 2.1). By replacing x with an equivalent representation, x can be assumed to be a continuous homomorphism of Rn into U(1). Let (ek)lrainbe the canonical base of Rn: el =(1, 0, ..., 0), ..., en =(0, ..., 0, 1). Then the mapping t k : R+Rn is defined by €=El.

Ck(Xk)

=&A

.

The mapping x°Ck is a continuous homomorphism of R into U(1). By Ch. I, Theorem 4.1, there exists a real number Ex such that ( p c x ) ( X k ) =e"k"k for any X&E R. So we have n

n

d x ) = x (zxkek)= x ( c d x k ) ) k-1

k-1

101

102

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

R"

- &elr1+...+Fnxn) -i d x ) . 3) Since U(1) is commutative, xe is equivalent to if and only if I F = xC,. Moreover xe =z E , if and only if ( x , c) ~ ( xE'), mod 2 x for all x E R". So putting x=ek ( l ~ & n ) we get & = & ' + h k Z (Igkgn)for some xel

. all x k E R , the integer n k must be integer n k . Since 2nkxkn=0 mod 2 ~for equal to 0 for each k. So xE= x F . if and only if E=f'. By Theorem 1.1, the dual (Rn)"(the set of all equivalence classes of irreducible unitary representations) of Rn is identified with Rn by the canonical bijection z e ~ €In. the same way as for the compact groups, the Fourier transformf of a function f on Rn is dehed as a function on (R")" = Rn. (See Definition 3 below.)

Definition 1. Let dm(x)=(2s)-"ladx= ( 2 7 ~ ) - " ' ~ d.x..~dx, be the normalized Haar measure on Rn. Then the Lp-norm llfllP is defined by

The space Lp (1 s p < + a) is the set of all complex-valued measurable functions on Rn with finite LP-norm. Two functions in Lp are identified with each other if they coincide almost everywhere. Then Lp is a Banach space with the norm 11f ]Ip. The space La is the set of complex-valued continuous functions f on R" satisfying lim f ( x ) = O with the uniform lxl-+norm llfll- = SUP If(x>l* xrRn Remark. The symbol Lmis often used for the space of all essentially bounded functions. Our space La is a proper subspace of the latter space. Definition 2. The convolution f * g of two functions f and g in L1 is defined by

De6nition 3. The Fourier transform f = T of f a function f in L1 is defined as (1.1)

f(c) =

1

f(x)e-'(=*t) dm(x), (E E R").

Rn

The conjugate (or inverse) Fourier transform N * f of f is defined as

103

RAPIDLY DECREASING FUNCTIONS

=f(-X) . Proposition 1.1. 1) The Fourier transform{ of a functionf in L1 is a bounded continuous function on R". It satisfies

Ilf 11- 4Ilf 111* (1.3) 2) Iffand g belongs to L1,then we have

( f *g>- =AProof. (1-4) we have

Since If(x)e-s(t.Ol= If(x)l

If (€)I5 1R ,If (x)ldm(x)=llflll for any E E Rn.For any Eo E Rn,we have l i d ( € )= f ( ~ o ) ('(0

because we can apply Lebesgue's dominated convergence theorem by (1.4). 2) By Fubini's theorem we have

Definition 4. For any multi-indexm =(ml, ...,m,) E N",and x E R", put 1m1=lm1l + ... Im,l, x"=x1"1 ... Xn"" and D"=(--)"l l a ... l a i axl i ax, *

+

(--)""

A complex valued Cm-functionf on Rn is called rapidly decreasing if

(1.5)

pN,m

cf)=suP (1 + Ixla)>"l(D"f)

+

O0

104

FOURIER TRANSFORM A N D UNITARY REPRESENTATIONS OF

R"

for all N E N and m E Nn.The set of all rapidly decreasing functions on R" is denoted by 9= 9 ( R " ) . Example 1. Any C"-function with compact support belongs to 9.For a concrete example of such a function, see Lemma 1 below. Example 2. The function u(x)=e-lsil belongs to 9,because D*u(x)= (2i)j"j xmu(x)and Jim xmu(x)=O for any m E N". Irl*+-

+

Proposition 1.2. 1) 9c Lp, for any p E [l, m]. 2) 9 is a metrizable, complete, locally convex, topological vector space (FrCchet space). €'roo$ 1) Since

+

If(x)l $ P N , O C f ) (1 Ixls)-N for any x E R", f is bounded and belongs to L". If 1 j p < Q) the above inequality implies (1.6)

+

Ilf IlP SPN,O(f) 1/-.(1+ Ixl2)-PNdm(x)1l ' P <

4-

Q)

for sufficiently large N (take N > n/2p). 2) 9 is a vector space by its definition. Since 9 is topologized by the I N E N , m E N")of seminorms, 9is metrizable. countable family { p N P m Then (pN,m(fn))n.N is a Cauchy Let (fn)n.N be a Cauchy sequence in 9. D ~ ~ converges .) unisequence of real numbers and ((1 I X ~ ~ ) ~ ((X))nrN formly to a function. Thus, for any 1, m E N", the sequence (Xz(Dmfn) ( x ) ) ~ , N converges uniformly on R" to a function fr,m(x)=x'(D*f) (x). q.e.d. Hence the sequence (fn),,eN converges to f in 9. The most important and interesting property of the Fourier transform lies in the fact that 5 transforms the operation of differentiation into the operation of multiplication of coordinates and vice versa. More precisely we have the following proposition.

+

Proposition 1.3. I f f is a rapidly decreasing function, then we have (1.7) (Dmf)"(()=Emf(€) for any m e N" and E E Rn. Moreoverf is a C"-function and satisfies

(W)(€I=(- 1)'"'(Xrnf(X))"(€). Proof. From integration by parts, we have

(1.8)

RAPIDLY DECREASING FUNCTIONS

105

because lim f(x)=O. By repeated differentiation, we get (1.7). s -is

Since xmf ( x ) belongs to 9 for any m E Nn, Ixkf(x)e-i("*oI= Ixkf(x)I belongs to L1(Rn).Hence we can differentiatef(E) under the integral sign and we obtain

k " We have proved (1.8)when m=(O, ..., 0, 1, 0, ..., 0).Repeating the operation of differentiation, we get (1.8) for every m E N m . q.e.d.

Proposition 1.3 implies that S and T *are continuous linear m a p 'into 9'. (See Theorem 1.2 below.) To prove that fl is surjecpings of 9 tive, we use the elementary solution uc of the heat equation au/at=Au.

Proposition 1.4.

1) Let u(x)=e-ltlalt. Then we have u E 9 and

(1 *9)

2) Let u@)

6=u. =(2t)-n"e-I="l4' for

t > 0. Then

rir (t)=e-glcla, y * f l u , = u t .

3 ) Iffis bounded and continuous, then we have

lim (ut*f)(x)=f(x) t*+O

for all x E R". Proof. 1) Since u(x)=

fi e-*k21aand e-"*.c' = fi e-(=ktk,it is sufficient

k-1

2) Since uc(x)=(2t)-n%(x/4/), we have

k-1

106

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

Rn

Sincef is bounded, the integrand is 5 (1 f Jl,e-I"l' which belongs to 9 ' c L'. So we can apply Lebesgue's theorem and get lim ( u t * f ) (x)=H-"~'

e-'*la f ( x ) d z = f ( x )

t-to

by the continuity off.

q.e.d.

Theorem 1.2 (L. Schwartz). Iff belongs to 9, then the Fourier invertion formula (1.19)

=I

f (x)

f(E)etCa,6)dm(0

RR

holds for any x. Namely we have (1.11) F * F f = f and F F * f = f for any f in 9. The Fourier transform fl is a topological isomorphism of 9 onto 9 and the conjugate Fourier transform fl*is the inverse mapping of fl. Proof. First we prove that

(1.12) F9c9, x*9c9. Let f E 9 and 1, m E Nn.Put g(x) = (- l)lml xmf(x). Then g E 9, and by (1.8) we have S=Dmf and Ez(Dmf)(E)=(Dcg)*([).Since D l g E 9 c L ' .

RAPIDLY DECREASING FUNCTIONS

107

(Dlg)*is bounded by Proposition 1.1. Hence f belongs to 9'.Since ( S * f )(()=f(-E), S*fbelongs to 9 '. We have proved (1.12). Proposition 1.4. 2) and Fubini's theorem prove that

=IRnGt(f)f(t)et(**C)dm(E).

Since f~ 9 c L 1 and lim Gt(E)=lim t*+O c++o (1.13) tends to S,nf(E)ef(**o

ME)

e-'Itla=

1, the right hand side of

=(.F*T~)

(XI

as t (>0) tends to zero. On the other hand, the left hand side of (1.13) tends tof(x) by Proposition 1.4, 3). We have proved (l.lO), i.e. S*flf=$ Since ( S *(€)= f) f(-E), we have

s

(flfl*f) (x) = f(-E)e-'(*.E) d m ( ~ ) = (fl*flf) (XI

=fO.

We have proved (1.11) and that flis a linear isomorphism of 9 'onto 9. Now we shall prove the continuity of R.Suppose a sequence (fn)nsN converges to f in 9'. The inequality (1.6) proves that (fn) converges to f i n L'. Then ( f n ) n . N converges uniformly tof by (1.3). Replacingf(x) by Hence 5 : fwfis, Dlxmf(x), we see that (f,JnCN converges to f in 9'. continuous. Since ( X * f )( E ) = ( N f ) ( - E ) = ( S X f ) (E) and S : ft+f (f(x) =f(-x)) is continuous in 9, fl*=f l - 1 is also continuous. Therefore fl is a homeomorphism of 9 'onto 9'. q.e.d. Theorem 1.2 is the basis of harmonic analysis on Rn. All the main results in this chapter are obtained from Theorem 1.2. As the first application of Theorem 1.2, we prove the following theorem.

Theorem 1.3. For any f in 9 'and g in L1,we have

-

(1.14)

/ R n g ( x ~ ) d m ~=x/Rn~(~)f(c) ) dm(E).

In particular, the Parseval equality

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF R"

108

IIfIIa=

(1.15)

IIPII,

holds for any f E 9'. Proof. By the inversion formula (1.10) and Fubini's theorem, we have

=/ s ) / g ( x ) e - i c z J )

dm(x)dm(t)

q.e.d. Putting g= f i n this equation, we get (1.15). Now we shall prove that .9' is dense in the Banach space L p for all p (1 s p s + a). We need some lemmas.

LEMMA 1. For any 6 >0, put

m)=[ e ~ p ( ~ ) i f , x E u a = { x €I ~ In<~~ I , if x # Ua,

0

1.n

and ha=Na-lfawhere Na=

fa(x)dm(x). Then the function ha satisfies

the following properties: 1) ha E C"(Rn), 2) h a ( x ) 2 0 for any x E R", 3) supp (ha)= Oa,4) S.,ha(x)dm(x)= 1. Proof.

Let

The n-th derivativef (n) ( x ) off at x +0 is given by if x>O if x
+

+

f(n'(+

+

0) -- lirnf'n) ( x ) = lim t b pn %-+O

=f ( n ) (- O),

I-.+-

(+)e-r = o

109

RAPIDLY DECREASING FUNCTIONS

f ("I (0) exists and is equal to 0. So f"' is continuous at 0 and f is n times continuously differentiable for any n E N . Hence f belongs to C" (R). Since

ga ( x ) = 62 -

1x12

=62

- (XI2 + . . . +

x.2)

is a C"-function on R", the composite function

f a ( 4 =fV2 - 1x13 is a C"-function on R". Hence 1 ) is proved. The properties 2), 3) and 4) of ha are obvious. Definition 5. The vector space of all complex-valued continuous functions with compact supports (support depending on the function) is denoted by L =L(R").

LEMMA 2. Iff E L and g E 9, then h=g* f belongs to 9'. Since supp(f ) =K is compact, the function

Proof.

h(x)

=I K

&-Y)f(Y)MY)

is differentiable arbitrarily many times and (1.16)

SX

(Drnh)(4"

( D 3 ) (x-Ylf(Y)dm(Y).

In particular h is a C"-function. Since K is compact, there exists a> 0 such that KcU,. If y € K and I x l > a + l , then ~ x - y l 2 ~ x l - l y l > l ,Iyl
ixICl(Drn4 ( x ) l s / + (= -Y )lf ( Y ) l M Y ) K

if 1x1 >a+ 1 . Since g E 9 ' and f is bounded, lim +(x-y)l f(y)l =O for any y E K. Ill+-"

On the other hand +(x-y)l f ( y ) I , being a continuous function of y, is bounded on the compact set K. So we can apply Lebesgue's theorem to (1.17) and prove that lim Ixl'l(Dmh)(x)l =O Ill-+-

for any 1 E N and m E Nn.The last equality proves that h belongs to 9. q.e.d.

110

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

R"

LEMMA 3. Iffbelongs to L, then we have lim Ilhd*f-fllp=O for everyp (1 r p $ d4O

+ a).

Proof. l ) p = + w . Sincef is uniformly continuous on R",for any E >0 there exists a positive number 6 ( ~ such ) that if Ix-yl
If(x)-f(v)l<~. Hence if 0 <6
'

If(x-y)-f(~)lha(y)dm(y)S e for all X E R".

[Ud

2) l ~ p < + w . Since supp(f) (the support off) is compact, there exists a positive number a>O such that supp(f)c U,. Then

+

supp (hd*f)c supp (hd) supp (f)c u d + U . If 0 c 6 < Min(G(E), l), then r

(1.18)

COROLLARY 1 to Lemma 3. For any fixed function f in L and E >0, there exists ij0>0 such that if 0 < 6 <6o then Ilhd*f-f IIP <E for allp E [l, + m]. Proof. We can take the positive number (I so large that m(UU+1)L1. Then m(Uu+l)lfpgm(Ua+l) for allpE [l, So the inequality (1.18) proves that

+ m].

Ih*f-fllp 5 m(Uu+l)llhd*f-fllfor allp E [l,

+ w].

q.e.d.

COROLLARY 2 to Lemma 3. For any f in L, there exists a sequence in 9 'such that

(f,JnrN

111

RAPIDLY DECREASING FUNCTIONS

lim Ilf-fnllp=Ofor allpE[l, +a].

n-+m

Proof. Put fn=hl~(n+l)*f. Then fn belongs to 9by Lemma 2. By Corollary 1 to Lemma 3, lim Ilf-fllp=O for anypE [l, a].

+

n-m

Theorem 1.4. The space 9of rapidly decreasing functions is dense in Lp (1 s p 5 00). Proof. Since L is dense in L p (cf. Appendix D. 5), for any E>O, and f E Lp, there exists a function g in L such that

+

Ilf-gllp <@By Corollary 2 to Lemma 3, there exists an element h in 9 such that Ilg-hllp

<4.

So we have

Ilf -hllP <6 q.e.d.

COROLLARY to Theorem 1.4. If 9 belongs to L1 n La, there exists a sequence (fn)n,nr in 9’ such that lim Il p -fnlll= n-m

lim IIp-fn

115

=0.

n-m

Proof. By the Appendix D. 5, for any integer n>O, there exists an element hn in L satisfying llp-hnlli
n-oo

Theorem 1.5 (Uniqueness of Fourier transform). Letfbe a function in L1.Thenf=O if and only if f = O in L1 (i.e. f(x)=O for almost every x). Proof. Iff =0 in L1 thenf= 0. Conversely let f be a function in L’ and suppose that f= 0. For any function h in L,there exists a sequence (hn)n,N in 9 such that lim llh-hnl[m=O n-m

by Corollary 2 to Lemma 3. Then we have

112

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF R"

On the other hand by Theorem 1.2 and the assumptionfl=O, we have (1.20)

p(x)mwx) =/f(e)fi.(E)dm(€) -0.

(1.19) and (1.20) prove Sf(xx)h(x)ciin(x) =o for an h E L.

By replacing h(x) by h(x-y), we get (1.21) h*f=O for all EL. For any e>O, there exists a 6 > 0 such that IILaf-flll<E (cf. Appendix D.6) Then

(1.21) and (1.22) prove that IIf l l l < e i.e.,

if Isl$6.

for any e>O,

[ I f 1I1=O and f=O in L1.

q.e.d.

In the above proof, we have proved the following.

s

COROLLARY 1 to Theorem 1.5. If a function f in L1 satisfies

f ( x ) h ( x ) h ( x )=0 for all h E L then f ( x )=0 for almost every x.

COROLLARY 2 to Theorem 1.5. Let f be an integrable function on Rn. Then f coincides with a function g in 9 almost everywhere if and only if the Fourier transformfl belongs to 9. Proof. Iff =g E 9 almost everywhere, thenf= g E 9by Theorem 1.2. Conversely supposedf E L1 andfE 9.Then g = Y * Y f belongs to 9 by Theorem 1.2. Since T g = S X * Y f = T f by Theorem 1.2, we get f =g almost everywhere by Theorem 1.5. q.e.d.

Theorem 1.6 (Riemann-Lebesgue). The Fourier transform fl maps L1 into L". The image f l ( L * )= el is dense in L-. Proof. Let f be a function in L1. Thenfis a continuous function (Proposition 1.1). By Theorem 1.4, there exists a sequence (fn)"
RAPIDLY DECREASING FUNCTIONS

113

Then by (1.3), we have (1.23)

i.e., (fn)nrN converges uniformly to f. Sincefn E 9 =9c L", and L" is a Banach space with the norm 11 !Im, (1.23) proves thatf belongs L". Since 9 c L', we get by Theorem 1.2 29 =.yc t1cL".

Since 9 is dense in L" (Theorem 1.4),

el

is dense in L".

q.e.d.

Theorem 1.7. Iff and g belong to 9,then 1) U g l A=f*k 2) f * g E 9. Proof: By Proposition 1.1, 2),

(1.24) 9-(f *g)=.Kf T g . Since 9 =9 (Theorem 1.2), we can replace f and g in (1.24) with f and t. We get by Theorem 1.2

9-(f*d)=F"f

- X'g=fi=(fg)* = S Y f g )

where {(x) =f ( - x ) . Applying N-l on both sides of the last equality, we f*g belongs to 9. get 1). Sincef g belongs to 9by the definition of 9, Since 9=9, we have proved that f * g E 9for any f and g in 9. We now give a sufficient condition for the inversion formula (1.10) to be valid. The result will be used later. Theorem 1.8. Iff and f are integrable, then f ( x ) is equal to h ( x ) = (x)= (9-.K *f)(x) for almost all x E Rn. Hence if we assume additionally thatfis continuous, then f ( x ) = h ( x ) for all x E R". Proof. Let g be a function in 9. Then by Theorem 1.3 and the assumption f E L', we have

(x*fl)

[ P(E)rnW€).

[ R n f ( x ) m d m ( x )= R"

Replacing g(E) by its definition and using Fubini's theorem we get

,.

c

Hence we have proved that

for any g E 9. Let (O be a continuous function with compact support K.

114

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

R"

By Lemma 3, ga = h a * p ~ converges 9 uniformly to p. Then by (1.25) we have

for any EL. Thus we have f ( x ) =fo(x) for almost every XER" by Corollary 1 to Theorem 1.5. Since ( F * f ) ( x ) = (Ff) (- x) = ( T f ) " ( x ) and Tf= F*f, we have FF* f = F ( 9 - f ) " =F * S f = f 0 . q.e.d. We give one more proposition which will be used in Ch. IV, f35. Let m and k be two multi-indices in N".The relation m 5 k is defined as m, 2 k, for all j~ { 1,2, . . . , n}. We put

For any two functionsf and g in C'"'(R"),we have the following generalized Leibniz's formula (1.27)

D" ( f g ) =

xc)Dkf -D"-'g.

K m

If the number n of the independent variables is equal to I , then this formula is proved by the induction on m. Once the one variable case is established, the general case is proved by the induction on the number n of the independent variables. Proposition 1.5. For any multi-indices m and k in N" and for any function f in C"(R"),the function D " ( 9 f ( x ) ) is a linear combination of the functions of the type (1.28)

xi (D'f (x), ( l , j ~ N "I , 5 m).

Proof. We have (1.29)

c(")

D" (x"f(x)) = I s m 1 (D"--'x") P'f)(XI

by (1.27). Since

PLANCHWEL THEOREM AND DECOMPOSITION

115

(1.30)

(otherwise), the right side of the equality (1.29) is a linear combination of the q.e.d functions of the type (1.28).

$2. The Plancherel theorem and the decomposition of the regular representation In this section, we shall study the Fourier transform of square-integrable functions.

Theorem 2.1 (Plancherel). 1) The restrictions 9-19' and fl*ISp of Fourier transforms S and S*on 9 can be uniquely extended to unitary operators F and F* on La.F* is the inverse of F. 2) F q = f l ( ~ a n d F * q = f l * q f o r a n y q o L ~ n L ~ . 3) For any (O E La,put qn(c) =

v:(c) =

1 1

q(x)e-*(", W m ( x )

Izlsn

p(x)e*(zs~ m ( x ) .

lzl sn

Then we have the limits in L': Fv=lim q,,

and

F*q=:p_qZ.

n-+*

Proof. 1) For any q in La,there exists a sequence (fn)nrN in 9 such that

(Theorem 1.3). Since I I S ( f n -fm)lla=

llfn-fmlla

N a Cauchy sequence in La and by Theorem 1.3, the sequence ( S f n ) n c is has a limit 4 in La.Define Fv=$=lim fin, n-or

116

FOURIER TRANSFORM AND UNITARY REPRESENTATIONSOF

Rn

Fp is independent of the choice of (fn). For, if (g,),,, is another sequence in 9 converging to p in L9, then the sequence (hn)nrN defined by h2n= fn and hln+l=gn converges to p in La and

Similarly F * is defined by putting F*p=lim X*fn. n-roo

F and F * are linear mappings of L2 into Lz.By the Parseval equality { 1.15), we have

(2.1)

-

IIFqlla=lim IIXfnll2=lim Ilfnll~=11~l1~ n--

n-m

Thus, F is an isometry of L2 into L2.Similarly (2.2) IIF*pII2= Ilplla Let +=F*p. Then we have

(2.3)

llp-F+llaS II~-fnll~+ llfn-F+IIa

IIFF*fn-FF*~lI~ as n+ + co by (2. l), (2.2) and the relation FF*fn = T X *fn =fn (Theorem 1.2). Hence p=F+=FF*p and = llp-fnlla+

=21/9 -fnll,+O

(2.4)

FF* = 1.

Similarly we have (2.5)

F*F= 1.

(2.4) and (2.5) show that F and F * are bijections of La onto itself and that F* =F-l. So F and F * are isometries of Ls onto La, namely, unitary operators on La. Since 9is dense in La, the unitary operators F and F* are uniquely determined by respectively. and .F*~.Y 2) By Corollary to Theorem 1.4, for any p E L1 n Lz, there exists a sequence (fn)n,N in 9satisfying lim

119-fnIlr

n-m

By Proposition 1.1,

=lim Ilp -fnIIa

II$-fnll.sSIl~-fnlll

and we have

Em Il$-fnll-=O. n-m

Therefore,

=O.

n-m

PLANCHEREL THEOREM A M ) DECOMPOSITION

117

Since lim IIFp-Ffnlls =lim Ilp-fnlla -0, n-00

(1-01

there exists a subsequence (Ffnk)krN of (Ffn)n.Nsatisfying

(2.7)

lim (Ffnk) (E)=(Fp) (€) for almost every € E R".

k-+oo

Since f n = X f n = F f n , (2.6) and (2.7) prove that (Fp)(€)=&) for almost every f E R". We have proved F q = g = S p in L'. Similarly we can prove F*p = F * p .

3) Let

Then fn belongs to L1nLpand

Hence by 2), we have

' Fp=lim Xfn=lim p,, in L n*oo

**01

Similarly we have F*p=lim pn* n-oo

.

. q.e.d.

Definition 1. The unitary operator F in L' =L1(Rn)defined in Theorem 2.1 is called the La-Fourier transform or Fourier-Plancherel transform. For the compact group T,the Parseval equality gives us the irreducible decomposition of the regular representation (Ch. I, $4). Now we shall study the irreducible decomposition of the regular representation of R". The Fourier transform enables us to pick out the irreducible components. The essential part of the analysis lies in the following theorem.

Theorem 2.2. defined by

Let a € Rn. Put S, and R , be the linear operators on L' ( s a p ) (0=etCa. p(€)

(Ray) (x)=p(x+a).

Then we have

(2.8) 9 R a f = S a F f for any f E Y and (2.9) FR, =S,F for any a E R" . S : a H S , is a unitary representation of R" which is equivalent to the right regular representation R".

118

FOURIER TRANSFORM

Proof.

AND UNITARY REPRESENTATIONS OF R"

Let f be an element in 9' Then . we have

=/Rnf(x)e-i(z-".

h(x)

=eSca* ()(Sf) (E)=( S a Y f ) (El.

We have proved (2.8). Since S, is a unitary operator on La,both sides of (2.9) are unitary operators on La and coincide with each other on a dense So we get (2.9). The last part of Theorem 2.2 is clear from subspace 9. (2.9). q.e.d. For the compact group T , a relation similar to (2.9) is valid. For any element a E T , let S a be the unitary operator on L a (Z )defined bY (2.10) (Sap) (4=etnap(n). Then we have

La(T) (2.1 1)

FRa =S aF

F1 Lyz)

R.

La(T) .lF LyZ)

Sa for any a E T . Let (on be the element in L a ( Z )defined by ~ , , ( m=)a.,, Then (p,Jnrzis an orthonormal basis of L a ( Z )and S, is a diagonal matrix So S is a direct sum of irreducible representations with respect to ((o,,),.~. ( X & ~ Z : S = @ x,,. The Fourier transform F transfers the regular reprenrZ

sentation R to S = @

Xn.

ncZ

Similarly the relation (2.9) gives the irreducible "decomposition" of the regular representation. However there exist some difficulties in carrying out the "decomposition" for a non-compact group R". For example, the analog of the function pn on Z=f cannot be constructed for R". If we put yE(7)=aE1then yE=O almost everywhere and (pe=O in L2. In fact we can prove that the regular representation R of R" cannot be decomposed into a direct sum of irreducibIe unitary representations as the following proposition shows.

Proposition 2.1. The regular representation R of R" is not irreducible, but the representation space La of R contains no closed irreducible in-

PLANCHEREL THEOREM AND DECOMPOSITION

119

variant subspace. R can not be decomposed into a direct sum of irreducible representations. ProoJ Since Rn is commutative, every irreducible unitary representation of Rn is one-dimensional (cf. Theorem 1.1). Hence an infinite-dimensional representation R is not irreducible. Now suppose that V is a closed irreducible invariant subspace of L2. Then dim V= 1. Let f be a basis of V. Then, by Theorem 1.1,RI Vis equivalent to xc for some E E Rn. Namely, we have (2.12) Raf = X44f for all U E Rn. The equality (2.12) implies (2.13) f = cxP for a certain constant c , by Ch. I, Theorem 3.8. Sincef is a basis of V, c#O, (2.13) implies

I l fIl:=[

R"

14' W x ) = +

which is a contradiction, becausef belongs to La. The last part of Proposiq.e.d. tion 2.1 is clear from what we have proved.

To decompose the regular representation R, we shall introduce the notion of direct integral of Hilbert spaces which is a generalization of direct sum. In this book, the most general notion of direct integral is not necessary. So we restrict ourselves to the case where the component spaces are identical. In this case, a direct integral of Hilbert spaces is nothing but a vector-valued La-space. Definition 2. Let H be a separable Hilbert space, X a set and p a positive measure on X. Let 8 be the set of all functionsf satisfying the following two conditions 1) and 2): 1) f is a function on X with values in H for which the C-valued function X H ( ~ ( X ) , a) is measurable for every a E H. (Such a function f is called measurable.) r

Let (an)nr~ be an orthonormal basis of H. Then the Parseval equality (x, V ),=n%(~, an)

(v,4

is valid for any x, y E H. Let f and g be two functions in 8. Then (f(x), Ax)>=C(f(x), 4 (g(x), an) nrN

120

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF Rn

is a measurable function. In particular Ilf(x)lla=(f(x),f(x))is measurable. So condition 2) is meaningful. The inequality I(f(x), g(x))l S2-l {llf(x)ll'+ Ilg(x)ll'} proves that (f(x),g(x)) is integrable for anyfand g in Q. Hence an inner product

(2.14) is defined on 8. @=L;(X, p) is a Hilbert space. (The proof of completeness is same as in the scalar-valued case.) The Hilbert space .6 is called the direct integral of the Hilbert spaces H, =H, and this is denoted by

8=

SI

@HZ dp(x).

An elementf in 8 is denoted symbolically thus: X

DeMtion3. Let B be the set of bounded linear operators on the Hilbert space Q=J@H,dp(x). A function x n A ( x ) on X whose value A(x) is a bounded operator on H , is called measurable if the function xwA(x)f(x)is measurable for allfE 8. Suppose that IIA(x)ll is essentially bounded and put ess sup IIA(x)ll =M.Then J@4(x)j'(x)dp(x)belongs to ,#X 8, because r

r

+

=M'llflla< 03 . In this case a bounded, linear operator A on 8 is defined by

(2.16)

Af =

@ A ( x ) f W dP(X)* /X

Such an operator A is called decomposable and is denoted by r

(2.17)

@A(x)dp(x).

A= J X

Proposition 2.2. Let (Us),.x be a family of unitary representations of a topological group G on a Hilbert space H = H s , and assume that the mapping X H U : is measurable for each s E G. Then the mapping X

PLANCHEREL THEOREM AND DECOMPOSITION

121

s

is a unitary representation of G on 8 = @ H,dp(x).

Proof.

For any f and g in 8, we have ( u r f t g)=(ft

u8

g)=(uJ g s f )

dp(x)= 1. Similarly U,U:= 1. So U, is a unitary operator on 4. It is easily seen that

u,u,= u,, . Since (U:f(x), g(x)) is a continuous function of s and ((U;f(x), g(x))\ s Ilf~~)llllg(~)ll s2-1(llf(~)lla + Ilg(~)ll’h we can apply ht=gue’s dominated convergence theorem to the integral

It follows that the function s ~ ( U , fg), is continuous. Hence, U :s n U , is weakly continuous. Then U is strongly continuous because (2.18)

Iluif- UtfII’= IIU,-l,f-fll~ = (2 llfll*-2 Re

(ug-~,f,f)l.

q.e.d.

In the course of the above proof, we have obtained by (2.18) the following proposition which will be used later.

Proposition 2.3. Let U :SH U,be a weakly continuous homomorphism of a topological group G into the unitary group U ( H ) of a Hilbert space H. Then U is strongly continuous and is a unitary representation of G. Definition 4. The unitary representation U in Proposition 2.2 is called the direct integral of the family of unitary representations (Us),,,and this is denoted by

U=/

@ Uzdp (x)

.

H

Now we can analyze the situation described in Theorem 2.2 as follows.

Theorem 2.3. Let x&)=e*(a*e) Theorem 2.2. Then

( E E P ) and let S be the same as in

122

FOURIER TRANSFORM A N D UNITARY REPRESENTATIONS OF

R"

Proof. The representation space La of S is a direct integral, Ls(Rn)=

[email protected]($, where H,=C for all (Sa ID) (E)=xe(a)p(€) for any

(P 6

XE

Rn. Theorem 2.2 shows that

La and S a =

IRnexr(a)h(c)for

any

q.e.d.

aE:R".

$3. Positive definite functions and Stone's theorem In this section we shall prove Stone's theorem which enables us to decompose any unitary representation of Rn into a direct integral of irreducible representations. Stone's theorem is equivalent to Bochner's theorem on positive definite functions in the sense that one can easily prove either of them from the other. So we start from the positive definite functions.

Definition 1. A continuous function on a topological group G is called positive definite if for any finite set of elements g l , ...,g , in G and any complex numbers B1, ..., En, the inequality n

1

(3.1)

'p

(g&)

EtEjSO

1. j - I

holds. Example 1. If U is a unitary representation of a topological group G and x is a vector in the representation space H ( U ) of U, then the function dg)=(U, x , x ) definite. Such a function 9 is called a diagonal element of the unitary representation U. That in fact it is positive definite follows from IS positive

ID W g A ( 6

(ug;lgjx, x ) El

Ej=

El

t .j=1

t.j-1

n

n

n

1-1

t=1

t-1

=(E EjUQ,X,1e'uQ,x)=IIcetuQ,xll"o. Example 2. If G is a locally compact unimodular group and f is a square-Haar-integrable function on G, then p =f ** f is positive definite,

POSITIVE DEFINITE FUNCTIONS AND STONE'S THEOREM

123

where f*(g) =f(g-'). In fact p is a diagonal element of the right regular representation R as is seen from a(g)=/omf(&?)dh=(&Lf 1.

Hence, p =f* * f is positive definite by Example 1.

Proposition 3.1. If

(o

is a positive definite function on G,then we have

1) pO(e)ZO,

-

2) p(g-l)=dg)9 3) lp(g)l4p ( d for any g E G. (e is the identity element of G.) 4) p(g) is also positive definite. Pro05 Let A be the matrix of degree n whose (i, ))-element is p(gT1gj). Then the inequality (3.1) implies (3.2) (AE, €)LO for any 6 E C".Putting E='(1, 0, ..., 0), we get p(e)gO. Since (Ae, e) is real by (3.2), we have (A e, t)=(t,A* ()=(A* e, 6) for all E C". Polarizing the last equality, we get (A e, v)=(A* e, 7) for any e, 7 in C". Therefore A = A * and A is Hermitian. This implies that p(g-l)== Since the Hermitian matrix

is positive semidefinite, its determinant, which is the product of its eigenvalues, is non-negative. So we get Ip(g)lSp(e). 4) is clear from Definition 1 (3.1) and (3.2). q.e.d. It is a remarkable fact that the converse of above Example 1 is valid. 'Namely every positive definite function is a diagonal element of a certain :unitary representation. Proposition 3.2. If p is a positive definite function on a topological group G, then there exist a unitary representation U of G and a vector x in H = H( U)such that p(g)=(Uvx,x) for all g E G. Proof. Let V be the vector space of all finite linear combinations of elements in {R,(olsE GI ,where (R,p) ( x ) = p(xs). For any two elements

c

m

n

(3.3)

f=

(-1

we put

a&,p

and g = c ,BjRt,p in V, j-I

124

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF R" n

m

n

Then (f,g) is independent of the expressions off and g given in (3.3) and is a Hermitian form on V x V. Moreover ( f , f ) Z O for any fc V by the positive definiteness of p. Let W be the subspace of V defined by W = (fe V l ( f , f ) = O } and let p :fwJbe the canonical mapping of V onto the factor space V/W. Now an inner product is defined on V/W by (A g) = (f,g). Let H be the completion of V/W with respect to this inner product. Then H i s a Hilbert space. The operation of right translation R is a linear representation of G on V. The operator R , preserves the inner product, because (R,f, R.g)=(qa' R,,, p, C,8j&, p)=ga&p(t;'s-' sst)=(f, g). j Hence, R , defines a linear transformation V, on V/W by V8f=(R,f )-. Since V, preserves the inner product and satisfies V,Vt= Vat, Ve= 1, V, can be extended uniquely to a unitary operator U,on H. U,satisfies U,Ut = USt. Moreover, the function n

1atg (ssr)

C O (= ~( ) M g>=

t-1

is a continuous function on G. Hence the function e(s) =(U,x, y ) is continuous if x, y e V/ W. If lim xn =x and lim y, =y in H, then e&) =(U,xn, yn) converges uniformly to B(s)= (U,x, y). This can be seen from the inequality

18(s)-en(s)lsI(U,x, ~ ) - ( U , XYn)l , +I(Utx, Y n ) - ( U , X n , yn)l d llxll Ily-Ynll+ IIX-Xnll IlYnll * Thus e(s) =(U,x, y) is continuous for any x, y in H. So U is a weakly continuous homomorphism of G into the unitary group U ( H ) on H. Therefore U is strongly continuous and is a unitary representation of G (Proposition 2.3). Let x be the coset in V/W containing p. Then we have q.e.d. p(s) = (U,x, x ) because p(s) =(R,p, p). As applications of Proposition 3.2, we give two propositions. Proposition 3.3. Any positive definite function p on a topological group G is uniformly continuous. More precisely (D satisfies IpO(g)-p(~)l'S2p @)-Re dg-'h)I. Proof. We can assume that 40 is a diagonal element of a unitary representation U of G (Proposition 3.2). Put Y(g)= (Ugx,x). Then we have

POSITIVE DEFINITE FUNCTIONS AND STONE'S THEOREM

125

Proposition 3.4. For a continuous function (O on a locally compact unimodular group G,the following two conditions 1) and 2) are mutually equivalent : 1) (O is positive definite, 2) for any bounded complex Radon measure p on G, <(o,p * * p > 2 0 , i.e.,

(3.4) ProoJ 2)- 1) Let 81,

..., g,

be a finite number of elements in G and

2

61, ..., En be complex numbers. Then p = €1601(6, is the Dirac measure 1-1 at g) is a bounded complex measure on G.Hence

1)*2). By Proposition 3.2, we can assume that p(g)=(U,x, x) for a unitary representation U on a Hilbert space H. Since p is a bounded Radon measure on G, we can define a bounded linear operator Upon H such that

Then we have

Definition 2. Let Hl and Ha be two Hilbert spaces and Ho be the (algebraic) tensor product of Hl and H2.There exists a unique inner product on Ho satisfying

(xi@xxa, yi@YS=(xi, ~ i(xi, ) yr). The completion of Ho with respect to this inner product is called the tensor product of the Hilbert spaces HI and H, and is denoted by Hl@H,. If A t E B(HJ (i= 1,2) is a bounded linear operator' on Hi, then there

126

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

R"

exists a unique bounded linear operator A1@A2 on HI@& satisfying (3.5) (Ai@Aa) ( x ~ @ x ~ ) = A ~ x ~ @. . A ~ x I Ai@Aa is called the tensor product of A1 and Alr. If Ui is a unitary representation of a topological group G on H' (i= 1, 2). Then U :g H U ; @ U i is a unitary representation of G on Hl@Ha. U is called the tensor product of U' and Us and is denoted by U1@Ua.

Proposition 3.5. Let p1 and pa be two positive definite functions on a topological group G. Then the product p(g) = p1(g)(pl(g) is also positive definite. Proof. By Proposition 3.2, we can assume that pt is a diagonal element of a unitary representation U' and has the form pr(g)=(U:xc, xt) (i= 1 , 2). Let U = U1@U2and X = X l @ X a . Then the function (p.(g)=pi(g) pa(d=(u:xiy xi) (u: X I ,

XS

=((U1@U')g ( ~ i @ ~ s xi@xa)=(Ugx, ), is positive definite (Example 1).

X)

q.e.d.

Now we pass to the study of positive definite functions on Rn. First, the irreducible unitary representation xt(x)=(xc(x)l,1) is positive definite. Since the sum of two positive definite functions and a positive constant multiple of a positive definite function are positive dehite, a linear combination with positive coefficientsof xc's is positive definite. More generally we have the following proposition.

Proposition 3.6. If p is a bounded positive Radon measure on R", then the Fourier transform p of p defined by (3.6)

p(x)=

1

et(Z.6)

dp ( E )

R"

is positive definite. Moreover p(0) = p(Rn). Proof. A unitary representation U of Rnis defined on H = La(Rn,p) by

(UJ)(E)=e'(s.F)f(t). Then the function p(x)=(U=l, 1) is positive definite (Example 1). By the q.e.d. definition (3.6) of 9,we get p(0) = p(R"). The important fact is that the converse of Proposition 3.6 is valid. That is, every positive definite function on R n is the Fourier transform of a bounded positive measure. To prove this result, known as Bochner's theorem, we need two preliminaries.

POSITIVE DEFINITE FUNCTIONS AND STONE'S THEOREM

127

Proposition 3.7. Let e-1z1'r4c ( t > O and X E Rn). ut(~)=(2t)-"'~

Then 1) ut* I(, =ut+, for any r, s>0, and 2) for any positive definite function p on Rn, we have JRnp(x)ur(x)

for any t > O .

Proof: 1) By Proposition 1.1,2) and Proposition 1.4, (uc* u , ) A ( ~ ) = l i c ( ~ ) l i , ( ~ ) = e - ( t + ' ) l " a =( U t + , )

YE).

Since ut E 9c L1, we get ut * u, =ut+, by Theorem 1.2 and Theorem 1.7. 2) Using the result in l), we get

The positive measure p defined by dp(x)=u&)dm(x) p(Rn)=(2xt)-n/a

e-ltlaWx=z-n/a

is bounded by

e-l~lady=1. So we can apply

JBn

Proposition 3.4 and it follows that the right hand side of the above equality is positive. q.e.d. Proposition 3.8. Let (p,,JnzfNbe a sequence of positive Radon measures on Rn with total measure 1 and let

be the Fourier transform of pm.Assume that the sequence (p,,Jnr~converges to a function p uniformly on every compact set. Then p is positive definite, and there exists a positive Radon measure p with total measure 1 such that p(x) =JUn e*(=* dp (€1 for any X E Rn. Proof: The function p is continuous because it is the limit of a sequence of continuous functions which converges uniformly on compact sets. Since pmis positive definite, it satisfies the inequalities

128

FOURIERTRANSFORM AND UNITARY REPRESENTATIONS OF

Letting m tend to

R"

+ a,we get

c n

. , f

(0 ( X j - x o

t' €/LO.

f-1

Hence p is positive definite. Let T be the linear form on 9defined by

T(f)=/ f ( x ) ( 0 ( 4 d m W .

(3.7)

R"

Then T is a bounded linear form on 9. In fact, iff belongs to 9, by Fubini's theorem and Theorem 1.2, we get

~ ( f=Jim )

m*+-/Rn

f ( x ) (on(x>dm(x)

=m-fimIRnY(E)d p m (E).

Since

ISf(E)dpm(BI

5 pm (Rn)llfll- = Ilfll-,

T(f)satisfies

IT(f)lS llfllfor every f in 9. Since 9 is dense in L" (the space of continuous functions f on Rn which vanish at infinity) by Theorem 1.4, the linear form T

on 9 can be uniquely extended to a bounded linear form on L" which is also denoted by T. In particular T is a bounded linear form on L. So T is a Radon integral on Rn. Since pmis a positive measure, T is positive o n 9 (fZO+T(f)LO). If an elementfin L is positive (fZO), then there exists a sequence (fn),,.~of positive elements in 9(e.g., takefn=hlln*f) converging to f i n the uniform norm 11 I]_. Hence T is positive on L. So there exists a positive Radon measure p on Rn such that

for any f E L" (cf. Appendix D, 4). Using the inversion formula (Theorem 1.2), (3.7), and (3.8), we get

POSITIVE DEFINITE FUNCTIONS AND STONE'S THEOREM

129

s

for a n y f ~9.Since p(x) and e*(=,e)dp([)=4(x) are bounded and belong

to the dual space of L1,and since 9 =9 is dense in L1,the last equality implies that erCr*f)dp ( E )

(o(x)=/

(3.9)

R"

for almost every X E R". Since both sides of (3.9) are continuous, (3.9) is valid for all X E R". Puttingx=Oin (3.9), we get p(R")=p(O)=lim ym(0)= m-lim pm(Rn)= 1. q.e.d. m--

Theorem 3.1 (Bochner). If (O is a positive definite function on R", then there exists a unique positive Radon measure p with total measure p(R")= p(0) such that

for all x E R P . Proof. Existence of p. If p(0) =0, then y =0 by Proposition 3.1. In this case, we can take p=O. So we assume that p(O)>O. By replacing y with q(O)-'p, we can assume that ~(0) = 1. Put ut(x) =(2t)-n~ae-i"isr4t and Vm(x) = ( ~ ~ ) ' " ' U ~ ( X ) ( O ( X )

for any m B N and dp,(E) = @,(€)dm((). Then pmis an integrable function 'and cp is bounded, and p,,, is a bounded Radon because urnbelongs to 9 measure on R". If g E .si' is positive, then the function ~(XX =)/Rned(i.6 )

g(E) d m ( ~ )

-

is positive definite by Proposition 3.6. So the functionf(x) =g(x) is also positive definite. By Theorem 1.3, we get P

P

130

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF R"

s

=(2m)R12

2 0.

-

umW (DWf(x) dm(x)

-

The last inequality is valid by Proposition 3.7 because v(x)f(x) is positive definite (Proposition 3.5). The inequality (3.10) implies for all E E Rn.

@,,,(E)TO

(3.11)

Suppose @,,,(to) <0 for a point Eo E Rn. Since Q~ E is continuous, there exists a neighborhood Ua= {E E RnI ( E - (01 < 6 } of tosuch that iF E belongs to Ua,then Q&) 5 -E , where E = -cpm(€o)/2.Put g(€)=ha(€-40) in (3.10), where ha is the function defined in $1, Lemma 1. Then we have the inequality 0s S R n @ m ( E ) ha(€-€0) ME)

5 -6

1

ua

=

La

grn(e) ha(€- € 0 ) dm(E)

ha(€-fo) ah(€)=- E < O

which is a contradiction. So we have proved (3.11). The measure p,,, is positive. Since z2c(()=e-t1f1'(Proposition 1.4), we have

=(ulrk

* (om) (0)

by Theorem 1.3. Taking the limit of (3.12) when k tends to

+ m , we get

-1

(3.13)

pm(R ) -

~ ~ (dm(E) € 1 =vrn(0)= 1

by Proposition 1.4, 3). Since

Icpm(x)Id (2m).laurn(x)E 9 c Li n La, L1n La. Therefore, Q , , , = F ~ ,belongs to La by Plancherel's theorem (Theorem 2.1). On the other hand (3.11) and (3.13) prove that q m E L1.Then by Theorem 2.1, we have pm belongs to

vm(x)=(F*l;iam)

J.*(-*

=

(4=(F*Qm)

( x ) =(.3*0m) (x)

t ) Q m ( ~cirn(6) )

=/et(-*c)dpm(e)

for all x E R"

Since cp,(x) =e-121a14mcp(x),we have

.

POSITIVE DEFINITE FUNCTIONS AND STONE'S THEOREM

(3.14)

131

lim qm(x)= p(x) for all x E Rn m-gD

Since (ym)mtNis a monotone increasing sequence of continuous functions and the limit q is continuous, the convergence in (3.14) is uniform on any compact set by Dini's theorem. Therefore 'pm and (D satisfy the assumption of Proposition 3.8 and there exists a positive Radon measure p with total measure 1 such that

s

q( x )= et(r.e)dp(E)

for every x E Rn. Uniqueness of p. Suppose there exist two positive Radon measures pl and pa satisfying (3.15)

q(x)=

s

etcz." dp, ( E )

(j= 1,2).

Let p = pl - p. Then p is a bounded Radon measure and defines a bounded linear form T on the space L" by

s

T(f)= f(8dp(E). For any g E L1, g belongs to L" by the Riemann-Lebesgue theorem (Theorem 1.6). By Fubini's theorem and (3.15) we have

=/ [ / e - t ( z .

I1

dp(c) g(x) dm(x)

=/[a(-x)-~(-x),g(x)dm(x)=O.

Since t1is dense in L" (Theorem 1.6), the last equality implies that T=O. Therefore p =0 and pl= p a by the uniqueness in Appendix D. 4. q.e.d. Definition 3. Let Xbe a set, B be a o-algebra of subsets of X,and H be a Hilbert space; let P be the set of all projections in H. (Pis the set of all idempotent Hermitian operators on H.) A mapping E : B+P is called a spectral measure on B in H if it satisfies the following two conditions: 1) E(X)=l. 2) If (An)nrr is an at most countable family of disjoint sets in 23, then

where the right hand side converges strongly.

132

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

Rn

Proposition 3.9. If E is a spectral measure on !I3 in H , then it has the following properties, where A and B are any sets in 23: 1) If A 2 B, then E ( A - B ) = E ( A ) - E ( B ) . 2) If A n B = # , then E(A)E(B)=E(B)E(A)=O, and

IIE(A u B)ulla= IIE(A)ull’+ IIE(B)u[la for any u E H. 3) If A JI B, then E(A)E(B)=E(B)E(A)= E(B). 4) E ( A u B ) E ( A n B ) =E ( A ) E(B). 5 ) E ( A n B ) =E(A)E(B)=E(B)E(A). 6) If ( A n ) n cis~ a monotone increasing sequence in 23, then E( u An)= lim E(An) (strong limit).

+

+

n--

nsN

7) If (A,,),,
nrN

(strong limit).

8) Let A be an element in 23. Then b A= [A n BIB E B} is a o-algebra on A and F :B w E ( A n B) is a spectral measure on bd in E(A)H. Proof. 1) Since A = B u (A -B) is a disjoint union, E(A)=E(B)+E ( A B) by condition 2) in Definition 3. 2) Since (3.16) E ( A ) E(B)= E ( A u B) =E(A u B)’ =E(A) E(B) E(A)E(B)+ E(B)E(A), we get

+ +

+

(3.17) E(A)E(B)= -E(B)E(A). Multiplying (3.17) by E(A) on the left and on the right respectively, we get (3.1 7)‘ E(A)E(B)= -E(A)E(B)E(A) = E(B)E(A). (3.17) and (3.17)’ prove that E(A)E(B)=E(B)E(A)=O.(3.16) and the last + IIE(B)ulla. equality prove that IIE(A u B)ulla= IIE(A)UI/~ 3) Multiplying E ( A ) = E ( B ) + E ( A - B ) by E(B) on the left and on the right and using 2), we get E(B)E(A)=E(B)= E(A)E(B). 4) Since A u B is the disjoint union of A n B, A -(A n B), and B-(A n B), we get E ( A u B) =E(A n B) ( E ( A )-E(A n B))+(E(B)-E ( A n B)). 5) Multiply 4) by E(A) and use 3). 6) Let B,=A,-A,-I (n21) and Bo=Ao. Then ( B , J n eis~ a family of

+

disjoint sets and

m

u

n-0

97l

B, = u A , =Am. So by condition 2) of Definition 3, n-0

E(UAn)=E(UBn)=kE(Bn) nrN n-0 nrN

POSITIVE DEFINITE FUNCTIONS AND STONE'S THEOREM

=lim m-+-

m

133

m

C E(Bn)=lim E( u Bn)=lim E(Am). m--

n-0

m--

n-0

7) Let A,' = X - A, be the complement of A,. Then is an increasing sequence and (n A,)" = U A,'. So we get by 6), 1 - E (n A,) nfN

n€N

n€N

=E( u AnC)=lim E(AnC)= 1- lirn E(An). ntN

n--

n--

8) is easily verified.

q.e.d.

Let E be a spectral measure on b in Hand u, v be elements in H. Then the mapping pu,v : b+C defined by p,,,(A)=(E(A)u, v) is a finite complex measure on b. Iff : X 4 C is an integrable function with respect t o the measure p.,,,, we denote the integral

fdp... by

/X

In particular, the measure pr =p.,

.

s

is positive and the integral f d p , is

denoted by (3.19)

Here we recall some basic definitions and results on complex measures. A complex measure p on a 0-algebra b is a completely additive set function on b with values in C. The total variation 1p1 of p is the function on b defined by

c Ip(At)I n

IpI(A>= SUP

9

6-1

where the supremum is taken over all decompositions of A into a disjoint union of a finite number of sets in b.The total variation 1p1 of p is a finite positive measure on 123. Let p1 and p, be the real and imaginary parts of p. Then the positive and negative parts p; of p k are defined by pk+(A)= SUP P k ( B ) BcA

and

p;(A) = -inf pk(B). BcA

p; and pi are finite positive measures on b and pk = p l - p ; (k= 1,2). A complex-valued measurable function f on X is called p-integrable if I f 1 is Ipl-integrable. Since Ogp:(A)g [pI(A) for any A E 8 (k= 1,2), f is then

s

&integrable. The integral f d p is defined by

134

FOURIER TRANSFORM A N D UNITARY REPRESENTATIONS OF

R"

and satisfies (3.20)

Proposition 3.10. Iff and g are square-integrable functions on X with respect to the measures pu and purespectively, then the product f(x)g(x) is integrable with respect to pU,,,and satisfies

Proof.

Because of the general inequality (3.20), it is sufficient to prove

So we can assume fro and g2O by replacing f and g with respectively.

If I

and lgl

n

If A = u At (disjoint union), we have 1-1 n

n

= I I ~ ( 4 u l lIIE(A)vll by the Schwarz inequality in Hand in Rn and by Proposition 3.9,2). Thus we have proved

(3.23) Ipu,ul(A)~pU(A)*~u(A)* for any A E 8 Now we prove (3.22) for simple functions n

n

i=1

J-1

.

and (Bj)lsjsn are two disjoint families in where A r 0. Since X A X B= x A B , we have by the inequality (3.23)

B and ar20,

POSITIVE D ~ M T FUNCTIONS E AND STONE'S THEOREM

n

m

135

n n l

Iff 20 and g L 0 are square-integrable with respect to p, and p,, then there exist two monotone increasing sequences (fn ) and (g,) of positive simple functions which converge to f and g respectively. Then we have

Since I fnl d f and lgnl $ g, we can apply Lebesgue's theorem on both sides of the last inequality. Letting n tend to infinity, we get (3.22). q.e.d. The following proposition will be used frequently. Proposition 3.11. 1) Let B be a fixed element in b.Then the complex measure p:," defined by p t , , ( A ) = (E(A)E(B)u,v ) satisfies p:,,(A) = p.,,(A n B)= (E(B)E(A)u,v). For any p:,,-integrable functionf,

,.

c

(3.25)

is a complex measure. Iff is p-integrable, then we have c

c

(3.26)

Proof. 1) Since E(A)E(B)=E(An B) (Proposition 3.9), we have p:,, (A) =pu.,(A nI?)=(E(B)E(A)u,v) and (3.25). 2) Since p is an indefinite integral with respect to the measure gp,.,, (3.26) holds. (cf. Halmos [l] p. 134, Theorem B). Proposition 3.12. Let E be a spectral measure on b in H, f and g be

136

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

Rn

complex-valued measurable functions on X. 1) Then the set D( f) of all vectors u in H satisfying (3.27)

Jxlf(x)l~dllE(x)ull' < +

is a subspace of Hand there exists a linear operator T(f):D ( f ) + H such that (3.28)

for all u E D ( f ) and v E H. 2) D ( f ) = D ( f )and for any elements u, v in D(f), ( T ( f ) u ,v> = (4T(P)V). 3) For any set A in 123 and u E D ( f ) , E(A)u belongs to D(f) and we have T ( f ) E ( A ) u= E ( A ) T ( f ) u . 4) For u E D(f)and v E D(g),

f ( x ) r n ( E ( x ) u ,9 .

( T ( f ) u , T(g)v)= J x

In particular (3.29)

llT(f)ulI'=/

X

If(x)12dllm)ull'

5 ) For any a E C,D ( f ) c D ( u f ) . For any u E D(f),T(uf)u=aT(f)u. 6) D(f)n D(g) c W+d.For any u E N f )n D(gh

(3.30) T(f+g)u = + TWu. 7) Let u E D ( f ) . Then T ( f ) u belongs to D(g) if and only if u E D ( g f ) . If this condition is satisfied, then T ( g ) T ( f ) u= T(gf)u. 8) Iff is a bounded measurable function then D ( f ) = H and T(f)is a bounded linear operator on H. Proof. 1) If u, v belong to D(f),then Proposition 3.10 shows thatfis square-integrable with respect to p,,, and /I,,,,, because 1 is ,u,-integrable. Hence the equality

+

1I-W)(u+ v)l12 = IIE(A)ul12+(E(A)u, v) +(E(&, u) IIE(A)vlla shows that f is square /I,+,-integrable, namely, u + v belongs to Wf). Define the mapping B : D(f)x H+C by

N u',) V

f(x)d(E(x)u,v). /X

Then UHB(U,v) is a linear mapping and VHB(U,v) is an anti-linear mapping. Moreover, we have by Proposition 3.10,

POSITIVE DEFINITE FUNCnONS AND STONE’S THEOREM

137

So by Riesz’s theorem, there exists a linear operator T ( f ) : D( f )+H satisfying B(u, v) = ( T ( f ) u ,v) for any u E D(f)and v E H. 2) By the definition of D(f),D ( f ) = D ( f ) . If u and v belong to D(f), we have

P

because E(B)E(A)=E(A)E(B). 4) By 2), 3) and Proposition 3.11.2), we have

5 ) is trivial.

6) Since I f + g l a 9 (If12+lgla),D(f)n D ( g ) c D ( f + g ) . We have (3.30) from (3.28). 7) For any u E D ( f ) , we have by Proposition 3.1 1,

138

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

R"

Hence T(f )u E D(g) if and only if u E D(gf ). If this condition is satisfied we have

8) Iff is a bounded, measurable function, then f is square p,--integrable for any u E H, because pu is a finite positive measure. That is D ( f )=H. The equality (3.29) proves that IIT(f)ull S Ilfllmllull, where 11f 11- is the supremum of If I. So T(f)is bounded.

Definition 4. For the linear operator Tcf> :D O - H defined by (3.28) we write

=I

T(f)

f(x)dE(x).

X

Theorem 3.2 (Stone). If U is a unitary representation of Rnon H, then there exists a spectral measure E on b (the set of all Bore1 sets in R") in H such that U, =[Rnei[z.el dE(E) for every x E Rn . A bounded linear operator T on H commutes with every U, (XE Rn) if and only if T commutes with every E ( A ) (A E b). In particular U,E(A) = E(A)Ux. Proof.. For any vector u E H, the function v,(x) = (Uxu, u) is positive definite (Example 1). So there exists a unique bounded positive Radon measure pu satisfying

(3.31) and

(3.32) pu(R") = llul12 by Bochner's theorem (Theorem 3.1). For any two vectors u and v in H, put pu. 0

(A)=4-' Ip*+u ( A )-pu-u ( A )

+

ipu+to

(4- ipu-iv ( 4 1 .

POSITIVE DEFINITEFUNCTIONS AND STONE'S THEOREM

139

Then pU,,,is the unique complex measure on R n satisfying

(3.33)

(U,u, v) =

1

Rn

ef("*a) dpu.0 (8

for every x E R". For any fixed Bore1 set A, the mapping (u, v ) n p u . , ( A ) is a positive Hermitian form on H x H. It is bounded, because by (3.32) Ipu,v(A>I 5 pu(~)"2pu(A)"'

5 llull llvll -

Hence there exists a unique, bounded, Hermitian operator E(A) such that

(3.34) pu,.(A)=(E(A)u, v) for all u, v E H. The mapping E : A n E ( A ) is a spectral measure on b in H. In fact, by (3.32) and (3.34), we get

(B(R")u,V ) = p u , u ( R n ) = ( u , v); hence, E(Rn)= 1. If B= u B, is a countable disjoint union, then n

(E( U Bn)u, v)=pu.u(

B.)=Cpu.u(Bn)=C(E(Bn)u, V) n

n

because pu,uis a complex measure. Hence

It remains to prove that E(A) is a projection. Since E(A) is Hermitian, it suffices to prove E(A) is idempotent. More generally, we prove (3.35)

E(A)E(B)= E(A

We have

n B) for any A, BE b.

SRnf?-

%f(E(c)u,u, v) = (U,U,u, v)

(3.36)

et("+u.E)d(E(€)u, v)

S n

where p(A)=

ef(gsE)d(E(c)u, v) for any A E 9.For any fixed A E b,put

A

F(B)=E(A

B). Then we can write p(A) as p(A) =/R,et(u. E ) d(F(c)u,v).

By the uniqueness of the measure pUyu,,, the equality (3.36) yields

140

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF R"

(3.37)

J

(E(A)U,,u,v)=p(A) = Rne*(Y*t)d(F(e)u, v)

for any A E 23. On the other hand, we have

(3.38)

SI.

(E(A)U,u, v) =(Up,E(A)v)=

et(v*e)d(E(E)u,E(A)v).

Again by the uniqueness of p,,,E(A),,, (3.37) and (3.38) yield (E(A n B)u, V ) =(F(B)u, V ) =(E(A)E(B)u,V ) for any B E b. Since A , u, and v are arbitrary, we get (3.35). The uniqueness of E is clear from the uniqueness of pu,u. Let T be a bounded linear operator on H. Then we have for any u, V E H and X E Rn, (3.39)

(TU,u, v)=

e((=." d(E(c)u, T v)-

*

I-R"

and

(3.40)

e*(=.W(TE(e)u,v)

-/Rn

)-IRn

( U,Tu, v -

e*("*W(E(€)Tu, v).

Hence if TE(A)=E(A)T for every A E 23, then we have TU, = U,T for every x E Rn. Conversely if TU,= U,T for every X E Rn, then by the uniqueness of pu,T.u,we get TE(A)=E(A)T for every A E b .

q.e.d.

Remark. We can easily derive Bochner's theorem from Stone's theorem. Let 9 be a positive definite function on R". Then there exists a unitary representation U of R" and a vector u in H = H ( U ) such that cp(x)=(U,u, u) (Proposition 3.3). By Stone's theorem U can be written as

s

Us= e"**"dE(E). So &c)= finite Borel measure.

s

e*("."dp,(E)where pu : Aw(E(A)u, u ) is a

Theorem 3.3. If U is a cyclic unitary representation of Rn on H, then there exists a finite Borel measure p on Rn such that

.

where ~ ( ( x=)e*(=.b ) Proof. By Stone's theorem (Theorem 3.2), there exists a spectral measure E on b in H such that

POSITIVE DEFINITE FUNCTIONS A N D STONE'S THEOREM

U,=/Rnet(s.

141

dE(€).

Let u be a cyclic vector in H and define a finite Bore1 measure p by

.

p(A) = IIE(A)ulls=(E(A)u,u) for any A E b Let f be an element in L'(R", p). Now u belongs to o(f) in the notation of Proposition 3.12, and the linear operator

T ( f ) = / R n f ( €dE(€) ) :W ) - + H

is defined. Then the linear mapping 0 : L' (R", p)+H defined by 0 ( f )= T ( f ) u is an isometry by (3.29). The image H' of 0 is invariant under U because we have u,u=Jlnet(s*~)d ~ ( € ) u = T(x.)uE H'

for any X E R". Since u is a cyclic vector, H' is dense in H. O n the other hand, H' is closed because it is isometric to the complete space La(R",p). So we have proved that H ' = H and 0 is an isometry of L'(Rn, p ) onto H . Since

L1(R", p)=J

R"

Cl3 Cedp(€1,

where Ct = C for all 6E R", a unitary representation U oon La(R",p ) is defined by

U'=/

R"

a3 xedp (8*

Then we have (3.41)

U p 0 =00Uj for every x E R"

.

In fact, for any f E La(Rn,p ) and v E H, we have by Proposition 3.1 1,

u:>f,v) =s R p"f(€)d(E(OU, . v)

((00

=IRn

f ( € ) d e (R/" e'(=*?)d/(E(7?)~, E(€)V))

=Sf(g)d,(U=u, E ( W =

s

f(€)d(E(E)UdJ,v )

142

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

Thus we obtain (3.41).

R" q.e.d.

Remark. We can drop the assumption that U be cyclic in Theorem 3.3.

In the general case, an irreducible representation xr can be contained in U with some multiplicity. Let U be an arbitrary unitary representation of Rn. Then U is a direct sum of cyclic representations Ua(aE A):

u=

(3.42)

@

u a

atA

(Ch. I. Proposition 2.5). Each cyclic representation U a is decomposed in the form

ua=[R"

(3.43)

@ xdpa (E)

by Theorem 3.3. Let J= A x Rn. Define a measure p on J by p(a x B)= p a ( B ) and put 7j)=(a,6) and x,,=xr. Then (3.42) and (3.43) yield

$4. The Paley-Wiener theorem

In this section we give a characterization by means of the Fourier transform of the infinitely differentiable functions on Rn with compact supports. This result due to Paley-Wiener, reveals a remarkable relation between a certain class of holomorphic functions and harmonic analysis of functions with compact support. First we give fundamental definitions and results on holomorphic functions of several complex variables. Let Cn be the space of all vectors z =(zl,..., z,) with n complex conponents. The norm IzI of z is defined by IZI =(Izlp+ ...+ IZnla)l'a. The canonical inner product in Rn is extended to a bilinear form (z, w)= n

c.?ewk

on C" x c".If zk =

+ivh and &,

vk

E R,then the real and imaginary

k=l

parts of z are defined by Re z=(ul, ..., u,) and Im z=(vl, ..., vn). For any multi-index m E N",we put m ! = m l ! ... m,!, We introduce an order in N" by defining p hq if and only if pk2 qk for all k e {1,2, ..., n}. Z ~ = Z ~ ~ . , . Zand ~ "

143

PALEY-WIENER THEOREM

Definition 1. A complex-valued continuous functionf on a connected open set A in Cn is called holomorphic if the function f, : cwf(a1, .-.,ak-1, c, aX+l, .-*,an) is a holomorphic function of the complex variable C on an open neighbork of a&in C for every u=(al, ..., a,,) E A and k E {I, 2, ..., n}. hood u We recall that a functionf, of a complex variable C is holomorphic on a connected open set UX,if (4.1)

exists for every

To e UX.The limit in (4.2) is called the derivative off, at

To and is denoted byf,'(Co) or by

(Co).&'(aX)is the partial derivative of

f a t a with respect to the k-th coordinate and is denoted by (ayaz,) (a). Let f=u+iv and z=x+iy be the decompositions off and z into real and imaginary parts. Iff is holomorphic on A, then the functionf, defined in (4.1) is a holomorphic function of a complex variable. Hence the Cauchy-Riemann equations a v =-- au and (4.3) axX

hold on A for any k E { 1,2, bY

auk

...,n} . The partial derivatives off

are given

(4.4)

Let rr be a piecewise smooth, simple, closed curve in the complex plane C and let Dk be the domain interior to r r(1 5 k 5 n). Iff is holomorphic on an open set A containing 6, x ... x fin,then from the Cauchy integral formula for one complex variable we obtain

for any z=(zl, ..., z,) E D1 x ... x D,. (4.5) is the Cauchy integral formula for a holomorphic function of several complex variables. Put r =rlx ... x rn, D=D1 x ... x D,, and r;=(Cl, ..., then (4.5) can be written in the form

m);

f(z) =( 2 ~ i ) - * / ~ -dC

for z E D.

If r k is the circle lzX-ukl =rk and f is holomorphic on an open set A containing the polydisc 6,then f ( z ) can be expanded into a power series with the center a = (al, ..., a,). In fact, put

1 4

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

f (C) (C -a)-'"+" dc

a, =(27r9-n

(4.7)

R"

s r

..., m,+1)

foranymENn, wherem+l=(ml+l, and

C a,(z

gN(z)=f(z)-

-a)" for N E N".

NL,>O

Then we get

If c is a real number in the interval (0,1) and z belongs to the polydisc 0,= { zI I zk -akI 5 crk (1 5 k i n) }, then we have a constant M such that IgjV(z)I

5 M {1 - (1 - &+1)

...(1 - &+')}.

Hence when Nk+ + 00 (1 $k$n), gN converges to 0 uniformly on D,. We have proved the following theorem. Taylor's Theorem. Iff is holomorphic on an open set containing the polydisc D with the center a, then f can be expanded in a power series: (4.9)

f(z)=

c

a,

(Z-u)m

.

mrNn

The coefficients a, in (4.9) are given by (4.7). Since the power series converging in a polydisc D can be differentiated arbitrarily many times, a holomorphic function can be differentiated arbitrarily many times. Differentiating(4.9), we get the following Corollary. COROLLARY to Taylor's Theorem. The coefficient a,,, in (4.9) is given by (4.10) where D:=(=)"'a

... (-)"" a azn

Hence the power seriesexpansion of a holomorphicfunctionf is uniquely determined by f and a.

Definition 2. A function f which is holomorphic on the whole space C" is called entire. LEMMA1. If an entire function f on C" vanishes on Rn,then f=O on Cn. Proof. All the coefficients a, in the Taylor expansion (4.9) off with the center a=O are equal to 0, because by (4.4) and (4.10) we have

145

PALEY-WIENER THEOREM

q.e.d.

Definition 3. Let r 2 0. Then an entire function ~pon Cnis said to be of exponential type 5 r if there exists a constant C> 0 such that lq(z)ld CerlIrnz' for all z E Cn . We denote by S r = S r ( C n )the space of all entire functions f on C" such that zmf (z) is of exponential type r for every m E Nn.The vector space p,is topologized by the family of seminorms:

The topology of g f , just defined coincides with the topology defined by the family of seminorms:

(4.12)

sm ((.)=sup {e-,lIrn*I Izmp(z)l], rrCn

(mE N") .

n

In fact, since 1zm15(1+(z(')N if I m ( l = ~ r n r 5 2 Nsm(v)dqN(q) , if ImllS k-1

2N. Conversely, since (1 + lzla)" is a linear combination of the monomials

the inequality qN((o)d

1

CP S%P (9)

IpllSN

holds for any (o E s

r ,

where the cp's are certain constantsz0.

Definition 4. The space of all complex valued C"-functions on R" with compact supports is denoted by g = g ( R n ) . The subspace of 9 consisting of those functions whose supports are contained in B, = { x E RnI 1x1 $ r ) is denoted by 9,.9,is topologized by the family of seminorms:

(4.13)

I(omf) (XI, (m€Nn) pm(f>=SUP zdn

9

where Dm is the partial derivation defined in Difinition 4 in $1.

Proposition 4.1. The spaces S,and g rare Frechet spaces (i.e. complete, metrizable, locally convex vector spaces) if r >0. Proof. They are metrizable, locally convex spaces, because their topologies are defined by countable families of seminorms. Now we prove their completeness. Let ( f & ~be a Cauchy sequence in the space Br.Then (f,JnrN and (Dmfn),,.N converge uniformly. So f = lim fi is a C"-function

146

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

R"

whose support is contained in B,, i.e. f E 9 , . Let ( q & . ~ be a Cauchy Then for any m E Nn and E >0, there exists N, E N such sequence in Sr. that

.

then S, ( y p- ~ , J < E (4.14) if p , q 2 N,,, Since the inequality

(4.15) 12" ( p p (z)-pq(z))I S sm (pp-pJer'Irna' holds for all z E C", the sequence ( Z ~ ~ ~ ( Z ) ) , , ~ Nconverges to a function g , and uniformly so on every compact set. Hence the limit function g, is entire and g,(z)=zmgo(z) for every m E N" and z e Cn. Since Is,(pp)sm(pq)lSs,(pP-pq), the real sequence ( S , ( ~ ~ ) ) , , ~has N a limit c,zO. Taking the limit when p-+ + to in the inequality

lzmpp(z)le-,

Irnlrl

ds, (pp),

we get lzmgo(z)le-'Ixrn 11 I:C ,

for any z E Cn. Hence gobelongs to Sr and go =lim pn in the topology n-+-

of ZI.

q.e.d.

Definition 5. The Fourier-Laplace transform Sc f of an integrable functionf on Rn with compact support is defined by (4.16) flc f

Hc((f>

f ( x ) e -t(*,*)h(x).

(z)=/ R"

is a function on Cn.

Theorem 4.1 (Paley-Wiener). The Fourier-Laplace transform X cis a topological linear isomorphism of 9, onto s r for any r >0. The inverse mapping of X cis given by the conjugate Fourier transform on Rn. and p =S C f b ethe Fourier-Laplace Proof. Let f be a function in gr transform off. Then p is given by

j'

(4.17)

p(z) =

f(x)e-'(s*a) h ( x ) ,

Br

where Br= { x e RnI Ixl$r}. Since le-*(z.a)l=e(*Jrna)<erlXm for any x E B, and since a continuous function f ( x ) is integrable on the compact set B,, the integral (4.17) converges for every z E C" and p is a continuous function of z by Lebesgue's theorem. Since e-'(***' is an entire function of z and

-I a

azk

(f(x)e-'(*S*))l=I-ixk f(x)e-'(".')I 5 Ixkf(x)lerlxm'l

147

PALEY-WIENER THEOREM

is integrable on B,, p(z) is holomorphic at every point with respect to the k-th coordinate z k for any k. Hence p is holomorphic on Cn and is an entire function. Moreover, by repeating integration by parts as in the proof of Proposition 1.3, we get

(DY)(x)e-6('".a ) dm(x)

(4.18)

foranyzECnandkENn. Hence we have (4.19) Izk'p(z)l5 IIDy:flller1Im *I 5 rn(Br)llD"fll&lm 'I for any z E C" and k E N n . (4.19) proves that 'p belongs to g?? and that RC is a continuous mapping of sT into z,. Sc is injective on 9,.In fact, if flc f=O for a function f E .gr, then the restriction flf=f of F c f to Rn is equal to 0. Hencef =0 by the uniqueness of Fourier transform on 9 2 3 (Theorem 1.2). It remains to prove that S c 3=,p,and ( f l c ) - l . is continuous. Let Q be a function in Z1 and put (4.20)

f ( x )=

SpRn

p(t) e$(=* E) d m ~ .

Since Ip(€)ISqN(y)(1 +I€I')-" for every (E Rn and N E N, p is integrable on Rn and f can be differentiated arbitrarily many times under the integral sign. Hence f is a C"-function on Rn. Let z k = t k + i v k , ( k , ?lk E R ; then the integral (4.21)

Im'p(z)e'('"*

d&

is independent of 77r.Let r be the rectangular path in the &-plane given in Fig. 4. Since p E Zrsatisfies

l'p(z)I ICl (1 + Izl2)-l e' the vertical integral tends to 0 when b-+ have

Fig. 4

, and

IIrn 00

c+-co.

In fact, we

148

FOURIER TRANSFORM AND UNITARY REPRESENTATIONSOF

Similarly lim C*--

(4.22)

s'

R"

=O. So we have proved

C+%

+

f ( x ) = I R n y(t ir/)e(( 2 , t+w WE)

satisfies for any x and '1 in R".Since (o E S+ lp(z)ls Cn (1 + IzIl)-" erlxmrl 5 Cn (1 I€l')-" erlrl for any z = € + iv E C", (4.22) yields

+

(4.23) 5 Coerlql - ( 2 . d for any x, 7 E Rn , where Co= C n h l iI ~ l a ) - ~ d m>(0~is) a constant. For any x E R" sati sfy ing 1x1>r, put '1=rx/]xl, where r is an arbitrary real number >O. Then (4.23) gives

If(x)lS Coet(+-121) for any r>O. Making t- + co in the last inequality, we have proved that f ( x )=O if Since g+c.Y,we have jxl > r and f belongs to g+. (4.24) T*csf=f by theorem 1.2. On the other hand, let 4 be the restriction of Then the definition (4.20) off can be written as

(o

to R".

9-4 =J: (4.25) (4.24), (4.25) and the uniqueness property of the Fourier transform (Theorem 1.5) implies that ( T f - 4 ) (x)=O for every x E R" because f - 4 is continuous. Hence the entire function sc f - p vanishes on Rn. So s c f - p = O on Cn by Lemma 1. We have proved that X cis a mapping of sr onto sr and the inverse mapping of T c is given by the conjugate Fourier transform T *on Rn. Now we prove the continuity of ( S C ) - ' . Let k E Nn be a multi-index. Replacing cp(z) in (4.20) by z*p(z), we get (4.26)

(DY) (x) =

J

R"

P ~ Ge') ( 2 , o h ~ .

149

PALEY-WIENER THEOREM

If 2N> Max (lkl, n), then (o E py satisfies

Izip(z)le-r I*m 5 -(1+ Izl')Nlp(z)le-"lm'f

(4.27)

81

5 (1+ 1Z1')-"qlN

($0)

for any z E Cn.(4.26) and (4.27) prove that (4.28)

IIDyfl-$Aq2N (p)

J+

where A = (1 lCls)-Ndm(€)is a positive constant. (4.28) proves that (Fc)-l

:pwfis continuous.

q.e.d.

Now we extend the result in Theorem 4.1 to the space 9 = u gr. It r>O

is necessary to introduce a topology in g which is related to the topology in grin a reasonable way. Although the family (4.13) of seminorms defines a topology on 9,this topology has the disadvantage of not being complete. We shall give a complete locally convex topology on 9 which induces the original topology on 9 7 . This topology is also important for defining a distribution which will be discussed in the next section. The is likewise to be equipped with a topology in a way space p =u Sr r>O

similar to that for 9.So we shall give a general way of introducing a locally convex topology on a vector space starting from the topologies of certain subspaces. A subset B of a complex vector space E is called disced if ax E B whenever xEBand l a l s l .

Proposition 4.2. Let E be a complex vector space and let (E&N be a family of subspaces of E satisfying the following conditions: (a) Enc (b) u En=E, (c) En has a locally convex topology T,, (d) the n topology induced by Tn+lon En coincides with T,. Then we have: 1) There exists a unique strongest locally convex topology T on E which :En+E continuous for all n E N. makes the injection in 2) A linear mappingfof E into a locally convex topological vector space F is continuous (in T) if and only if fn =fo in :E,+F is continuous for every n E N. Proof. 1) Let N, be the system of neighborhoods of 0 in En, and for any element ( Vn)nrN of 17 N,, let U((V,)) be the smallest convex, disced nrN

set containing u Vn. Then the set B= { U((V,)) I

(Vn)nrNE

17n.N N,} is a

ntN

fundamental system of neighborhoods of 0 in a topology on the vector space E with respect to which E is locally convex. Since V, c U((V,)), the injection inis continuous for every n E N. Let T' be a locally convex topology on E in which every in is continuous and let V be a closed.

150

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

R"

convex, disced neighborhood of 0 in T'. Since in is continuous, i;'(V)= V n En belongs to N,, for any n E N. Therefore U((V n En)) = V belongs to B. Hence T is stronger than T'. 2) If f is continuous, then fn =f .in is continuous. Conversely, if f n is continuous for every n E N , then for any convex, disced neighborhood V of 0 in F, f ;l( V )= i;l( f - l ( V ) )= f l (V )n En belongs to N,. Hence U = U((f-'(V) nEn)) belongs to B. Since f - l ( V ) is convex and disced, U = f - l ( Y ) andf-l(V) belong to B. Thusfis continuous. q.e.d. Definition 6. The locally convex topological vector space E with the topology T defined in Proposition 4.1 is called the strict inductive limit of En and we write E=lim En. -----,

Remark. It can be proved that the topology on En induced by T coincides with T,, and that E is complete if all En are complete. (cf. N. Bourbaki [2] Ch. 11, !$I.)

Definition 7. The vector space 9 = 9 ( R n ) is topologized as the strict inductive limit of d m where 9, is the subspace of 9 consisting of those functions in 9 whose supports are contained in B, = {x E RnI 1x1

-

5 m} . The topology of the space S = u S ,= u g?, of all entire funcT>O m-1 tions f on Cn for which z*f(z) is of exponential type for every m E N n is defined as that of the strict inductive limit of g?,. With these definitions, the first result in Theorem 4.1 can be stated as follows. Theorem 4.2. The Fourier-Laplace transform isomorphism of .g=g ( R " ) onto p= u Sr.

P is a topological

7>0

Proof. Since 9 =

6 sm and Z = 6 g?,,and ,, X cinduces a linear

m-1

m-1

onto Z,,,by Theorem 4.1, SC is a linear isomorisomorphism of sm be the restriction of N C to B,,,, regarded phism of 9 onto S . Let 5; as a mapping of 9, onto Zm, and let imandj, be the injections gm-+ 9 and S,-+Srespectively. Then S; is continuous by Theorem 4.1. So j , , , ~ ~ ~ = ~is ccontinuous. o i m Hence is continuous by Proposition 4.2. Similarly ( . F c ) - I is continuous.

$5. Tempered distributions and their Fourier transforms In this section we define distributions on Rn and discuss their Fourier

TEMPERED DISTRIBUTIONS A N D THEIR FOURIER TRANSFORMS

15 1

transforms. Since the mechanism of Fourier transform does not work smoothly on the space 9' of distributions, we introduce the subspace 9' of tempered distributions. Definition 1. Let 9 = 9 ( R n ) be the space of complex-valued C"functions on Rn with compact supports, 9 is equipped with the topology Then, a of the strict inductive limit of g n = 9 ( B n ) introduced in continuous linear form T o n 9 is called a distribution on Rn. By Proposition 4.2, a linear form T on 9 is continuous if and only if the restriction T, of T to s m = d ( B m )is continuous for every m>O. The space of all distributions on Rn is denoted by 9 ' = 9 ' ( R n ) . The canonical bilinear form on 9 x 9' is defined by

w.

=T(p)foryE9and T E ~ ' .

Example 1. The Dirac measure aa at a is defined by

=CpW. It can be easily verified that 6, is a distribution. Example 2. Any Radon measure p on Rn may be viewed as a distribution if we take

For, the inequality

I <~ p , p > I 5 I l ~ l l - l(Bmh ~l which holds for p in gm, proves that p is continuous on 9". Example 3. A measurable function f on Rn is called locally summable iff is integrable with respect to the Lebesgue measure dm(x) on every compact set. Then f defines a distribution Tf by <(D,

S.I

Tf > =

f ( x ) y(x) dm(x),

because the measure p j defined by dp (x)=f(x)dm(x) is a Radon measure (Example 2). Since the mappingfi+Tf is an injection of the space of locally summable functions into g',f is identified with T,.

Definition 2. If T is a distribution on R" and a ... then the linear form i axl i ax,, p+(- 1)'"' is a continuous linear form on 3, i.e., a distribution on Rn. By definition this distribution is DmT:

(- -)"'

D m= i

(IL)",

152

FOURIER TRANSFORM AND UNITARY REPRESENTA~ONS OF R"

=(-1)lm1

Example 4.

<9,Dm6,> =(- 1)Irn1(Drn(p)(a). Proposition 5.1. Let 9 = 9 ( R n ) be the space of rapidly decreasing functions (cf. Definition 1 in $1). Then we have: 1) The injection i : d + y is continuous. 2) d is dense in 9'. 3) If T is a continuous linear form on .9' (i.e. T E.Si"), then TwToi is an injection of 9'into g'. = g ( B , ) , then Proof. 1) If p belongs to drn PL,N(V)=SUP I(1-k lXI')N(DXp) s ( 1 +m')NIIDkv!lfor any k E N" and N E N. So the injection i, : gm+9 is continuous for every m. Hence i is continuous by Proposition 4.2. 2) Choose a function (0 in g such that $(x) = 1 if 1x1 5 1. For any function f i n 9, put &(x) =f(x)$(rx) for r > 0 and x E R" Thenf; belongs to 9.Now we prove that fi converges to f in 9 when r-+ +0. If P is a polynomial and m E N" is a multi-index, then by Leibnitz' law of differentiation we see that (5.1)

P(x)Dm( f - f i ) ( x ) = P ( x )

,C8(Dm-kf)(x)r'"Dk(l-$ (rx)).

esm

1 By the choice of (0, Dm(l-$(rx)) =0 if 1x1 5 - for every rn E N".Sincef r belongs to 9, P(x) (Dm-Lf)( x ) is bounded on R" and the right hand side of (5.1) converges to 0 uniformly on R" when r++O. Hence we have proved that fr converges to f in 9. 3) By l), Toi is a continuous linear form on 9.Therefore the mapping I: THToi is a linear mapping of 9' into 9'.I is injective by 2).

Definition 3. If T Ey',then the distribution Toi defined in Proposition 5.1 is called a tempered distribution. Since I : T w Toi is an injection of 9'' into g',we identify T E9' with Toi a 9'. Example 5 . The Dirac measure 6, and its derivatives D6, are tempered distributions. Example 6. A positive Radon measure on R" is called slowly increasing if

TEMPERED DISTRIBUTIONS AND T H ~ R FOURIER TRANSFORMS

153

for some N E N . A slowly increasing measure p is a tempered distribution. In fact, we see that

I <(D,

lu > IS

Po, N ( d J R n ( + l Ixl2>-"4 (4.

Definition 4. The Fourier transform f ' = T T of a tempered distribution T is the tempered distribution defined by (5.2) cp, f'> = for all p E 9. Since the Fourier transform N : C ~ H is Q a topological isomorphism of 9 onto 9 '(Theorem 1.2), the linear form f defined by (5.2) is continuous and is actually a tempered distribution. Similarly the conjugate (or inverse) Fourier transform N * T of T is defined by <(D, N

* T > = <**ID,

T> for all p E 9.

Example 7. Iff E L', the space of summable functions, then we have f,=T;. Namely, the Fourier transform off regarded as a distribution (Ex. 3) is equal to the Fourier transform of the functionf. In fact, by Fubini's theorem we get

Example 8. Let 6=a0. Then 6 = 1 and I = S . In fact,

= <@, 6>

and <$D,

I > = <@,1>

=@(O)=

s

s

y(x)dm(x)= c v , I >

= @(x)dm(x)=(O(O)=
a>.

Theorem 5.1. The Fourier transform N is a topological automorphism of 9 '' equipped with the weak*-topology (Appendix F). The inverse of fl is the conjugate Fourier transform fl*. Proof. 5 and N*are linear mappings of 9' into 9".Moreover for any T E9" and p E 9, we have <(a,

and

5 X * T > = < X * 5 p , T> =

154

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

R"

T * X T > = < T T * p , T> = by theorem 1.2. Hence 9 * T T = T and 9 9 * T = T for every T E9'.So fl is a linear automorphism of 9' and 9-l=S Take *. a finite number of elements pl,..., pmin 9 and a positive number E , and and put
.

&I, ..., p m : E ) = { T E ~ I I < e for l d k s m } Then the family of all such subsets of 9' is a fundamental system of neighborhoods of 0 in 9'' equipped with the weak*-topology. Since

-

pm: e)) = V ( T * V ~..., , F * V m : 81, and 5 is an automorphism of 9, both 9and X - l = S *are cononto itself. q.e.d* tinuous and hence homeomorphisms of 9' T(v(~1y

y

CHAFTER IV

The Euclidean motion group

In this chapter, we study harmonic analysis on the motion group M(2) of the Euclidean plane Ra. The group M(2) is the simplest example of a non-commutative, non-compact Lie group. First we construct a series of irreducible unitary representations Ua parametrized by a positive real number u>O. A function on M(2) can be expanded with respect to the set of representations { U“}. More precisely, if we introduce the Fourier transformf of a functionf i n L1(M(2))by means of the formula {(a)=/

arm

f(x)U,-P dg,

0-0,

then the analogues of theorems of Schwartz (Ch. 111. Th. 1.2), Plancherel (Ch. 111. Th. 2.1), and Paley-Wiener (Ch. 111. Th. 4.1) are valid for the “Fourier transform” on M(2). For example, iff is a “rapidly decreasing” function on M(2) (for the precise definition, see 43), the inversion formula Ng)= p f ( u ) U , a ) a h and Parseval equality

I f(g)12&=/;-I1

P(U)lls2U~

hold. These formulas are natural extensions of the corresponding formulas for a compact, non-commutative group (Ch. 1I) and for the non-compact, commutative group R” (Ch. 111).

$1. Construction of irreducible representations The Euclidean plane R3 is identified with the complex plane C by the mapping ( ; ) n x + y i .

The Euclidean inner product

given by 155

(2,

w)=xu+yv is

156

EUCLIDEAN MOTION GROUP

(z, w) = Re(z@)

for z = x + y i and w=u+vi. The mapping of C into itself which keeps invariant the distance d(z, w) = Iz - wI between any two points z and w in C is called an isometry of Rz = C . The group Z(R2) of all isometries of C is generated by three kinds of transformations: 1) translations t(z) : W H W + Z , 2) rotations r(a) : wHeiuw, and 3) the reflection s : wnw. The isometry group Z(Ra)is a 3-dimensional Lie group with two connected components. The connected component G containing the identity 1 is generated by the set of all translations and rotations. The group G is the motion group of R 2 = C . In matrix notation, the motion group G can be given in the following way.

Definition 1. The subgroup G of GL(2, C ) consisting of the matrices (1.1)

h(Z,a)=(c

f>

for any a € R

and any Z E C

is called the motion group of the Euclidean plane and is denoted by M(2). Since

h(0, a) represents the rotation r(a) and h(z, 0 ) represents the translation t(4.

Proposition 1.1. Put h(0, a)=r(a) and h(z, O)=t(z). Then we have the following relations 1) r(a)r(B)=r(a+p),t(z)t(w)=t(z+w), 2) r(a)-l=r(-a), t(z)-'=t(-z), 3) h(z, a )= t(z)r(a>, 4) r(a)t(z)r(a)-l=t(r(a)z), 5) h(z, a)-'=h(-r(-a)z, -a), 6 ) h(z, a)h(w, B)=h(z+r(a)w, a+@. Proof. These relations are easily proved by direct calculation of products of matrices. For example, 6) is proved by the relation

1) and 3) are particular cases of 6). 2) follows from 1). As transformations on C, these elements satisfy the relations

CONSTRUCTION OF IRREDUCIBLE REPRESENTATIONS

157

+

r(a)t(z)r(a)-lw=r(a)(r(a)-'w+z ) = w r(a)z=t(r(a)z)w for any w E C. So we have 4). 5 ) is proved by using 2), 3) and 4).

q.e.d.

Remark. Proposition 1.1, 5 ) and 6) show that G=M(2) is actually a subgroup of GL(2, C ) . Proposition 1.2. Let T = {t(z)JzE C) and K = {r(a)laE R} Then 1) T and K are subgroups of G= M(2) and we have T=Ra, KrU(1)zT. 2) G=TK, T n K = { I ) . 3) T is a normal subgroup of G. 4) G is the semidirect product of T and K. G is homeomorphic to each of the product spaces T x K and Rax T. Proof. Proposition 1.2 is a direct consequence of Proposition 1.1.

.

To construct the irreducible unitary representations of G= M(2), we use the method of induced representations. This technique was discovered and effectively used by Frobenius for finite groups. This is a method of constructing a representation of a group from a representation of some subgroup of it. The theory of induced unitary representations of locally compact groups was initiated by G. W. Mackey [3]. We shall give a brief account of induced representations of topological groups in the last part of this section. Here we construct directly the irreducible unitary representations in the form most convenient for later use. The representation to be constructed is induced by an irreducible unitary representation za :t(z)I+ efCXsa) of the subgroup T. This fact will be proved later. Since K G U(1)z T =R/2xZ, (1 -2) dr =da/2n is the normalized Haar measure of the compact group K. In this section, the Hilbart space La(K)=L3([0,2x],da/2x) is denoted by H.

Theorem 1.1. Let a be any element in Ra. Then there exists a unitary representationU" of G=M(2) on H = La(K)defined by (U,"F)(s) =ef(*.'"'F(r(a)-'s), (1.3) where g=f(z)r(a)=f(z)rand F E H. Proof. Since

158 ( 1.4)

EUCLIDEAN MOTION GROUP

(Ug"F,UguF')=

F(r-ls)F'(r-'s)ds

SR S K

=

~ ( s ) ~ m=d(F,s F')

for any F, F' E H , UgUis a unitary operator for every g E G. If gt = firi, fr=t(zt)ET, r t E K ( i = l , 2), then glgs=t(zl +r1z2)r1ra. Thus we have (uVl92"F)(s) =ei(si+riza, s@)F(ra-lrl-ls) -eKX1, IU),+a. ~ l - l ~ u ) F ( r ~ - l ( r ~ - l ~ ) )

UgaUF) (r1-ls) = [uol'(ug,uF) I($> and =ei(zls'u)(

Uglgaa= UglaUgza* It remains to prove that the mapping U" :g n Uguis strongly continuous. It is sufficient to show that for any I; in H and any E>O, there exists a neighborhood N of 1 in G such that

(1.5)

(1.6) IIU,"F-F\I<E for any g E N . If F=O, (1.6) is trivial. We assume that FzO. There exists a continuous function p E C ( K ) satisfying

IIF-v11
IILrp-pll..<e/6

for any r E V .

Similarly since I(x, sa)l d 1x1 )a1 for any s E K (1.9) there exists a neighborhood W of 0 in Ra such that

(1.10) let(a,ra)- 11<~/61[p11for every x E Wand every s E K. Let N = t ( w ) x V. Then N is a neighborhood of 1 in G. If g = t(z)rE N , then we have (1.11) 11Ug"p- yII.. =sup ler(*. '")p(r-ls)-&)I 5 Iletcx*au)(p(r-ls) -p(s))llm Il(e*(". m - l)p(s)ll, d IILp - pll- + Ile'('. 1 ll~llpll<€16 el6 =€13

+

-

+

159

CONSTRUCTION OF IRREDUCIBLE REPRESENTATIONS

by (1.9) and (1.10). The inequalities (1.7) and (1.11) together with the relations \IUgaFII= llFll and llplls Ilpll.. prove that

IIUgaF-FII IIIU9"F- Up"pll+ IIuaa(4-$4f Ilp-FII <&/3+&/3+&/3=&.

Theorem 1.2. Then we have

q.e.d.

Let R, (r E K) be the right regular representation of K .

R, Uga Rr-'= Ugra for any r E K,g E G and a E Ra . In particular if la1 = Ibl, then U a g Ub. Proof. For any roE K, g =f (z)rE G and F E L2(K),we have (1.12)

0

0

(RroUgaRro-'F)(s) =(UgaRr0-'F) (sro)=et(** TO - l F ) (r-lsr0) =&(Zs rroa)F(r-ls)=(Ug'oaF)(s1.

If la1 =lbl, then there exists a rotation r E K such that b=ra. Hence we get Ugb=R,UaaRr-l for any g E G. Since R, is a unitary operator on L2(K),we have proved that U a is equivalent to U'. q.e.d. By Theorem 1.2, it is sufficient to study the representations U a for aLO.

Proposition 1.3. If we identify L2(K)with L2([0,24, da/2a), the unitary operator Uga(ar0) for g = f(pe(p)r(a)is represented by (U,aF) (0) =etapcos(p-e)F(e -a) . (1.13) Proof. Since (z, a) = Re(zri), we have (z, r(0)a)= Re(apet(p-8))=apcos(~ - 0) . Hence (1.3) can be rewritten in the form given by (1.13). Proposition 1.4. 1) Let F be an element in H=LZ(K).Then UTaF=F for every r E K if and only if F=co for some constant CO. 2) Let pa(g)=(Ugal, 1) for U E R and gEG. Then we have p a ( g ) = Jo(ap), for g = t(pe")r(a), where Jo is the Bessel function of order 0. Proof. 1) Since Ura=L, for each r E K, we get

1

urcm>a(

ne2

Cnxn)

(0) =

1c n e - ~ n ~ x n (.~ ) nci

Hence F= CC,,~,, E H satisfies UraF= F for every r e K if and only if F= cox0 =cO.

160

EUCLIDEAN MOTION GROUP

2) By (1.13), we get

=L/ax etap eosede

277 0 =Jo(ap). The last equality is the definition of the Bessel function Jo. (Bessel's integral formula). Theorem 1.3. Let a, b B R1.Then Ua is equivalent to U bif and only if la1 = lbl. Proof. By Theorem 1.2, it is sufficient to prove that if U a s U e for a, b z O , then a=b. If U az Ub, there exists a unitary operator T on H such that Tuga= /. UgBTfor every g E G. Therefore UTbT1= TUral = T1 for any r E K, hence T1= c by Proposition 1.4. Since T is unitary, IcI = 1. So we have pa(g) = (Ug'l,

1)=(Tuga 1, T1) =(UgbT1,T1)=lcla(Ugbl,1) = q ~ b ( g ) for any g E G.

Thus we have (1.14) Since

Jo(ap)=Jo(bp) for every

oER

.

we have (1.15) Let fa(x)=Jo(ax).Then we have

.

(1.16) fa"(0)=aaJo"(0) Therefore (1.14) implies that aaJo"(0)=baJo"(0),hence, aa=bs by (1.15). So we have proved that if U a sU band aLO, b z O , then a=b. q.e.d.

Theorem 1.4. If la1 >0, the unitary representation U a is irreducible. Proof. We can assume that a>O by Theorem 1.3. It is sufEcient to prove that if a projection operator P on H=L'(K) satisfies

CONSTRUCTION OF IRREDUCIBLE REPRESENTATIONS

161

Puga = UgaP for any g E G , then either P=O or P = 1 (Ch. I. Proposition 2.3). For any integer n, put Xn(e)= eine and Px,=f n . Then we have Sn(8-a) = (Ur(m)aPXn) (8) = (PUr(a)aXn)(8) -P(efn(e-u)) =e-tnufn(e) for any a

.

Hence we have fn(a) =cnetnu

i.e. (1.17)

for almost all a ,

PI,,=cnzn for every n E Z .

(Ch. I. (3.15)). If x E R, then

P(etascoseetne) =(PUt(.)'Xn)(B)=(Llt(=)"PXn)(8) =Cn(Ut(rlaXn)

(0) = Cneiaf

coseeins

.

Hence

Since

by Taylor's theorem, we have

(1.20) =cniacosBetne,so that (1.18)and (1.20)prove that P(iu cos8etn@) P(cos8 ein8)=cn cose efne; (1.21) considering t(xi) instead of t(x), we get similarly P(sin8 erne)=cn sin&P . (1.22) (1.21) and (1.22)prove (1.23) Pxn+l =CnXn+l for every n E Z (1.17) and (1.23) prove that Cn+l=Cn and hence Cn =CO for every n E 2. Since (x,,),,,~ is an orthonormal base of H, we have

162

EUCLIDEAN MOTION GROUP

P=col.

(1.24)

Since P a=P,coa= co and co= O or 1. We have proved that P=O or 1.

q.e.d.

Definition 2. The set of representations P= {U"la>O} is called the principal series of irreducible unitary representations of M(2).

Induced representations There exists a canonical method of constructing a representation T of a group G from a representation r of a subgroup H of G. The representation T is said to be induced by T. In its essence, the induced representation T is the natural representation of G on the space r ( E ) of sections of the homogeneous vector bundle E associated with T. Let V be the representation space of r. Then E is constructed as follows. An equivalence relation R is defined on the product space G x V defining (g, x)R(g', x') if and only if there exists an element h in H such that g'=gh and x'=r(h)-'x. Let E= (G x V ) / R be the quotient space of G x V by R . If H is a closed subgroup of a Lie group G and r is a continuous representation of H , then E is a vector bundle over the homogeneous space G/H associated with the principal bundle G(G/H, H ) . But here we are giving the description of the induced representation in an abstract setting and we make no such additional assumptions. Let ( g , x ) be the equivalence class containing (g, x) E G x V under the relation R and p : (g, x) W g H be the canonical projection of E onto G/H. G acts on E from the left by the rule

mo,

g(g0, x>= x> * A section f of E is, by definition, a mapping f:G/H+E satisfying paf = Ido,x. Let r ( E ) be the set of all sections of E. Then r ( E ) has the structure of a vector space. If fi,& E r ( E ) and fi(gH) = ( g , xi> (i= 1,2), then fl+fa and ufl (a is a scalar) are defined by

(fi+fa)

= ( g , xi

+xa>,

(ah)( g H )= -

G acts on r ( E ) canonically from the left and r ( E ) is a G-module in virtue of the representation T of G on r ( E ) defined by ( T d )(goH)=g *f(g-lgoH). The representation T of G is said to be induced by the representation t of H. The induced representation can be realized in another way. Let W be the space of all mappings F: G-, V satisfying F(gh)=~(h)-lF(g) for any g E G and any h E H . Then a representation S of G on W is defined by

setting

CONSTRUCTION OF IRREDUCIBLE REPRESENTATIONS

163

-

( S l a (go)=F(g-lg0) The representation S is equivalent to the induced representation T. Let F E W and r be the mapping of G into G x V defined by 7 ( g )=( g , F(g)). Then r(gh)= (gh, r(h)-lF(g)) is equivalent to r(g) for any g E G and h E H. Hence we get a mapping f : G/H+E by passing to the quotient:

G - G xI V

4-

Ip

G/H

f

E

-

f k H ) =( g , F O ) The mapping 0 : F H f is a linear isomorphism of W onto r ( E ) . In fact it is clear that 0 is injective. Let f be an element in r ( E ) and put f ( g H ) = ( g , x ) . Then an element F in W is defined by putting F ( g ) = x and it satisfies 0 ( F ) = f . Hence 0 is surjective. The linear isomorphism 0 is an intertwining mapping between T and S, i.e. 0 satisfies O O S , = T , O O forevery

gEG.

In fact, we see that "0 SJFI(goH)= (go9 (&F) (go)) =g(g-lgo, Fk-'go)> =&W) 1(g-'goH) = U l 7 @IF1 (gem. O

O

Detailed studies of the induced representations have been made under the assumption of additional conditions on G , H and r by taking a suitable subspace ro of r ( E ) . For the case where G is a finite group, see, e.g., Curtis-Reiner [11. For the case where G is a compact group, see Weil [l]. For the case where G is a complex Lie group and rois the space of holomorphic sections, see Bott [l] and Kostant [2]. For the case where G is a Lie group, see Bruhat [l]. For the case where G is a semisimple Lie group and rois the space of square-integrable sections, see Okamoto-Ozeki [11, Narashimann-Okamot0 [l], and Hotta [l]. Here we give the definition of unitary induced representation of a locally compact group due to G. W. Mackey [3]. Definition 3. Let G be a locally compact group, H a closed subgroup of G and assume that GIH is o-compact and that there exists a positive Ginvariant Radon measure AL, # O on G/H. Let r be a unitary representation of H on a separable Hilbert space V

164

EUCLIDEAN MOTION GROUP

and let H be the space of all functionsf on G with values in V satisfying the following three conditions: 1) gt+(f(g), v) is measurable for every v E V, 2) f(gh)=r(h)-'f(g) for every g E G and h E H,

where g=gH. Then W is a Hilbert space with the inner product

(fi,fa)= ~ G , H ( x ( g ) . $ d ( g ) ) d P ( 9 ) *

A unitary representation T of G on H is defined by setting (1.25) ( T d )(go) =f(g-'go). The unitary representation defined by the last equality is called the representation induced by r and denoted by T=Ind t. HtQ

Proposition 1.5. The principal series representation U aof the motion group M(2) is equivalent to the representation Ta induced by the irreducible unitary representation

x a : t(z)i+ef(s.a) of the translation subgroup T. Proof: Define the linear mapping A of La(K)=H(Ua)into H = H(Ta) by setting (AF)(t(z)r)=xa(t(-r-'z))F(r) for F E L s ( K ) .

Since (AF)(t(z)r t(zo))=(AF) (t(z+rzo)r) =xa(t( -r-'z-zo))F(r)= Xa(Zo)-'(AF)(t(z)r) A F actually belongs to H. Moreover, since dr is the G-invariant measure on K = G / T , we see that

Thus A is an isometry of La(K)into H. A is surjectivebecause A C ( K ) is dense in H. A is an intertwining operator between U aand T a ,i.e. A satisfies A 0 U g a = T g a o A forany g e G . In fact, if g = t(w)s and s E K, then

CLASSIFICATIONS OF IRREDUCIBLE UNITARY REPRESENTATIONS

165

In the above definition of unitary induced representation, we assume that-there exists a positive G-invariant measure p # O on G/H. However no such a measure p exists in many important cases. In general case, a quasi-invariant measure p plays the same role as the invariant measure in the above discussion. A positive Radon measure p on G / H is called quasi-invariant if the measure gp ( ( g p ) (A) = p(g-lA)) is absolutely continuous with respect to p for any g E G. It can be proved that there always exists a quasi-invariant measure # 0 on G/H where G is a locally compact group and H is a closed subgroup (cf. Bourbaki [3] Ch. VII. $1. no 9). The measure gp has the density (Radon-Nykodimderivative) d(gp)/dp with respect to p. Using the quasi-invariant measure p, we can define the induced representation T o n the space H in Definition 3 by setting

(Tuf)ko)=(dp(g-'b)/~p(80) )'f(g-lgo) .

52. Classification of irreducible unitary representations There exist irreducible unitary representations other than the principa) series representations constructed in $1. Let p :G H K be the projection of G= T K onto K : p(t(z)r)=r. Then p is a homomorphism of G onto K whose kernel is T. Since G I T z K , any irreducible unitary representation x of K defines an irreducible unitary representation x op of G. We know that 12- ( ~ ~ : r ( c u ) ~ e ' " - I n ~ ~ ) is the set of all irreducible unitary representations of K (more precisely l? should be called a complete set of representatives of the set of classes of irreducible unitary representations).

Theorem 2.1. (It0 [I]-Mackey [3]). Any irreducible unitary representation U of the motion group G=M(2) is equivalent to one of the elements in the set

e=(U"la>O) u (XnoplnE21. No two elements in

6 are equivalent to each other.

166

EUCLIDEAN MOTION GROUP

Proof. Since Ua is infinite-dimensional and zn op is one-dimensional, Ua is not equivalent to zn op. Hence the last part of Theorem 2.1 is clear

from Theorem 1.3 and Ch. I, Theorem 4.2. Let U be an irreducible unitary representation of G=M(2) on a Hilbert space H and put Urtr)=V, and

Urta)=W,

.

Then V is a unitary representation of R a g T.Hence by Stone's theorem (Ch. I11 Theorem 3.2), there exists a unique spectral measure E on the family B of Borel sets in R1such that

For any Borel set A E: b and any x E Ra, we have (2.2) V,E(A) =E(A)v, (cf. Ch. 111. Theorem 3.2). Since r(a)t(x)r(a)-l=t(r(a)x), we get

s

e"". W (W,E(y)W,-I) = W, V , W,-'

= Vr(a)s = = /e'(*.

s

e"".r(a,-'ff)&(y)

u)dE(r(a)y).

Hence for any Borel set A E 23 and a E R,we have WaE(A)Wm-'=E(r(a)A) (2.3) by the uniqueness of E. In particular, we see that (2.4) if r ( a ) A = A , then W,E(A)=E(A)W,. By (2.2), (2.4) and the irreducibility of U, we see that (2.5) if r(a)A= A for every a E R, then E(A)=O or 1. In particular an open disc Br= { X E R21 1x1 < r } of radius r>O is invariant under every rotation, and we get (2.6) E(B,)=O or 1 for any rZO . Since u Bn=Ra and E(Ra)= 1, E(B,)= 1 for sufficiently large c by the n

strong continuity of E(Ch. I11 Proposition 3.9.). Put a = sup [r IE(B,) =01 . Then we have O$a< + a. Since u Ba-l,n=Ba, we have n

CLASSIFICATIONSOF IRREDUCIBLE UNITARY REPRESENTATIONS

(2.7)

167

E(B,)=O.

Put Dr= { x e Ral I x l s r } . Then E(Dr)=l if r > a . Since n Da+l,n=D,, we have n

(2.8) E(D,) = 1 by Ch. 111, Proposition 3.9. (2.7) and (2.8) show that the spectral measure E is concentrated on the circle S,= { x e RaI Ixl=a). There exist two possibilities: a=O and a>O. We examine the two cases separately.

Case I, a =0. In this case, we get ”

for every X E R a . Hence the kernel of U contains the subgroup T and U can be regarded as an irreducible unitary representation of K. Therefore U is equivalent to xn o p for some n E Z by Ch. I. Theorem 4.2. Case 11, a>O Since W is a unitary representation of the compact group T E K, W is a direct sum of irreducible unitary representations (Ch. I. Theorem 3.1). In particular there exists an integer n and the representation space H of W contains a unit vector u satisfying (2.9) Wau= einau for any a E R . Let 9 be the mapping r(a)nae*”.Then 9 is a homeomorphism of the group K onto the circle S,. Put F(B)=E($(B)) for B E b(K), where b(K) is the set of all Bore1 sets in K. Then F is a spectral measure on B(K). (2.1) can be re-written as

Let p be the measure on b(K) defined by p(B)= IIF(B)ull’ for B E b(K). Then by Ch. I11 Proposition 3.12, for each function f E La(K,p ) , there exists a linear operator

defined on the space

168

EUCLIDEAN MOTION GROUP

The mapping CJ : f ~ H ( fis) an isometry of La(K,p ) onto a subspace M = M ( u ) of H {Ch. 111. Proposition 3.12). M is isometric to L'(K, p) and is complete, hence is closed. Now we prove that the closed subspace M is invariant under the representation U.By (2.3) and (2.9), we see that ( ~ u ~ ( f )v)u=,r>(n)dA(waF(l)u, v) =

J)(~)Mw +a) ~ . u v) ,

=/ ; e t n q f ( l

-a)d(F(R)u, V) .

Putting

f=W=etm-f(A-a) , we get (2.11) W , H ( f ) u = H ( f " ) u for any a E R and f E La(K,p ) . We recall here that the integer n in (2.11) depends on the choice of u in (2.9). (2.10)

If we put ~ ~ ( r ) = e ~ ( * .then ' " ) ,we may write V,=H(x.). Hence if we put g,(r) = ~.(r)f(r) = e((".ra)f(r then by Ch. 111, Proposition 3.12, 7), we get 1

9

.

V,H(f)u=H(g,)u for any z E C and f E L'(K, p ) (2.12) (2.11) and (2.12) prove that M = { H ( f ) u l f L2(K, ~ p)) is invariant under U. Since U is irreducible, M must be 0 or H . Since M contains u= H(l)u+O, M must be the whole space H. Put e(l)=e6n2;then since emn(;c)= efnaeinCa-a' =&), (2.1 1) shows that uo=H(e)u satisfies (2.13) Wuuo=uo for all a E R . Hence we can take uo instead of u in (2.9). Starting from uo, the above argument shows that the isometry CJ is the desired intertwining operator between U and U".In fact, the measure po(B)=IIF(B)uoll' is invariant under the translation r ~ r o rin the group K by (2.3), (2.13) and satisfies p o ( K ) = 1. Hence dpo(l)=dR/2n. Furthermore, the isometry CJ : f ~ H ( fis) an isometry of L'(K) =La([O,2x1, d2/2n) onto M(u0)=H. Moreover, the function fqo(l) in (2.10) for uo is given by fa0(A) =f(l -a) =(U,c.)"f)(A). Hence (2.11) and (2.12) can be re-written as KO-1

urCu, WI (8) = f ( e ) = (urcU)af) (0) a

[(CJ-l o

Utca)o @)f](0) =g.(8) = =

e'("9

r(s)a)

(utca)af) (0) .

f(e)

d

FOURIER TRANSFORMS OF RAPIDLY DECREASINGFUNCTIONS

So we have proved that O-l.UvaO=Uga forevery g E G , and U is equivalent to Ua.

169

q.e.d.

$3. Fourier transforms of rapidly decreasing functions In the following three sections, we discuss the Fourier transform on the group M(2) which is always denoted by G. In this section, we introduce the space Y(G) of rapidly decreasing functions on G=M(2). It turns out that any rapidly decreasing functionf has an expansion with respect to the principal series representations Ua(a> 0). It is a remarkable fact that the one-dimensional irreducible representations x,, ap of G play no part in the expansion. It is also noteworthy that there exists a close relation between the Fourier transform on G=M(2) and the ordinary Fourier transform on R'. For a function f on G =M(2) we write f ( g ) = f ka )= f kr )

if g = t(z)r(a)= t(z)r. Proposition 3.1. Let g = t(z)r(a)= t(x+yi)r(a). Then dg =dzdr =dxdyda is a left invariant Haar measure on G =M(2). It is also right invariant and G is a unimodular group. Proof. Since t(zo)rot(z)r= t(zo+r0z)rOr and the Lebesgue measure dz on Ra is invariant under the rigid motion zHroz+zo,and since dr is the Haar measure of the group K, we have d(gog)=dg. So dg is a left invariant Haar measure on G =M(2). Similarly we have d(ggo)=dgo and G is unimodular. q.e.d. Definition 1. The measure on G given by dg=dzda =dm(z)dr (2d' is called the normalized Haar measure on G. In the following dg always denotes the normalized Haar measure given by (3.1). Definition 2. The Fourier transformf of a functionf E L 1 ( Q is a function on R*+=(O, + a) with values in the Banach space B(L'(K)) of aU bounded linear operators on L1(K)defined by

170

EUCLIDEAN MOTION GROUP

{(a) =

(3.2)

1

f(g)Ua,-,dg for a> 0 ,

Q

where U ais one of the principal series unitary representations of G =M(2) constructed in $1. Proposition 3.2.

Iff and h are integrable functions on G, then we have

1) IIf(~)lldllflllfor any a>O,

2)

and

(f*W=v,

3) (I*)* (a)=tf(a))*, where f * ( g ) = f j . Proof. 1) For any vector u E H, we have

Ilf(a)ull.~ ff If(s)l II~"0-144?

5 llflll llull * 2) By Fubini's theorem, we have

= h(a)f(a). 3) For any two vectors u, v E H,

Definition 3. A complex-valued C"-function f on G = M ( 2 ) is called rapidly decreasing if for any N E N and m E N 3 we have

(3.3)

where

m,m(f)=

SUP I(l+lZl')"(~"f)(Z,

arR, a c C

all < +

9

FOURIER TRANSFORMS OF RAPIDLY DECREASING FUNCTIONS

171

The vector space of all rapidly decreasing functions on G is denoted by

9=.y(G). 9 is a Frdchet space in the topology given by the family of seminorsms { p N , , I N ~N,m E N 3 }

.

Proposition 3.3. 9(G)cLp(G) for any p 2 1. Proof. Since f E 9 ( G ) satisfies

Ifk a>lIpiv.o(f)(1 + I z l Y for any N E N, Proposition 3.3 is a direct consequence of Proposition 3.1. q.e.d. Proposition 3.4.

Iffis a rapidly decreasing function on G,then we have

(3.4)

for any u>O and F E La(K), where

.

(3.5)

f ( z , rs-')e-*(".ra)dm(z)

Proof. If F and F' belong to La(K)and g = t(z)r, then g-l= r l t ( -z ) = t( -r-lz)r-l and n

n

,

.

n

n

Since F' is arbitrary element in La(K),we have proved that (3.4) is valid for almost every s E K . In the above proof of Proposition 3.4, we actually prove the following corollary.

COROLLARY 1 to Proposition 3.4. If YE Ll(G), then (3.4) is vaIid for almost all s E K. Proposition 3.4 shows that i f f € g ( G ) , thenf(a) is an integral operator on La(K)whose kernel k, is given by (3.5). COROLLARY 2 to Proposition 3.4. We denote the ordinary Fourier transform off(z, r ) , regarded as a function of z E Ra, by f ( f , r ) :

172

EUCLIDEAN MOTION GROUP

(3.6) Then the kernel k&; s, r ) is given by (3.7)

kf(a;s, r ) =f(ra7rs-l) .

Definition 4. A bounded linear operator A on a separable Hilbert space H is said to be of truce cZms if for any orthonormal basis ( p n ) n rof ~ H, the series

-

(3.8)

C

(On)

n-0

converges to a finite sum which is independent of the choice of (lon). The sum of (3.8) is called the truce of A and is denoted by TrA. If A is of trace class, the series (3.8) converges absolutely, because the sum is invariant with respect to any change of ordering among the pn's.

LEMMA1. Let H be a separable Hilbert space. If a bounded linear operator A on H satisfies (3.9)

for a fixed orthonormal base then A is of trace class. Moreover, if U and V are two bounded operators on H, then UAV, AVU and VUA are of trace class and have the same trace. Proof. Let urnn=(A~rn,pn), urnn=(Uprn, pn)7 and Vrnn=(Vprn, pn)Then we have

Hence we see that (3.10)

UA Vprn =

~(rnn&kvkl(bL n. k. 1

The Schwarz inequality in 1 9 proves that

-

FOURIER TRANSFORMS OF RAPIDLY DECREASING FUNCTIONS

by the assumption (3.9). Hence the series C

&nankVLm

173

converges abso-

m.n.k

lutely and we can change the order of terms without affecting the sum. So we have by the Parseval equality

Let (+,,),,.N be another orthonormal basis of H. Then the linear operator W on N defined by Wp, =$,, is a unitary operator. By (3.12) we have

Hence UAV is of trace class. In particular, putting U = V = 1, we see that A is of trace class. Similarly A VU and VUA are of trace class. (3.12) shows that Tr(UAV)=Tr(AVU)=Tr(VUA). LEMMA 2. If k(0, p) is a C2-functionon T', then the Fourier series m

(3.13)

C

um,nei(m8+n+

m.n---

converges absolutely and uniformly. Proof. Let

of k

174

EUCLIDEAN MOTION GROUP

Then the Fourier coefficient's bm,, of h satisfy bm,n =(Ak,e(m,n)) = (k, Ae(m,n)) = -(ma+n3)am,,.

Since h =Ak E C ( T a )c La(T),Bessel's inequality shows that

-

1 Ibm, s llhllsa< + n12

m. n---

Hence by Lemma 1 in Ch. 11, $8 we have

q.e.d. Proposition 3.5. If k(e, v) is a Ca-functionon K a=K x K s T a, then the integral operator L on La(K)defined by

satisfies the assumption of Lemma 1 for an orthonormal basis (x,,)~.z. Hence L is of trace class and has the trace

Proof. Let Xn(B)=eine.Then ( p J n r zis an orthonormal base of La(K). Lemma 2 proves that the Fourier series (3.13) of k converges absolutely and uniformly. Since

-am,

-n

it follows that

C

I(Lxm, xn)I < + 00, i.e.9

m ,n=--

L satisfies the assumption of Lemma 1 for (&.Z. Hence Lemma 1 proves that L is of trace class. Since the series k(B,O)=

C m . n=--

am,

FOURIER TRANSFORMS OF RAPIDLY DECREASING FUNCTIONS

175

converges uniformly on T,we can integrate it term by term and get

=Tr L by (3.14). Theorem 3.1. (Inversion formula) Any function f in p ( G ) may be recovered from its Fourier transform f,by the formula f ( g ) = r m T r (Ugaf(a))adu.

(3.15)

0

In particular, Uguf(a)andf(a) are of trace class. Proof. Since f ( u ) is an integral operator with C"-kernel (Proposition 3.4), it satisfies the assumption of Lemma 1 by Proposition 3.5. Hence Ugaf(u)and f ( a ) are of trace class. Let g=t(z)u, u E K. Then Ugaf(a) is an integral operator with kernel m f ,#(a;s, r ) =e"Z8 aa)f(ru, rs-1 u ) . In fact, we have (u,~~(u)F) (s) =e"'".a a ) ( f ( u ) ~ (u-1s) )

=J

e*('.Ja)f(ru,rs-1u)~(r)dr

by Proposition 3.4 and Corollary 2 to it. Hence by Proposition 3.5, Tr(u.mf(a))=J

X

mf.g(f.2;Y r r)dr

=j-ne".'"'f(ra, u)dr . If r is fixed, then the functionf, :zl-.f(z, r ) is a rapidly decreasing function on Ra. Since

f ( u , r )=

j-

f ( z , r)e-f(z+)dm(z)

Ra

is the Fourier transform of A E 9 ( R a ) on Ra, the inversion formula for

f7 E 9 ( R s )gives

176

EUCLIDEAN MOTION GROUP

=fm

Proposition 3.6. Iff and h belong to 9 ( G ) ,thenf * h and f * (g) belong to 9 ( G ) . Proof. Since(r(z)r)-l=t(-r-lz)r-l,f*(z,a)=f(-r(-a)z, -a). Hence f * E 9 iff E 9.Since f E 9is bounded and h E 9is integrable (Proposition 3.3), f(ggl)h(g1-') is integrable with respect to gl. Hence

(f* h) k ) = J

fkgl)h(gl-l)&l (I

is defined and has a finite value for all g E G. If g=t(z)r and gl =t(zl)rl, then ggl =t(z rzl)rrl Hence for any m E N 8and N E N,we have

+

.

l(1 +Izl')NDc,,r,mf(z+rZI, rrl)h( -r1-lzIy r1-91 JPN.m(f)Ih(-rl-'zl, rl)l * Thus we can differentiatef * h under the integral sign and have

(3.15)

(1 + Izl')"D"(f*

=so

4(g)

(1+ IZl")"(a"f)(ggl)hkl)&

*

Since

(1 + IZl')"l(D"f )(ggl)t s ( 1 + IZI')-'PN+%m(f) we see that (3.15) tends to zero when lzl++ m by Lebesgue's theorem. Hence the left hand side of (3.15) is bounded andf*h belongs to 9 ( G ) . q.e.d. 9

Definition 5. Let Hand H' be two separable Hilbert spaces. The set of bounded linear operators from H into H' is denoted by B(H, H'). Let (pn).,.,v be an orthonormal basis of H. For any element A of B(H, H'). Put

-

(3.16)

IIAII~~= 1IIApnIIf n=O

-

FOURIER TRANSFORMS OF RAPIDLY DECREASING FUNCTIONS

177

The value of (3.16) is independent of the choice of (+). In fact, if ( @ J n r N is an orthonormal basis of H', then, by the Parseval equality, we have

Hence the value of (3.16) is independent of equality shows that

Moreover, the last

(cp,).

(3.18) llAll2= IIA*llz An operator A E B(H, H') is called a Hilbert-Schmidt operator (abr. H-S operator) if IIAlla< 03. Let

+

+

(3.19) Ba(H, H')= { A E B(H, H')I llAlla< 00) . be the set of H-S operators from H into H'. Then Ba(H, H') is a subspace of B(H, H') and llAlla is a norm on Ba(H, H') which is called the Hilbert-Schmidt norm (abr. H-S norm). Moreover if A and B belong to Ba(H, H'), then the inner product (3.20) is defined. With the inner product (3.20), Ba(H, H') is a Hilbert space. A bounded operator A E B(H, H') is an H-S operator if and only if A*A is an operator of trace class on H. If A and B are H-S operators in Ba(H, H'), then B*A is an operator of trace class and we have (A, B) =Tr(B*A) .

(3.21)

Theorem 3.2. (Parseval equality) Iff belongs to 9 ( G ) , thenf(a) is a Hilbert-Schmidt operator on H = La(K)and it satisfies (3.22) Proof. Put h= f *f *. Then h belongs to 9 ( G ) by Proposition 3.6. Since &a) is of trace class (Theorem 3.1 1) and Tr &a) =Tr ((f*>"(a)f(a)) =

Tr(f(a) *f(a))= IIf(a)l12' by Proposition 3.1 and (3.21),f(a) is a HilbertSchmidt operator. Moreover, Theorem 3.1 proves that

:1

h( 1) =

Tr (&(a))a& =J:-Il

f(a)llAzda

.

On the other hand, by the definition of the convolution product, we have

h(l)=j-@fk)E)&=J We have proved (3.22).

Q

Ifk)l@r. q.e.d.

178

EUCLIDEAN MOTION GROUP

The image 9= { f If€ g ( G ) } will be determined in $5. As an application of properties of the Fourier transform on G we calculate the character of U a . As we have seen in Chapter 11, the characters determine the unitary representations of compact groups. The characters play the same role for the non-compact group G =M(2). However, here is a difficulty in defining the character of U",because an infinite-dimensional unitary operator U," does not belong to the trace class. In fact, the series "

(3.23) does not converge. This difficulty is overcome by considering (3.23) as a distribution on G. Definition 6. Let g ( G ) be the space of all complex-valuedC"-functions on G with compact supports. For any 00, put B5= {t(z)rE GI lzlsa} and

a u = g ( B u ) = { f ~ s ( G ) l f ( z ,a)=O if lzl>a} . a(&) is a Frkchet space with respect to the family of seminorms:

(3.24) IPm(f)= IP"f Il-lm w Then g ( G ) = 6 s ( B n ) is topologized as the strict inductive limit of n=1

9 ( B n )(cf. Ch. 111. Definition 6). A continuous linear form on the topological vector space a(G) is called a distribution on G. The space of all distributions on G is denoted by m G ) . Example 1. Iff is a continuous function on Raand p is a Radon measure on K E T,then the linear form f@,u on =(G) defined by

P H (lo, f @ p )

f (z)lo(z, a)dm(z)dl*(a)

=

(lo

Es

( G ))

S R J K

is a distribution on G. Now we consider (3.23) as a distribution x5 on G. This means that the character xu is the linear form on s ( G ) defined by

C J f(g)(Ugaxn, Xn)dg= C

Xu :fH

s---~l

where

Q

n--*

(U/axn,

Xn)=TrU/"s

PJANCHEREL THEOREM

179

Theorem 3.3. (Vilenkin) For any fixed 0-0, the linear form xu :fe Tr Ufais a distribution on G. In fact, xa is equal to JO(alzl)@(a) (cf. Example l), where JO is the Bessel function of order 0 and 6 is the Dirac measure at 0 on T z K. Proof. By Proposition 3.4 and 3.5, Ufais oftrace class and 1

Tr UfU=&J

Par

kh(a;8, e)de 0

q.e.d. Remark. From Theorem 3.3, we obtain another proof of Theorem 1.3. In fact, if U u z U b then we have U I u t z and X l o l = X l b l by Theorem 1.2. Hence we have la1 = lbl by Theorem 3.3.

$4. The Plancherel theorem In this section, we discuss the Fourier transform of the square-integrable functions on the group G=M(2). We shall prove an analogue of the Plancherel theorem which asserts that the Fourier transform fl ;f-f on G can be extended uniquely to an isometry of La(G)onto LaB,(R*+,ada) =/ : @ H ,

a&, where H , = Ba(La(K))is the Hilbert space of H-S opera-

tors on La(K)(Theorem 4.2). The analogue of the Plancherel theorem can be interpreted as the decomposition of the regular representation into irreducible representations (Theorem 4.3).

LEMMA1. Let H=La(X,p ) , H'=L'(Y, v ) , and let @ be the mapping of H"=L3(Xx Y, p x V ) into the space Ba(H', H) of H-S operators (cf. $3, Definition 5) which maps k E H" =L2(Xx Y, p x V ) into the integral operator K with the kernel k. Then @ is an isometry of H I onto Ba(H', H) = Ba. Proof. Let (p,,) be an orthonormal basis of H'. Then applying the Parseval equality to k(x, y), regarded as a function of y, we get

180

EUCLIDEAN MOTION GROUP

= CI(Ki2 (x)l'

*

I

Integrating the last equality over X,we get llkli2'=/

X

1

~ ~ ~ & b ' = ~ ~ ~ ~ ~

Ik(x, Y)l'dp(X)dp(Y)= f

Y

(G)

because is an orthonormal base of H'. The last equality proves that 0 is an isometry of H" into Br. Now we prove that Q, is surjective. Let K be an element of B, and define an operator K,, for each n E N by putting

Then K n is an H-S operator and ~~K-Knllzl= IlKv#+O

(4.1)

if n--r

+ . Q)

p>n

The operator Kn is an integral operator with the kernel n

kn (x, Y )=

C

*

(KPP)

P-0

in fact, since any f in H' can be written as

f= C ( f ,$Dk)$Dk ,

we see that n

c n

=

(f,v k ) ( K n p k ) ( x )

k-0

=(Kaf ) (4*

Since (4.2)

Ilkn-knrll,"~~Kn-Knrlln'=

IIKvkll'

if n > m ,

k-m+l

(K,) and (k,) are Cauchy sequences in Br and in H" respectively. Let k =lim k,, be the limit of (k,,) in H" and KO=@(k).Then since @[kn)= n-oD

K,, we have

181

PLANCHEREL THEOREM

limIIK~-KnII,=limIlk-knll,=O.

(4.2)

n--

(1-01

(4.1) and (4.2) prove that K = KO=Q(k).

q.e.d.

Definition 1. Let X and Y be two sets and T ( X ) , S ( Y ) , and ~ ( X Y)Xbe respectively the vector spaces of all complex-valued functions on X,Y, and X x Y. For any two functionf E f l ( X ) and g E .F(Y), Put ( f o g )(x, Y )=fW g ( Y ) * Then f o g E S ( X x Y). Since the mapping cf,g)Hf o g is a bilinear into ~ ( X Y), X there exists a linear mapmapping of x ( X ) x x(Y) ping @ : f l ( X ) @ T ( Y ) 4 f l ( X x Y)such that Q(f@d=fOg. Q is injective. In fact, let Q(h)=O. Then h can be written as

h=

Camnfm@gn

5

m. n

where

and (g,Jn are two linearly independent families in x ( X ) and

X(Y).Since

Cam.nfm(x)gn(Y)=o

for all x E x,Y E Y,

m. n

we have Cam,n fm=O for all n, by the linear independence of (gn).Hence U ~ , ~ = for O all m,n and h=O.

LEMMA 2. Let H=L2(X, p ) , H’=L’(Y, v), and H”=La(Xx Y, p x v ) . Then the mapping Q in Definition 1 can be extended uniquely to an isometry 0‘ of the Hilbert space tensor product H@H’ (cf. Ch. I11 53 Definition 2) onto HI‘. Proof. IfS, k E H and g , h E H’, then by Fubini’s theorem, we have

(fOg, k O h )=JX JYf(x)g(Y)koh(y)dp(x)dv(Y) = (f,k ) ( 8 9 h)

=(f@g, k0.h). Hence, if ((0,) and (+,) are orthonormal bases of H and of H’ respectively, (cpn@$m)n,m is an orthonormal system in H”. Since ( ~ n @ + m ) n . m is an orthonormal basis of H B H ’ , any element h in H@H’ can be developed in a series, (4.3)

h=

1(h, yn @ n. m

+m)pn

@ (In,

and

182

EUCLIDEAN MOTION GROUP

IW=

(4.4)

C~ ( h ,

-

pn@+m)Ia

n. m

The mapping 0,originally defined on the algebraic tensor product of H and H‘ can be extended to a linear mapping Of of H@H’ into H” by setting

~ ’ ( h=)

C(h,

pn 8 +m)vn

o

+m

n, m

for any h in H@H‘ (cf. (4.3)). The we have

z

IIO’(~)II~~= ~ ( hv,n B + m ) P

(4.5)

-

n, m

(4.4) and (4.5) prove that Of is an isometry of H@H‘ into H”. It remains is to prove that 0’ is surjective. It is sufficient to prove that (pn @,,,,q,m,) complete. To show this, it suffices to prove that if k E H” satisfies (k, (on @ +,J =O (4.6) then k=O. If k satisfies (4.6), then

for all n, m,

(k,f @g)=O for all f E H and g E H , because f = 2 anpn and g = C bm+m . Let K be the integral operator mapping H’ into H having kernel k. Then by Lemma 1 and (4.7), we see that llkllaa= llKIIaa=

1IIKG~II~= C IW+~,

vn)~2

n. m

7n

=

1

R.

I(k~n@+m)l’=O

m

q.e.d.

and thus k=O in H”.

By the isometry @’ in Lemma 2, we identify f @ g with f o g . So we write

(f@ g ) (x, v)= f (XI

*

With this notation, the following Corollary has been proved in the course of the proof of Lemma 2. COLLORARY to Lemma 2.

1) Iff, k E H and g, h E H ’ , then we have

(f@g,k@hh)=(f,k)(g,h), (4.7) 14.8) IIf@gIIa= llflla Ilglla 2) If ((on)n and (+m)m are orthonormal bases of H and of H’ respectively, then (cpn@gm),,, is an orthonormal basis of HI‘.

-

PLANCHEREL THEOREM

183

Theorem 4.1. Let G = M(2). Then 9 ( G ) is dense in La(@. Proof. By Lemma 2, La(Ra)@La(K)=La(G); hence, it is sufficient to show that for any f E La(Ra),g E La(K)and E >0, there exists an element h in p ( G ) such that (4.9)

Ilf@g-hIla<E

-

We can assume that f # 0 and g # 0. Since the set of trigonometric polynomials R(T) is dense in La(T)(cf. Ch. I. Theorem 4.3), there exists an element glE Cm(K)such that (4.10)

I I ~ - ~ i l l a < ~ / O l l f l l a ).

We can assume that ~/(211f11a)< Ilglla. Then we have gl #O. Moreover, since 9 ( R a )is dense in La(Ra)(Ch. 111. Theorem 1.4), there exists an element fi E . y ( R a ) such that

-

Ilf-fib <E/(2Ilgilld Let h =fl@gl. Then h is a C”-function on G = Ra x K and belongs to 9 ( G ) . By (4.8), we see that (4.1 1)

IIf@ g-hlli 5 If@ g-f@giIIa + IIf@ gi -fi @ gills = Ilf Ila Ik-gilla + IIf-fiII2 llgilla <E/2+€/2=E.

q.e.d.

Theorem 4.2. (Plancherel Theorem for M(2)). Let Ba = Ba(La(K))be the Hilbert space of all Hilbert-Schmidt operators on La(K). Put Ha=Ba for all a>O and H =

s:p

@Haadz. Then the Fourier transform

fl :fwfon .9’(G‘) (cf. 3.2)) can be extended uniquely to an isometry F of La(G)onto H. Proof. Theorem 3.2 proves that fl is an isometry of .9’(G) into H . Since g ( G ) is dense in La(@ (Theorem 4.1), 9can be extended uniquely to an isometry I; of La(G)into H . It remains to prove that F is surjective. Let L be an element in H . Then L is a function on R+* = (0,+ a) with values in B,. Since the value L(a) of L at a is an H-S operator on La(K), L(a) is an integral operator with kernel ka E La(Kx K) by Lemma 1. Put ka(s,r ) =k(a; s, r). Then we have

= / : 7 n J ~ / c ( u ;s, r)l’hdruh. Hence 0 :L w k ( a ; s, r ) is an isometry of H onto La(R+*x K x K). We identify H with La(R+*x K x K ) by the mapping 0.

184

EUCLIDEAN MOTION GROUP

Let (P :R+* x K+RP be the transformation (u,r)k+ru. Then the mapping gt+gov is an isometry of Ls(Rs)onto L2(R+*x K ) (the transformation to polar coordinates). We identify L’(R’) with L2(R+*x K ) and LS(R1 x K ) with Ls(R+*x K x K ) = H. Since F is an isometry, the image Im F is closed in H.Hence to prove the surjectivity of F,it is sufficient to show that Im 9is dense in If.Moreover since Im fl is contained in Im F, it suBces to prove that for any k E H =L’(R1 x K ) and E >0, there exists an elementf in P(G) such that

Ilk-k/Ila 0. Then u E La(Ra).Since y ( R a ) is dense in La(RS)(Ch. 111. Theorem 1.4), there exists an element v in 9 ( R a )such that

I ~ u-Vila <E . Let y * v = w be the inverse Fourier transform of v on Rs. Then w E 9 ( R a ) (Ch. 111. Theorem 1.2). Hence f=w@X-n belongs to Y ( G ) . This function f = w @ ~ - , satisfies , (4.12) for k=g&,,. In fact, since k/(a;S, r ) = ( X w ) (ra)X-n(rS-l) =v(ra) X-n(r)Xn(s)

(3.7)

9

we have

Ilg@Xn-k/lIx=II(X-nU)@ =IIX-n(U-~)llr

Xn-(X-nv)@ ~nll, IIXnlls=II~-~lll<E.

q.e.d.

Definition 2. The isometry F in Theorem 4.2 is called the FourierPluncherel transform on G =M(2). The Plancherel theorem for M(2) gives the irreducible decomposition of the regular representation in the same way as the original Plancherel Theorem gives the irreducible decomposition of the regular representation of Rn (cf. Ch. 111. Theorem 2.2). First we give a general proposition.

185

PLANCHEREL THEOREM

Proposition 4.1. Let U be a unitary representation of a topological group G on a separable Hilbert space H. Let Ba= B,(H) be the Hilbert space of all Hilbert-Schmidt operators on Hand define a unitary representation V of G on Bs by setting (4.15) V,(A)=U,A for A E B ~and g E G . Then V can be decomposed as the direct sum of countable copies of U. More precisely, let (pn),,*~be an orthonormal basis of H, P, be the projection on Cp,,, and B P = { AE BsIAP,=A). Then we have

-

B I = @ Bsn and

(4.16) (4.17) Proof.

n-0

V I B l n z U forall n e N . V, is a unitary operator on B,, because

-

c

(VAA), v~(B))= (UJpn,

uJ+n)

n-0

=

f (Ah, BPR)

n-0

= ( A , B) for any A, BE Ba. It is obvious that VPlrt= V,,V,, for any gl and g3 in G. Now we prove the strong continuity of V. It is sufficient to prove that for any A E Bl and E >0,there exists a neighborhood N of the identity e in G such that (4.18) IIV,(A)-AIIs<~ for any g e N .

Since 11 U,All, = llA1la< + m , there exists an integer m such that IIAqrllS= e l m

c II

U,Aq#.C (&/4Y*

klm

Then we have

(4.19)

1IW,A

Lam

--)pklla

5 2(

c II

klm

Uo4klls+

c

llAq*lI’)

klm

<Es/2. On the other hand, since U is strongly continuous, we can take a neighborhood N of e such that the m inequalities (4.20) IIU,Apk-Apkll<e/2 I/;;; for any g E N

186

EUCLIDEAN MOTION GROUP

are valid for each k E {0, 1,2,

..., m - 1).

c II

(4.19) and (4.20) prove that

0

I1 VAA)-AIl2

<m ( ~ / 2&$ + 212 = 2 .

=

U,Aw 44'

k-0

We have proved (4.18) and therewith the strong continuity of V. It is clear that Bsn is a closed subspace of Bat. Since for all n,k E N , B," is orthogonal to Barnif n#m. In fact, if A E Ban,B E B P and n#m, then Pnffk=

6n.kpn

c

m

m

1

(A, B ) =

( A v k , &k)=

k-0

( A P n v k , BPrnvk)

k-0

eu

= c6n,k&n,k(Aqon,

&m)=O

*

k-0

+ ...

Let An =A(Po+PI +Pn) for any element A E Ba. Then it is clear that lim IIA-A,IIa=O. Since A P , belongs to Ban, this shows that (Ban),cN n*-

generates Ba. So we have proved (4.16). The mapping On : Ba"+H defined by @,(A)=Avo, is an isometry, because if A E Ban,then llAllaa=

c-

m

IIAvklla=

L=O

= IIOn(A)IIa

C

IIAPnvtIIa

= IIAVnII'

k-0

.

Let P k n be the linear operator on H defined by P k n v m = 6 m , n ( o k . Then Pen belongs to Ban and @ n ( P k n ) =P k " v n = p a for every k E N. Hence On is an isometry of Ban onto H. Moreover, we have (A) for any A E Ban. We have proved (4.17), and hence (4.15) by virtue of (4.16). ( a n

0

Vg)( A ) = UgAvn = (up

0

@n)

Theorem 4.3. Let R be the right regular representation of G =M(2). Then R is decomposed as follows: R - I +- @Vaada (4.21) 0

187

and V5 is the direct s u m of countable copies of U": V 5 a $ U 5 forall u>O. (4.22) nrZ

ProoJ Let V5 be the unitary representation of G on Ha=B,(L'(K)) defined by Vg5(A)=Ug5A for A E H 5 and g E G . Since (x,,),,.z is an orthonormal basis of La(K), V" is decomposed as in (4.22) by Proposition 4.1. Let F be the Fourier-Plancherel transform on G. Then we have (4.23)

a [ (

Vgaudu) F for every g E G .

Fo R , =

0

0

Since both sides of (4.23) are isometries of L'(G) onto H =

1:'

$H5ada

(Theorem 4.2) and y ( G ) is dense in LYG) (Theorem 4.1), it is sufficientto show that (4.24)

( F oRa)f =

[(/+-a Vgaadu) F ] f 0

for anyf E 9 ( G ) .

Hence (4.24) is proved.

q.e.d.

$5. Determination of 9 ( G ) and &(G)

In this section, we determine the images 9 ( G )= {fIf E .9(G)] of (fIf E .g(G)) of . g ( G ) under the Fourier transform

9 ( G ) and &(G)=

188

EUCLIDEAN MOTION GROUP

fl on G=M(2). We shall see that the analogues of the Theorems of Schwartz (Ch. 111. Theorem 1.2) and of Payley-Wiener(Ch. 111. Theorem 4.1, 4.2) are valid for the non-compact, non-commutative solvable Lie group G=M(2). In $3, we have defined the Fourier transform{ off E L1(G)as a function on R+*=(O, + a). In this section, we regardf as a function on R'. Definition 1. The Fourier transformf of a functionf in L1(G)is defined for any C E R2 as

f(C)=/ f(g)Ucq-l&, C E R a 9

(5.1)

(i

where UC is the principal series representation corresponding to the parameter value c (i.e. (UCt(.)$)(s)=et(x.*c)F(r-ls)). If C=u>O, then.f(c) = f ( u ) coincides with the value given under the original definition (3.2). The extended Fourier transform f(C) satisfies the following functional equation. Proposition 5.1. Let f be an integrable function on G. Thenf satisfies R, o f ( C ) RV-I=f(rC) for any r E K and C E R9 . (5.2) Proof. Since U,C satisfies 0

R, U,' R,-I = UqrC for any g E G and r E K by Theorem 1.2, we get (5.2). 0

0

q.e.d.

Proposition 5.2. Let f E 9 ( G ) and E R*. 1) Then f ( C ) is an integral operator with the kernel k / ( s , r)=k,(C :s, r ) given by r

=f(rc, rs-9 Y

where f ( C , r ) is the ordinary Fourier transform of the function ZH f ( z , r ) on R f . k , is a C"'-function on R' x K x K. 2) The inversion formula (5.4)

fk)= J B f T r ( u q m1M

C )

holds. 3) (5.5.)

kf(uC;s.r ) = k,(C; su, ru) for any r, s, U E Kand CER'.

DETERMINATION OF

P ( G ) AND

&(a

189

Proof. 1 ) is Proposition 3.4. The proof of Proposition 3.4 is valid for any U E Ra. The expression (5.3) shows that k, is a C"-function on R a x K x K. 2) Put =ra, a 2 0, r E K. Then we have Tr( U,cf(C))=Tr(R,UgaRr-' R,f(a)R,-l) =Tr( U g a f ( a ) )

by Proposition 5.1. Hence by Theorem 3.1, we get f ( g )=["Tr(

Ugaf(a))ada= /:-SKTr( Ugraf(ra))adadr

0

-

= S R a T r ( U g m)dm(C)

=(f(c)R,-lF) (su) =

1

k,(C ;su, ru)F(r)dr.

kf(C;su, r)F(ru-l)dr=

K

S K

for any F in H = La(K).Hence we get 3). 4) Putting r = 1 in (5.3) and using the Fourier inversion formula on

R2

(Ch. 111. Theorem 1.2.), we have f(z, s-l) =

1

k,(C ;s, I)ei('.C'dm(C)

.

' R

q.e.d. In the sequel we always use the following notation. For function F on

K,we write F(r)=F(r(a))=F(a). Let Do be the differential operator Do=--I d i da' As an operator on 9'(G), Do should be interpreted by the formula Do= i-l a/aa. Then Do satisfies (DO& Fa)= (FI,DoF2) for any Fl, Fa E C 1 ( K ) . (A) Moreover, put Xp(a)=efpn for any p E 2. Then we have (B)

DOXP "PXP

-

190

EUCLIDEAN MOTION GROUP

Proposition 5.3. Let L be an integral operator on the Hilbert space H = La(K) with a continuous kernel k. Then the following five conditions 1)-5) are mutually equivalent. 1) The kernel k is a C"-function on K x K. 2) For any m and n in N, there exists an integral operator L,,,,, with a Cm-kernelk,,, such that

(Dorn L Don)F=Lm,,F for any F E Cm(K). 3) For any m,nE N, there exists a constant Cm,,2 0 satisfying II(Dom L Don)xpII5 Cm,n for any p E 2. 0

0

0

0

4) For any m,nE N, m

C I((DomoLoDo")~p,x31< +

-

PA=-"

5 ) For any m,nE N, the linear operator DornoLoDon on Cm(K)can be extended to a bounded linear operator L,,, on H = L2(K). Proof. Since the implications 2)==-5)*3) are obvious, it is sufficient to prove that the conditions l), 2), 3) and 4) are mutually equivalent. 1)*2). If k@, a) is a C"-function on K x K a T x T,then for any m,n E N, (5.7)

is well-defined. Let Lm,,,be the integral operator with the kernel k,,,. By differentiation under the integral sign and integration by parts, we easily see that

(Dom L Do")F= L m,nF for any F E C m ( K ) . 2)*3). It is clear that 0

0

II(Dom L Do")xpll= IILm.nxpllS IILm.nll= Cm,n for any p E 2. 3)+4). If L satisfies 3), then for any m,nE N, there exists a constant am,,,2 0 such that 0

0

~~(l+Doa)Dom~L~Don(l+Doa)~p~~Ium,n for any P E 2. (We can take Urn,, = Cm,n Hence we have

+ Cm+a.n + Cm.n+a +

Cm+2.n+a

-1

m

1 I(DomLDonX*,X J l P.P--" m

=

1 I(DomLDo"(l+Doa)xp,(1+Do')xq)I/(1+~')(1+q') 9,P-m

DETERMINATION OF

Q(G) AND &(G)

191

because

f p.q---

= (1

+ 43-'=(

(1 + P T ' (1

+ n2/3y < +

1 p---Ol

+2

P-1 2

09.

4) =-1). If L satisfies 4), then for any m,n E N, we have lim

Ipl--,lqi

SO

--I(DomLDon~-p,

Xq)l

=o

that for some positive b m , n I(Dom~Dony-p, ~31 S b m , n for any p , q E 2.

Since l(~omL~onX--9, X J I =IP"4"l KLX-p, X P ) l = IP"4"l I(k, X P 8 X q ) l 9

the last inequality implies that the Fourier coefficient LCp, q)= (k, xP@xq)of k is rapidly decreasing. Then k may be expanded in a uniformly convergent Fourier series

c ca

k=

(k,XP€3Xxp)XP€3XXo

p.q=--

and the series can be differentiated term by term arbitrarily many times. Hence k is a C"-function on K x K E T x T. q.e.d.

Remark 1. Since C ( K ) ,is dense in La(K),the bounded linear operator

Lm,,, in the condition 5) is uniquely determined by L and is equal to the with integral operator Lm,,,with the kernel (5.7). We identify DOmoLoDon Lm,,,.With this convention, the condition 5) can be written as follows: For any m,nE N,we have (5.8) 1 p o m L Don\( < + ca . Remark 2. By the equalities (A) and (B) in p. 183, the condition 4) in Proposition 5.3, can be written as follows: 0

(5.9)

c-

0

For any m,nE N, IP"4"(LXP, X 3 1 <

+ 00 -

Po I=--

Hence the condition 4) in Proposition 5.3 is equivalent to the following one :

3

192

(5.10)

EUCLIDEAN MOTION GROUP

For any m,n E N ,

c Go

l((1 +Doa)mL(l+ D o T x p ,

xq)l

c+

.

P. 'I=--'

Therefore the condition (5.8) is equivalent to the following one: For any m,n E N, (5.11) 11(1+Doa)"oLo(l+Do~)*1)<+to .

Proposition 5.4.

Iff is a rapidly decreasing function on G, then we have

(5.12) (Do"fnf)*(C) =(- 1)mDom o m =(- l)"f(C) Dom for all m E N and for all C E Ra. Proof. By Proposition 5.2, the operatorf(C) has the C"-kernel k,(C; s, r). Hence the operator DOm~J?(C)~DOn on C"(K) can be uniquely extended to a bounded linear operator on H = L 2 ( K )by Proposition 5.3. The middle and right sides of (5.12) represent the extended bounded operators on H following the convention made in Remark 1 above. It suffices to prove that three operators in (5.12) are equal to each other on the space C"(K). We note fist that (5.13) Ud=Lk for all k e K. The differential operator iDo=a/& is regarded as an element in the Lie algebra t of K. Put xt =exp itDo. Then we have 0

c

c

Hence if I:is a C"-function on K,we have r(R=,S)*tc)q (u) =m F )(x-tu) by (5.13). Differentiating the above equation at t = O , we get ((DOfY ( C ) F ) (4 = -( 0 0 (DofY (0= -Do Repeating differentiation, we get (~ot"f)*

(c) = (- I)'DP

o

9 (u) 3

m

of(c)

*

for a~ rn E N.

of(c>

Similarly, we get and

( L t f Y tc)=f(c)U=,-lc (Do"f)* (C)=(

=f(C)&,

- l)"'f(C)

0

Dom.

Proposition 5.5. Let k(C;s, r ) be a C"-function on R' x K x K and let

DETERMINATION OF

Q(G)

AND

193

&(G)

L(C) be the integral operator on H = L a ( K )with the kernel k(c;s, r). Then the function L :c=(cl, C*)HL(C)is a C"-function on R' with values in the Hilbert space BI of H-S operators on H. The partial derivative ( P L ) (c) is the integral operator with the kernel (Dmk)(C;s, r) for all m E A'*. Proof. Let h ( C ) be the integral operator with the kernel ak(C;s, r)/aCl. Then by Lemma 1 in $4 and the mean value theorem, we see that IW'W(C+h)-L(C) -Li(C)IIaa = Ilh-'(k(C +A) -k(C))- ak(C)/aCilla2

when h=(h, O)+O, where A is a compact neighborhood of C. Hence L(C) is differentiable with respect to C1 under the H-S norm and aL(c)/aCl is equal to Ll(C). Since k is a C"-function, the same argument as above is applicable to any partial derivative of L. Hence L is a C"-function under the H-S norm. Proposition 5.6. have

Iff@, r ) is a rapidly decreasing function on G, then we

(A*f) Yc) =( - 1)" ICl""fC)

(5.14)

and (5.15)

Proof. we get

for all EN and Cs R'. We write k,(C; s, r ) = k ( f ; C; s, r). Using integration by parts,

(A"f)(C)=(-l)"(l~1~"f)~(C)

n

=(rC)(l*O)k(f; I; s, r ) .

Similarly we get (5.16)

k(D(OB1)f;c ; s, r)=(rC)(osl)k(f;c; s, r ) W " fC;; s, r ) = W " k ( f ; C; s, r )

and

for every multi-index n E N*.Hence for the Laplacian A = we have (5.17) and (5.14).

k @ f ; C; s, r ) = -ICl*k(f; C; s, r )

-(D('.O' +

194

EUCLIDEAN MOTION GROUP

Since f belongs to 9(@we , can differentiate (5.3) with respect to ( under the integral sign arbitrarily many times. Hence we get the equality (5.18)

Dc"kcf; [; s, r) = k((- r-'z)"f; C; s, r).

Hence by Proposition 5.5, we get (5.15).

q.e.d.

Definition 2. Let H = L2(K)and B = B ( H ) be the Banach space of all bounded linear operators on H and B, = B2(H)be the Hilbert space of all Hilbert-Schmidt operators on H. Then a function L on R2with values in B2 is called a rapidly decreasingfunction with C"-kernel if it satisfies the following two conditions 1) and 2). 1) L is a C"-function on R2 with values in B,. 2) For any integers, j, k,n E N and m E N Z (5.19)

qj.k,m,n(L)

= sup(lDdo(1 CfR2

+ IC12)k(pL)(5)oDo"II < +

-

03.

The space of all rapidly decreasing functions L; R2 B, with C--kernels is denoted by 9 ( R 2 ,B,). The subspace consisting of all functions L in 9(R2, B,) satisfying the functional equation R, 0 L(5) R,-I = L(r0 for any r E K and T€Rz, 3) is denoted by Y ( R 2 ,B,). The vector spaces Y ( R 2 ,B,) and Y ( R 2 , B,) are topologized by the family of seminorms: 0

(1 5.20)

{qj,k,m,nlj,

k, n € N , m € N 2 j

Proposition 5.7. Both Y ( R z ,B,) and 3'-(R2, B,) are Frdchet spaces. Proof. Since the topology of Y ( R 2 ,B,) is defined by a countable family of seminorms, .9'(R2, B2) is a metrizable, locally convex, topological vector space. It is complete. In fact if (Ln)nfNis a Cauchy sequence in 9(R2, B,), then for any CE R2, the sequence (L,(r)) is a Cauchy sequence in B,. Since B, is complete, the sequence (L.(r)) has a limit L(5) in B2. Since (1 I 2)"(PL,(5))converges uniformly to (1 I('1 ')" (DmL)(5)for any m E N 2 and n E N , L is a C"-function and is rapidly decreasing. Similarly, it has a C"-kernel. Hence L belongs to Y ( R 2 ,B,) and 9 ( R Z ,B,) is complete. For any fixed r in K,the mapping L c-, M,where M(5) = R,-'oL(rr) 0 R,, is a continuous mapping of .9'(R2,B,) onto itself. Hence Y i s a closed q.e.d. subspace of 9 ( R Z B2) , and is complete.

+

Theorem 5.1. The Fourier transform F:f*f on the group G induces a topological isomorphism of 9 ( G ) onto S ( R 2 ,B,).

+

= M(2)

P(G)

DETERMINATION OF &(G) AND

195

Proof. Let f be a rapidly decreasing function on G: f€.Y(G). Then the Fourier transform = S f is a B,-valued C--function on R2 with C"-kernel by Propositions 5.2 and 5.5. Moreover by Propositions 5.4 and 5.6, we have

fl

llDd O (1 =11(~0""(1

(5.21)

5

ss K

+ I Cl3'(o"fl)(O

O

D0"lI

- d)'z2"f(2,r))"(Oll

IDo'+"(l - d)'(z"f(z,r ) ) Idm(z)dr.

12

By proposition 1.5. in Chapter 111, the last integral is smaller than a linear combination with positive coefficients of the integral of the type

sKJR,

Iz I I( ~ O " " D b (z, f ) r ) Idm(z)dr

(5.22)

'O

< + 03,

= cPo+Z.c b . I + m ( f )

where C =

s..(l +

121z)-2dm(z)<

+

03

andp,,,,

(b.j+n)

is the seminorm

(3.3). Hence we have proved thatfl belongs to 9 ( R 2 ,B,). By means of the inequalities (5.21) and (5.22) we see that the Fourier transform Finduces a continuous mapping from 9 ( G ) into 9 ( R 2 , B2).Since belongs to S ( R 2 ,B,) by Proposition 5.1, T induces a continuous mapping from 9 ( G ) into Y ( R z ,B,). We next prove that the image F ( 9 ( G ) ) coincides with .7-(R2,B,). Let L be an element in 9-(R ,, B,) and put

fl

(5.23)

c-

M l ;8 9 4 =

(L(r)X,, X P M P m J

PSI---

The series converges absolutely and uniformly on R2 x T x T.More generally, for any m E N Z and n E N , the series (5.24)

(1

+ IC l 3 "

f (PWOXI,

X P ) X P < S ) ~

p.,---

converges absolutely and uniformly on R2 x T x T. In fact since L belongs to 9 ( R 2 ,B,), there exists a constant C,,,," 2 0 such that 11(1

+ lC13"(1 + Do2)(D"L)(O(l + Doz)ll 5 C,,,,,, for

we have the inequality

any CER,,

196

EUCLIDEAN MOTION GROUP

PtP'--

1 + I C I 2y((1 + D o 2 X P w 3 0 + Do21X4,

I I(1

XP)

I

P, 4

x ((1 +p2)(1

+q2)}-'

5 cm,n

(1

+

p2)-1(1+

421-1

<+*

PIP---

for any C=R2. Hence k(C;8, a) is a C"-function of C. For any j , k N , we can replace ( P L ) (0by Ddo(D"L)(r) o D,,'in the above argument. So the series (5.23) can be differentiated termwise with respect to a and 8 arbitrarily many times and k is a C"-function on R2 x T x T. Moreover k satisfies the functional equation (5.5): (5.25)

MuC; s, r) = 1 ( L ( U O X P , x , l x p ( r ) x , ( s ) PIP

=

1 1(~(OXP,

(L(0Ru-'X4,

Ru-1XPlXP(42J3

PIP

=

XPlXP(4*)

= k(C; su,

PI 4

The integral operator Lo(o with the kernel k(C; B, a) is equal to L ( 0 , because for any FE L2(K),we have (5.26)

= L(C)F.

Put (5.27)

f(z, r ) =Ja2 k(c; r - l , l ) e ' ( z * C ) h ( o .

(cf. Proposition 5.2, 4).) By replacing L ( 0 by D,'oL(o in (5.24), we see that for any integers j, n e N and m E N 2 there exists a constant C,,",,,,such that l(1

+ I C12YJ(DrmD~'k) (C; 8, all 5 Cj,n.m

for all (C, a,B)€R2 x T x T. Hence the function ~ ( cr ,) = k(C; r - l , 1) belongs to the function space 9(G'). Therefore the function f which is the

DEITRMINATION OF

.@(a AND p(G)

197

inverse Fourier transform of Q, on Rz belongs to 9 ( G ) by Theorem 1.2 in Chapter 111. Moreover by the functional equation (5.25) of the kernel k(C; s, r ) and by (5.27) we have

k(r-1C; s, r ) = k ( < ;sr-1, I)=$

f ( z , rs-1)e-s(s*c)citn(z) lt=

Replacing C by rC, we get (5.28) =ki(C;s, r). Hence by (5.26) and (5.28), we have

.

f ( c )=L(C) for all c E Ra We have proved that S ( . . Y ( G ) ) = Y ( R * ,Ba). Since the mapping f l is a continuous mapping from a FrCchet space 9 ( G ) onto another Frkhet space . T ( R a , Ba), the inverse mapping f l - l is continuous by the open mapping theorem. q.e.d. Now we prove the analogue for M(2) of the Paley-Wiener theorem which gives a characterization of C"-functions with compact supports by means of their Fourier-Laplace transforms. Definition 3. The space of all complex valued C"-functions on G with compact supports is denoted by g ( G ) . Let B,= {t(z)rE GI IzJ$ a and r E K ] and g a = a(&) be the subspace of g ( G ) consisting of all functions in g ( G ) whose supports are contained in B,. For any m E Na,a seminormpm on g(G)is defined by (5.29)

pm(f>=SuPIDmf(g)l 3

iW

The space g,,is topologized by the family of seminorms {pmlmE W }. Then g , is a Fr6chet space. The space g ( G ) = ;gnis topologized n-1

as the strict inductive limit of g,,(cf. Ch. 111. &I. Definition 6). We have seen in Ch. 111. &I that g ( R " ) is characterized by its FourierLaplace transform. Similarly g ( G ) is characterized by its Fourier-Laplace transform. To define the Fourier-Laplace transform off in g ( G ) , it is necessary to complexify the parameter Z E R a of the principal series representation. The inner product (z, w) on Ra is extended to a bilinear form on Cax CS.

198

EUCLIDEAN MOTION GROUP

Then for any C E P,the function x C :t(z)wei(x.C)is a one-dimensional continuous representation of the translation group T. xC is not unitary unless C belongs to Ra. However, we can define the representation UC induced by xC in the following way. Proposition 5.8. Let g=t(z)r E G=M(2) and F E H = L a ( K ) and C E Cz, and put (UgcF)( ~ ) = e { ( * * ~ C ) F ( r - l s ) . Then U,C is a bounded linear operator on the Hilbert space H and the mapping U c : g ~ +U,C is a strongly continuous representation of G on H. Proof. Since Z E Ra, we have (,+a, rc)l =&x-11,ImC)jelxl - Ixmct . Hence we have (5.30) 11 UgCII5 e1.1 I X r n C l for g = t(z)r . So U,C is a bounded linear operator on H. The proof of the equality Ug,CU,,C = U,,,,C is same as in Theorem 1.1. Although UC is not a unitary representation in general, the strong continuity of UC may be proved in the same way as in the proof of Theorem 1.1, because the function gIIU,Cll is bounded on every compact set in G, as is seen from (5.30). q.e.d.

Definition 4. The Fourier-Laplace transform S c f on G of a function

e g ( G ) is the function on Ca with values in B=B(La(K)) defined by r

(5.31) Since U,C is a bounded continuous function andf is a continuous function f ) (C) is well-defined and is a bounded operator with compact support, (.Fc on H. In the same way as in Proposition 5.2, (.Fc f ) (C) is an integral operator with the kernel (5.32)

k,(C; s, r ) = [

f ( z , rs-l)e-i(','C'dm(z)=f(rC, rs-1) It=

for any C E Csand s, r E K, where f is the Fourier-Laplace transform on

R=. Theorem 5.2. A complex-valued function k(C; s, r ) on C ax K x K is equal to the kernel k,(C; s, r) for a certain function f in 9.= s ( B , ) if and only if k satisfies the following three conditions:

DETERMINATION OF

P(G)AND &(G)

199

1) k is a C"-function on Cax K x K. 2) For any m E N a , the function Ct+Cmk(C;s, r ) is an entire function on Caof exponential type Sa, i.e. there exists a constant CmrO such that (5.33) ICmk(C;s, r)lS CmealIrnCI. for any 5 E Ca and s, r E K. 3) k satisfies the functional equation (5.34) k(uC; s, I ) =k(C;su, ru) for any C E Ca and s, r, u E K. Proox Let f be an element in sa. Then k,(C; s, r) is an integral over the compact set B,, (5.35)

f (z, rs-l)e-s(z* +C)dm(z).

k,(C; s, r ) = !B

a

Since e-t(z*rc) is an entire function and the integral is over a compact set, the function Ct-+k,(C;s, r) is an entire function on C1.Moreover, since (5.16) is valid for f E g ( G ) and the complex vector variable C E Ca, we have (5.36) l(1 + lCla)nkf(C;3, r)l 1(1-A)"[ f ( r z , rs-l)]lea~lrn~~dm(z) 5S . a I;CealIrnCl for any C E Ca. Since I ~ ~ l d+ICla)lml (l for any m e N a , we have proved (5.33) in virtue of (5.36). Since the integrand of (5.35) is a C"-function of (s,r), k, is a Cfunction on Cax K x K. Moreover, UC satisfies R, U,' R7-l = UVrC for any 5 E Ca 0

0

and g E G and r E K in exactly the same way as in Proposition 5.1. Hence the Fourier-Laplace transform ( f l c f > ( C ) satisfies (5.37) ( X c f ) ( r z ; = ) R, ( X c f ) ( C ) Rr-l for all r E K and for all C E Ca 0

0

c

So the kernel k , satisfies (5.34),just as (5.5). Conversely let k satisfy the three conditions in Theorem 5.2, and define a function f ( g ) = f ( z , r ) on G by

Since for any m E N2,the function p+Cmk(C; r-l, 1 ) is an entire function of exponential type 5 a on Cay and f ( z , r ) is the inverse Fourier transform of the function &+k(C;r-l, 1) on Ra, the function zt-+f(z,r)is a C--function whose support is contained in B, by the Payley-Wiener theorem

200

EUCLIDEAN MOTION GROUP

(Ch. 111. Theorem 4.1). Since k satisfies (5.33) for any m E W , (5.38)can be differentiated under the integral sign, and we get (5.39)

(D*f)(z, r ) = /

<mk(C; r-1,l)e~(z.c)dm(c). R '

By (5.39), (D*f)(z,r) is a C"-function of r E K for any m E N'. Hencef is a C"-function on G and belongs to g a . By the condition 3) in the theorem, (5.38) can be written as

The two continuous functions Cnk(r-1C; s, r) and z n f ( z , rs-l) belong to L1(Ra)and the latter is the inverse Fourier transform of the former. Hence the inversion formula for (5.40), namely, (5.41)

k(r-lc; s, r ) =

SD.

f ( z , rs-l)e-<(*.c)dm(z),

is valid by Ch. 111, Theorem 1.8. Replacing ( by rC in (5.41), we get

k(C; s, r ) =

f ( z , rs-')e-"'. W m ( z )

/R%

= k f ( ( ; s , r ) forany C € R a and s , r E K .

Since Ct+k(C;s, r ) and Cwkf(c;s, r ) are entire functions on Caand coincide with each other on the real space R', they coincide on C* (Ch. 111. 54, Lemma 1 ) . Hence we have proved that k = k f for somef E 9.. q.e.d. We have defined the meaning of complex-valued holomorphic function on a domain D in Cn in Ch. 111. $4. In the same way we can define a holomorphic function on D c C n with values in a Banach space B. A B-valued functionf of n complex variables is called entire iffis holomorphic on Cn. A B-valued function L on C" is said to be of exponential type 5 a if there exists a constant C such that IIL(C)lld CeUlIrnCl for all E C" . Definition 5. Let a> 0 and Ba = BI(La(R)). Then the space of all functions L : Ca-+ Ba satisfying the following two conditions 1) and 2) is denoted by p a : 1 ) L is an entire function on Ca. 2) For any m E N a , C*L(C) is of exponential type 5 a, i. e., there exists a constant C,,, such that lIc"L(C)ll~ CmealIrnCl for all c E C'

.

DETERMINATION OF

p(G)AND

&(a

201

The subspace of g@, consisting of all functions L with C”-kernels k(c;B,a) is denoted by pa.For any m E N aand j , n E N , we define the seminorm (5.42)

qm,j,n(L)= SUP e-ailmCI1lDOjOCmL(C) oDonll E d 3

on P a pais topologized by the family of seminorms

(5.43) (qm,j,nlmEN a and j , n e Nl . For any m, n, j E N , put sm.j,n(L)=supe-allmC1llDoj(1 l ~ l * ) ~ L (Donll c) . (5.44) 0

Er 0

+

0

Then smn.j,n is a seminorm on p a . The topology of p a defined by the seminorms (5.44) is clearly equal to the topology defined by the seminorms (5.43). pais a metrizable, locally convex, topological vector space. It is complete, because the uniform limit of a sequence of entire functions is entire. Hence pais a Frkhet space. The subspace of paconsisting all functions L satisfying the functional equation R, o L(c)o R,-l= L(rC) for all r E K and C E Ca (5.45)

is denoted by go. As a closed subspace of p a , %a is a Frkhet space. The strict inductive limit of ( s n ) n z l is denoted by s. Definition 6. A function f on U with values in a Banach space H is called weukZy unulytic if the function x n ( f ( x ) , y ) is analytic for each ‘p in the dual space H* of H. LEMMA 1. Let M be a real or complex analytic manifold and U be an open set in M and H be a complex Banach space. A strongly continuous H-valued function f on U is analytic if and only iff is weakly analytic. Proof. The “only if” part of the lemma is trivial. A power series f ( x ) of real variables x = (xl,..., x,) converges for any real numbers x4 satisfying Ix,-atl
202

EUCLIDEAN MOTION GROUP

for any rectifiable closed curve C in U.Since p is an arbitrary element in H*, the above equality implies that

scl.),=o for any rectifiable closed curve C in U. Hence by Morera's theorem, f is analytic on U.

Theorem 5.3. The Fourier-Laplace transform X c on G=M(2) is a topological isomorphism of g aonto xufor every a>O. Hence 9" is a topological isomorphism of 9 =s ( G ) onto Proof. Let f be a function in Saand k,(<; s, r ) be the kernel of the integral operator ( X f c)(C). Then k,(C; s, r ) is a C"-function on Ca x K x K and an entire function of 5. Hence for any Fl and F2in H=La(K),

x.

is an entire function of C. Therefore X c f is weakly analytic on C a and hence analytic on Ca by Lemma 1. Namely X c f is an entire function. f is exponential type 5 a, because for any F in H = La(K) Moreover Tc and for any C E C2,we have by Schwarz inequality

=W(B,)~ llfllaaeaalIbCl ,

-We note that Proposition 5.4 is valid for any complex vector variable E C a and that (5.14) is valid for any E Ca when f belongs to g(@. Hence by (5.46), we have (5.47)

Po'

0

(1

+ I Cl2)"(~=n(r) Do71 O

= l l ~ c ( ( l - 4"Db ")(r)ll

$rn(Ba)ll(l - A)"Do'+"fllz@'*mC' 5m(BJ2 I I( 1 - d)"D$+"flI-@' ImC'.

Hence X cf belongs to pa.Since .Fc f satisfies the functional equation (5.37), S C f belongs to xu.Moreover the inequality (5.47) proves that Tc f induces a continuous mapping from gainto z,. Conversely let L be an element in xa.Then the series (5.23) converges

DETERMINATION OF

p(c)AND &(G)

203

absolutely and uniformly on Cax K x K.The proof of this fact is same as the corresponding part of the proof of Theorem 5.1. Since i is an entire function on C a , the function b k ( C ; p , a ) is an entire function on Ca, because the sum of a uniformly convergent series of entire functions is entire. Replacing L(c) by Do"oL(C)o DoRfor any m , n E N , we see that p, a ) is an entire function. In particular the function Cw(dm+nk/aandBm)(c; k is a C"-function on C 2x K x K. Since cmL(c) is an entire function of exponential type 5 a, the function cwgrnk(C;p, a ) is an entire function of exponential type 5 a. Since L satisfies the functional equation (5.37), k satisfies (5.34). Hence k satisfies the conditions of Theorem 5.2, and there exists a function f in g asuch that k = k , . Thus we have proved that L(c)= ( S c f ) ( c )for any c E C aand that Y eis a surjection of gaonto %a. Since Y e is an injection by the inversion formula (Theorem 3.1), F C is a continuous bijection of g aonto %a. Hence YC-l is also continuous on .Fa by the open mapping theorem. is a topological isomorphism of Therefore, we have proved that .FC 9 a onto %aSince 9 is the strict inductive limit of gn and % is the strict inductive limit of %, X C is a topological isomorphism of 9 onto X (cf. Ch. 111, Proposition 4.2). q.e.d.

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CHAPTER V

Unitary representations of SL(2, R)

51. The Iwasawa decomposition In this section, we give some group-theoretical preliminaries which will be necessary later. First we introduce the subgroups K,A,N, and show that any element g of SL(2, R)is uniquely and continuously decomposable in the form g = kan, k E K, a E A , n E N . This is the Zwasawa decomposition of g which plays an important role in the construction of unitary representations. Second, we show that SL(2, R) is isomorphic to another linear Lie group SU(1,l) by means of the Cayley transform. SL(2, R) acts naturally on the upper half-plane, while SU(1, 1) acts on the unit disc. The realization of SL(2, R)as SU(1, 1) is convenient and will be used frequently in the later sections. In the last part of this section, we show that SL(2, R) is the two-fold covering group of the 3-dimensional proper Lorentz group G+(2). Definition 1. The multiplicative group of all 2 x 2 real matrices with determinant 1 is denoted by SL(2, R). Proposition 1.1. 1) The Lie algebra GI(2, R) of SL(2, R) is given by 11(2, R)= { X E Ma(R) I TrX=O}. 2) The elements

form a basis of ~ l ( 2R). , 3) The Lie products among Xo, XI, and Xa are given by (1.2)

1x0,X1I=Xo-X2,

[Xo, X*]=X1, [XI, X,]=X,.

Proof. 1) Ch. 11. Proposition 5.7. 2) is clear from 1). 3) is easily obtained by direct calculations. 205

206

UNITARY REPRESENTATION OF

SL(2,R)

Definition 2. Let 8, t, E be arbitrary real numbers and put

ue=exp 0x0, at=exp t X l and nE=expEX%. Then the subgroups K,A,N of SL(2, R ) are defined by

K=

{uh9 E R ) , A = {atltE

R} and N = {nEIEE R } .

Proposition 1.2. We have

K E R/4sZ E T , A = R, N = R . Proof.

Since xS3=0, nF=exp C X= ~1

+ EX^ = (o1

1>. 5

Hence N E R by the isomorphism &me. We have

Hence A z R by the isomorphism ?Hat. Since Xoa= - 1,

cos el2 sin 812 -sin 812 cos 812

n-0

n-0

=

Hence, by the isomorphism eHUe, K G R/4nZ T. Remark. K is a maximal compact subgroup of SL(2,R).

Proposition 1.3. Any element g E SL(2, R ) is uniquely decomposable in the form (1.4) g =U d Z t n ( . If g=

(:

i)

E SL(2, R), then the triple (8, t, E ) in (1.4) is given by the

relations

Proof. Taking the product of matrices we have

e

e

Hence we get a= ella cos- c = -ella sin-, from which we derive a - ic = 2’ 2

207

IWASAWA DECOMPOSITION

e(tlS)+(‘@I2).

Hence la- icl

and we get

a-ic = --=== Jaa

+

and et =a’

CS

+ca.

On the other hand, we also have ab +cd= etE from the matrix calculation above, and this gives us [=

ab + cd

~

aa+ca

*

We have proved that every element g in SL(2, R ) can be written uniquely as in (1.4) where (0, t, E ) is given by (1.5). Proposition 1.4. The mappingf:(u, a, n ) ~ u a is n an analytic diffeomorphism of K x A x N onto SL(2, R ) . In particular, SL(2, R ) is connected. Proof. f is a bijection of K x A x N onto SL(2, R ) by Proposition 1.3. The mapping f is real analytic because the operation of multiplication is a polynomial mapping. The inverse mapping off is also real analytic as is seen by (1.5). (Note that if g belongs to SL(2, R), then (a, c) # 0 and aa+ ca>0.) Since K, A , and N are connected, SL(2, R ) is connected. Proposition 1.5. For an element g in SL(2, R ) and 0 E R,let (1.6)

W e =Ug.eat(p.

e ) k g , el

be the Iwasawa decompositicjn of gue. Then the following holds: 1 ) For g, g‘ in SL(2, R ) we have (1.7)

(gg’).O=g.(g‘.O) mod 477,

(1.8) (1.9)

t(gg’, 0) = t(g, g’-e>+ t(g’, 8) g . ( e + 2 K ) = g . e + 2 ~mod 477,

2) If g =

I;]E SL(2, R), then we have e

(a- ic)cos(1.10)

2

e((~.e)12 =

1-8=0,

t(g, e + 2 ~ ) = t ( g 0). ,

+(- b +id)sin-62

e

e s

I(a-ic)cos-+(-b+id)sin--l 2

e

2

e ,

(1.11)

et(QSe)=I(a-ic)cos-+(-b+id)sin-~a

(1.12)

E(g, e) =e-‘(p.

2

2

1 {(ab+ cd)coso +z(u3

-b2+c2-ds)sin8}.

K

N (because N is normal in AN).

A

By the uniqueness of the Iwasawa decomposition, we have (gg') 8 = 8" = g - 8' =g.(g'. 8) mod 411,

-

t(gg', 8) =t"

+t'= t(g, g'4) +t(g', 8).

To get the second half of l), we observe that uB+lr=-uI, and hence that guo+s.= -gu.9=

-Ug-eat(g,~)nt(g,e)

=U g . s + a P t c g , a+il)nc(g, e+ad,

so that again by the uniqueness of the Iwasawa decomposition, we have g . ( 8 + 2 7 ~ ) = g . 8 + 2 ~mod 411 and t(g, 8 + 2 a ) = t ( g , 8). 2 ) The assertion 2 ) is obtained by a straightforward application of Proposition 1.3. to

3) From 2) we obtain the relation ef(g.@)lPet(& @)is

" +(-b +id)sin-.2

= (a- jc)cos-

0

2

Differentiating the both sides of this equality with respect to 8, we obtain

By taking the imaginery parts of the both sides of this relation equal, we obtain

IWASAWA DECOMPOSITION

209

To construct a realization of SL(2, R), we consider the group SL(2, C) operating on the Riemann sphere C = C u [ 00 ] (the one point compactification of the complex plane C). As is well known, an element g =

[:: "d]

of SL(2, C ) acts on C as a linear fractional transformation (D(g),defined bY az+b +(g) : ZHgZ=( zE C ) . cz+d It is easy to see that, for g,g' E SL(2, C ) and z E C , we have {(gg') z=g(g'z), lz=z.

The map $ is a homomorphism from SL(2, C) to the group A ( C ) of automorphism of C,where, by an automorphism of C, we mean a bijective holomorphic transformation of C whose inverse is also holomorphic. One can prove, that 4 is surjective (cf. H. Cartan, Elementary theory of analytic functions of one or several complex variables, Hermann-Addison-Wesley, 1963, pp. 180-181). Hence, we have SL(2, C ) / { + l ]EA(C). If we let C+ denote the upper half-plane, { z=x + iy E C Iy >01, then, for g E SL(2, C ) , we have g c + = c + s g E SL(2, R ) and sL(2, R)/ { 11 EA(C+) (ibid. pp. 182-183). Let D = { z E CI IzI < l} be the unit disc. The Cayley transformation z-i c :Z H C Z = z+i transforms C+ onto D. [Observe that Z E C + o l z - i l < l z + i l . ] We have an isomorphism from A(C+)onto A ( D ) given by gncgc-'. Let C be the 2 x 2 unitary matrix given by c=-[1 1 E U(2). a1 We now have the following proposition.

*

-3

Proposition 1.6. 1) Let G=C-SL(2,R).C-l. Then G =

dl

If C[C a b C-l=["

B

"1, @

then

I["

"]I B @

la12-1p12=1).

210

SL(2,R)

UNITARY REPRESENTATION OF

a=

1

+

{(a+d) i(b-c)}

1

p = [(a- d )- i(b +c)] 2) We have G=SU(l, l), where SU(1, I ) = { g E M 2 ( C )I g * e g = e l , and

[:

det g= 1) and el=

-3

If H is a Hermitian form on C’, given by H(z)=zli-l-zzlzs,

.=“‘I

za

E C’,

then we can write G=SU(l, 1)= { g E M 2 ( C )I H(gz)=H(z) for all Z E C’,det g=l}. 3) By the isomorphism h :gnCgC-’ from SY2, R ) to G, we have

Proof. 1) Let g=@

[:ii, -it].

:] A

ne=[:

E

i]

E M2(R).Then we have

=-[ 1 (a+ d ) + i(b -

c)

2 (a-d)+i(b+c)

Hence CgC-1 is of the form lolalz-

i],

3

E SL(2, C), we have

+

(a- d ) - i(b c)

(a+d)-i(b-c)

1 ’

where a, B E C. We have det g =

1 ~ 1 ’ = 1. The assertion 1) follows.

2) For g =

1-iy

21 1

IWASAWA DECOMPOSITION

1;

p

-l 6 6 1 = L

-/I a1

Using this, we can easily deduce (g*elg=el and det g = 1) o (a=& ,8=p and

Ia12 -I,811=1).

Hence we have the first assertion in 2). The condition g*elg = el is nothing but the matrix form of the condition H ( g z ) = H ( z ) for all Z E Ca, so the second assertion in 2) follows. The assertion 3) follows directly from the truth of assertion 1). q.e.d. Proposition 1.7. Identifying corresponding object by the isomorphism h : SL(2, R)+SU(l, l), we write h(ue)=ue, h(a,)=ac, h(ne)=ne and

.

h ( g 4 = u, eat(,, 8) nt(,.e).

If g = [ "

1'

B a

E SU(1, l), then we have

(1.13) (1.14)

etQ*

= lac

+

a = Id

+pCla, where

= ete.

Proof. The proposition can be obtained from Proposition 1.5,2) via the conversion formula of Proposition 1.6, 1). It can also be proved by observing that from Proposition 1.6, 1) we get a - ic = a ,9; so Proposition 1.3 now reads

+

(1.15)

Now we calculate and obtain (in SU(1, l), using Proposition 1.6, 3))

Then we apply (1.15) to a'=aereIa, $=,!?e-telato obtain (1.13) and (1.14).

Definition 3. The 3-dimensional Lorentz group G(2)= O(2, 1) is the subgroup of GL(3, R) defined by G(2)= ( g e GL(3, R)l'gBg=B} where B=

(- 0 la ). The connected component G+(2)of G(2) containing

identity 1 is called the 3-dimensional proper Lorentz group.

212

UNITARY REPRESENTATION OF

Proposition 1.8.

SL(2,R)

Let

and eo=X(l,O,O), el =X(0,l ,O), e, =X(O,O,1). For any g E SL(2, R ) let s(g) be the matrix of the linear transformation x w g 2 g in W with respect to the basis (e,). Then s is a continuous homomorphism of SL(2, R) onto G+(2) with the kernel Z = { -+ 1). Proof. The vector space W is the subspace of M,(R) consisting of all symmetric matrices. Therefore, s(g)x=gxtg belongs to W for any x E W. Hence x-gx’g is a linear transformation on W. Since - det X(xo,xl, x,) = -xoa + xlS+xsa is invariant under s(g), s is a continuous homomorphism of SL(2, R) into G(2). Since SL(2, R) is connected, the image s(SL(2, R)) is connected, and contained in G+(2). To find ker s, we let g =

(z ”d>

E ker s.

Then the equations eo =s(g)eo=gcg =

(.I +” ac+ bd

c2+ d a

and

show that a’+bz=l, a’-.bS=1 ca+dS=l, cS-d’= - 1, from which we derive b=c=O, a = k 1, d= k 1. Since det g=1, we conclude g = k 1. Since s(- 1)= 1, ker s is actually equal to { k l}. It is easily seen that the kernel ker s’ of the differential representation s’ (Ch. 11. 55) is equal to the Lie algebra of ker s= { k 1). Hence ker s‘= {0} and s’ is a linear injection of Bl(2, R) into the Lie algebra g(2) of G+(2). Since dim g(2) = 3 =dimBK(2, R),s‘ is a linear isomorphism of GK(2, R) onto g(2). Since expg(2) is a neighborhood of 1 in G+(2) and the connected group G+(2) is generated by every neighborhood of the identity, every element h in G+(2) can be written in the form h =exp tl Yl ......exp tr Yk where YrE g(2) and tr E R. Since s’ is surjective, there exist k elements Xl, ..., Xk in Gl(2, R) such that s’(Xt)= Yt (1 S i S k ) . Put g=exp tlXl ... exptXXk.Then g belongs to SL(2, R ) and s(g)=h, because s(exp t X ) = exp rs’(X) for every XEBI(~,R), by Ch. 11, Proposition 5.9. We have proved that s is a continuous mapping of SL(2, R) onto G+(2),with kernel q.e.d. equal to Z = { k l}.

PRINCIPAL CONTINUOUS SERIES

213

Remark. Let g,,(O 5 i, j g 2) be the (i, j)-element of g E G(2). Then the relation lgBg=B yields goo'=l +g,O2+g~o2~1 and det g= & 1. Hence either gooZ1 or good - 1. It can be proved easily that G+(2) consists of all elements g in G(2) satisfying (1.16)

det g= 1 and goal 1.

$2. Irreducible unitary representations

I. PRMCIPALCONTINUOUS SERIES Let G= KAN be the Iwasawa decomposition of G = S U ( l , 1) (Proposition 1.3). We can identify K with the torus T = R/4 7~ Z by ue= btela e-,@:] we (mod 4 ~ ) .As we have seen in Proposition 1.5, G x K-tK, defined ~, (gg').e =g.(g'-O) and 1.8 = 0, hence, by identifying by (g, U ~ ) H U ~ . satisfies K with T, we can say that G acts on K from the left. On the other hand, we can map K onto U = {C E C I = 1) by (I, :ubw e(@.The map (I, is a homomorphism from K onto U with ker (I,= { t l } = {uo,ua.}, or we can say that (I, is a homomorphism from T onto U with ker Q = (0,27~(mod 47~)).If we deiine the action of G on U so that the following diagram: GxT+T

. 1 1 G X U-+U

is commutative, then we have

In fact, if C=efo, then using Proposition 1.5, 2) we have

We define a function t ; G x U+R by t(g, t;) = t(g, 81, where I: =(I,(us). This function is well-defined, because t(g, e +2n) = t(g, 19)(Proposition 1.5, 1)). By Proposition 1.7, we have e"g*C)= IbC+al'. Let the function u : G x U-+U be defined by

2 14

UNITARY REPRESENTATION OF

and for s E C and j = O ,

SL(2,R)

1 -, set 2 vf. 8 ( g ,

c) =u(g, @Ve-st(g*C).

Let p be the normalized Haar measure of the compact group U,i.e., 1 dp(C)=-dO 2z

We usually denote the integral

(C=ete).

1

f(C)

dp(C) by

1

f(C) dC.

U

U

Then we have the following proposition.

Proof. By Proposition 1.7, we have

Taking the complex conjugate of this relation, we get

from which we derive u(g, <)=e'(-o.@+@)/a,for =el@. Using (gg')-O=g-(g'-8)mod 4a, the first assertion of the proposition now follows from the simple calculation -(gg')-8+8= { - g . ( g ' 4 ) + g ' - e ] ( - g ' - e + e } mod 4 ~ , and from g'C =e*(g'.@). The second assertion follows from the first and from r(gg', 8) = t(g, g ' 4 ) t(g', 8 ) (Proposition 1.5, 1)). By Proposition 1.5, 3) we get

+

+

Proposition 2.2.

Let Q=La(U,p ) and j E

and s E C. For any

g E G, we define the operator Vgf.'on $j by

(vg$. af) (c) =e i.e.,

-a'(g-l,

C)

(uk - l ,

C) "(g-'.C)

(fe$1,

PRINCIPAL CONTINUOUS SERIES

215

Then the following hold: 1) V/' is a bounded linear operator on 9.For any compact subset A = A-' in G, there is a positive constant C(A), such that llVgj**ll IC(A)IRe*l+lla for all g E A, where Re s denotes the real part of s. 2) The mapping Vj.' :gt-+Vgj.* is a representation of G on 8 : * = v,j,* V,J. * and Vlj. = 1. a) vgoJ~ b) V'.* is strongly continuous. 3) V'pa is a ~ n i t a r yrepresentation of G if and only if Res= 1/2. V&' is a unitary operator for any k E K and for any j, s. 1

4) Let ~ ~ ( u ~ ) = (e~* E ~ z~ Z , andf,(c)=p(pE ) Z).

Then V./ 'fp = xp+f(ub)fp. Proof. 1) Since e-t(9-1*C)is a positive continuous function on G x U, it follows that for any compact subset A of G, there exists a constant C(& 1, such that C(A)-' ge-L(g-l.C) 5 C(A), for all g E A, c E U. Then, for any s E C,we have (2.1) le-*t(g-l. C ) I 5 C(A)lReal. We compute the norm of V,J."f: IIVpj.'flla=

le-rC(g-l*C)laIf(g-lc)la J U

dt:

216

UNITARY REPRESENTATION OF S42,R)

IIVg’.”l5 C(A)’=e*’+l’a (2.2) 2) a) By Proposition 2.1, we have

kE 4

(VOOJ.If)(C) = vJ*“(gg’)-’C)f((gg’)-’C) = v’* a(gI-1,g-10 v’, yg-1, C) f(g‘-’(g-’C))

C) (VO.’.’f)(g-’C) i.e., =( Vg’l’( VOJ* f)) (C),

= v’qg-1,

VgrA = V,j. * V,J. I. J

From the definition of t(g, 8 ) (Proposition 1.5) and u(g, C), we have t(l,c)=l andu(l,C)=l, so weget V1jpJ=l. b) We fix j and s and first prove, that Vjl * is strongly continuous at 1, i.e., (2.3)

for given f E 8 and E >O, there is a neighborhood W of 1 in G, such that

i

IIVgJ.If-fll<~ for g E W. Iff=O, then (2.3) is trivial. So we assume llfll +O. Let N be any compact neighborhood of 1 in G and let C=C(N), the positive constant for the compact set N,given in 1). The set of continuous functions is dense in 8 (for example, by the Peter-Weyl Theorem), so there exists a continuous function $ on U,such that

Ilf- $11 <Min ( ~ / ( 4 C i ~ ~ *1,l llfll, + l ’ ~44). (2.4) Note that, in particular, (2.4) means $+O, ll$ll+O. We shall prove that there is a neighborhood M of 1 in G such that

-=

1 vj,‘(g-l,c) - 11 ~/(411$]1).for any g E M and any C E U. (2.5) In fact, since vj. ’(g-l,c) is continuous on G x U,for any c’ E U,there are an open neighborhood Bc# of c’ in U and a neighborhood ME’ of 1 in G, such that, if c E BE,and g E Mcl, then (2.6) Iv’*’(g-l,0- 11-=~/(411$11). Since U is compact, there exist a finite number of elements CI,

such that

..., Cn,

u” B,, = U.Let M = n” Mci;then M is a neighborhood of 1 in

i-1

i-1

G, satisfying (2.5). Since the function (g,c)l-.$[g-lC)is continuous, this function is uniformly continuous on the compact set N x U. So there exists a neighborhood L of 1 in C, such that Il$(g-1C)-$(t;)ll.,<E/(4Clse’1), f o r g e L and C E U. (2.7) Since I1 II 5 II I\-, we get

217

PRINCIPAL CONTINUOUS SERIES

~~~(g-'C)-~(c)ll<&/(4CIRea1), f o r g e L and C E U.

(2.8)

Let W = N n M n L (a neighborhood of 1 in G). If g s W, then, by (2.1), (2.2), (2.4), (2.5) and (2.8), we have

II V,j.Y-fll s II V,j* Y-

V,'* *#I1

+ llvj~a(g-lY C ) K C ) -

+ Ilv'* "g-',

C)qW1C)

- Wg-l, C)KC)II

#(C>ll+ 119-fll

- CIReal+l/a Ilf-~II+C!Realll~(g-lc)-KC~ll+Maxlv~~*(g-l, 5 0- 11 11dI C*U

+Ilf-411

<44+ &/4+ &I4+ €14=E . Hence VJ.' is strongly continuous at 1. Now let go E G and consider

II V,,r*'f- V,'* Yll = II V,,J*Tf- V170-'Pj'Y)ll 5 C({go})'Rer'+I'4 Ilf- V,o-ld.Yll * If g is in goW, where W is the neighborhood of 1 obtained above, then go-lg E W,-so ]If- Vg0-1,j~ 'fll < E , and we get

IIV,,J.'f- V*j*yll
(2.9)

€2

which shows that Visais strongly continuous at go. 3) Since d(gC)=e-f(g.C) dC (Proposition 2.1) and t(1, C)=O (Proposition 1.5), we have, f o r f s 8,

e-sReWo-l.

oF)-tCu. c)I f(c)l'dc

=~;G'R~~-L)I(L% C)lf(,")lsdc,

Since any non-negative function $ in L1(U) can be written as &)= If(C)ll for somefe L2(U),and any function in L1(U)is a linear combination of non-negative functions in L1(U),we have the following equivalences :

IIV,'*YIl=IIfll

for a l l f s L 2 ( U )and g e G

r

(by (2.10) below) ee(*Re*-1)t(g*

c)=

1 for almost all 5 E U and g E G- K.

218

UNITARY REPRESENTATION OF SL(2,R)

(Note that the dual space of L1is the space of essentially bounded, measurable functions.) Moreover, we have that the last condition holds -(2Res-l)t(g, C)=O for all C E UandgEG-K -2Res= 1. This equivalence, together with 2), shows that V'.' is a unitary representation of G if and only if Re s= 1/2. We have (2.10)

t(g,C)=OforallCEUogEK. In fact, if g E K , then t(g, C)=O is clear from the definition of t(g, c). Conversely, suppose t(g, C) = O for all C =eie E U for an element g=

[z i]

in SU(1, 1). Then by (1.14), we have

for all 8 E R . Putting e = O andnin(*), we get la+fi12=I-a+p13 andhence Re(aB=O. [ aet0 + 8 1 2 =ett(g. 8) = 1

(*)

Putting

e=- 2R

and

--n2

in (*), we get lia+fila=[-ia+p[z and hence

+

Im(aS>=0. Therefore we have afl= 0. Since la[a = 1 [ p[a 2 1, we can conclude that ,8 =0 and g E K. From (2.10) it also follows that V&* is a unitary operator for any k E K. 4) We have (V,,@j. a f p ) (c) =e-rt(ue-', c) (u(ue-', c))aYP(ue-lc). Since ue-l E K, from the Iwasawa decomposition we have t(ue-l, c) =O. From the definition of the function u, we have u(ue-l, c)=ede'2. Further-

ac+p we get ue-lc=e-tec. Putting this in the right more, from g.c=----pc+a member of the relation above, we get (V,,j*.S,)(0= e j f e + p r e ~ - p = x P + j ( u e l f P ( C ) . Definition 1. The set of unitary representations 1 1 {Vj.a(j=O,-; s ~ C , R e s = - } 2 2 is called the principal continuous series. The subset { Vo,t+i2117 E R} forms E R} forms subseries called the first series, and the subset {V***+t21R subseries called the second series. To study the irreducibility of the representation W a ,we introduce a representation (Vj*a)'=dVj.*of the Lie algebra g of G which can be obis a generalization tained by differentiating W The representation of the differential representation U' of the finite-dimensionalrepresentation U defined in Ch. 11, $5. However for an infinite-dimensional representation V " s 8 , the derivative dVj*a(X) =(dVe,,r,j.*/dt)t,o is an un#.

PRINCIPAL CONTINUOUS SERIES

219

bounded operator and its domain cannot be the whole space 8.Hence it is necessary to introduce a sufficiently large subspace of 4 which will be contained in the intersection of the domains of the operators dVj*'(X) for all X E g. Several subspaces of 8 are suggested as the representation space of dVj.'. We use the space of differentiable vectors. Definition 2. Let U be a strongly continuous representation of a Lie group G by bounded operators on a Banach space H. Such a representation is called a Banach representations. A vector v in H i s called a dr;trerentiable vector or a C"-vector if the mapping gHU,v is a C"-mapping on G with values in H. Proposition 2.3. The space H, of C"-vectors for U is invariant under

U, for any g E G and under U,=

So

f(g)U,dg for any complex valued

C"-function f with compact support. Proof. Let go E G. Then the mapping g H U,V,,v is the composition of the right translation gt+ggo and the mapping v" :gHU,v. Hence if v is a Cm-vector,then U,,v is also a C"-vector. Iff is a C"-function on G with the compact support C, then (2.1 1)

g H u g u f v = J f ( g o ) U , ~ , ,v dgo

is a C"-function on G . In fact, gHU,U,,v is a C"-function by the first half of the proposition and f is integrable on C, hence, (2.11) can be differq.e.d. entiated arbitrarily many times and is a C"-function on G. Definition 3. Let U be a Banach representation of a (linear) Lie group G on the space H. Then for any X E g (the Lie algebra of G) and v E H, (the space of C"-vectors for U ) , a linear mapping U ' ( X ) = d U ( X ) of H , into H i s defined by d (2.12) U'(X)v= [-dt u e x p txvlt-0. Since the mapping tHUexpt x v is a C"-mapping of R into H, the derivative in (2.12) exists. Clearly U ' ( X ) is a linear mapping of H, into H. As the following Proposition 2.4 will show that the mapping XI+ U ' ( X ) is a representation of the Lie algebra g of G on H,, the representation U' of G is called the dzferential representation of U and is denoted by U' or dU. The representation U' of the Lie algebra g may be uniquely extended to a representation of the universal enveloping algebra U(g) of Q. The extended representation of U(g) is also denoted by U' or dU.

220

UNITARY REPRESENTATION OF

SL(2,R)

Proposition 2.4. U'(X), defined by (2.12), is a linear transformation on H,, i.e. U'(X)H,cH,. Moreover U' is a representation of the Lie algebra g on H,. Proof. Let v E H,. Then 9 :gHU,v is a Cm-function.Since the mapping p : (g, t ) n g exp tXis a C"-mapping from G x R into G, it follows that the composed function 9 =v" p and its partial derivative a9/at(g, 0) = U,U'(X)v are C"-functions. Hence we have proved that U'(X)H,c H, for any XE g. Now we prove that U' is a Lie algebra homomorphism of g into gI (H,). Let XI, .., Xnbe a basis of g and v be an element of H, Put U,= U(g) to avoid complicated suffices, and put 0

.

.

n

p(t) = p(ti,

..., tn) = U(exp CtJt)v. $=I

Then p is a C"-function on Rn and

%O)

(2.13)

atr

= U'(Xf)V.

n

Let X=

1t&.

Then we have

f-I

f-1

by (2.13). We have proved that U' is a linear mapping of g into gI(H,). In particular, for any fixed v E H,, the mapping XHU'(X)V is a C"mapping of g into H. Now we prove that U' preserves the Lie product. Let X and Y be elements in g and x(t)=exp tX. Since the differential representation of the adjoint representation Ad (Adg :XHgXg-') of G is the adjoint representation ad (ad X : Yt+[X, y1) of g (cf. Ch. 11, $5, Example 4), we have (2.15) Ad (exp t X ) =exp (t ad X ) (Ch. 11, Proposition 5.9). We see that for any v E H,, d W A d g ) W = LxU(exp(t(Ad g) Y))VI~.=O

PRINCIPAL CONTINUOUS SERIES

d =[,U(g)U(exp

and (2.16)

221

tY)U(g)-lv]r=~= UgU'(Y)U,-l v

[ U'(X), U'(Y)]v=(U'(X)U'(Y) - U'(Y )U'(x))v

d =[-U2~t~U'(Y)U2(-t)vlt=o dt

d =[xU'((Ad x(t))Y)v]t-o. By (2.15), we have (Ad x(t))Y= Y + t [ X , Y J + O ( t a )

(t+O).

Hence t-' { U'((Ad x(t))Y)v- U'( Y)v) = U'([X, Y])v+ U'(O(t))v.

Since XI+U'(X)v is continuous, we have lim U'(O(t))v=O.

(2.18)

8.0

(2.16), (2.17) and (2.18) prove that [ U'(X), U'(Y)]v= U'([X, Y])v

and U' is a representation of the Lie algebra g on H,.

q.e.d.

Proposition 2.5. Let Vj.' be the principal series representation of G = SU(1, 1) on the Hilbert space 8 = L a ( U ) . Then the space g = C " ( V ) is a #) (c) = contained in the space 8, of C"-vectors for Vj.a. Put (Vexpcxf~ @(C, t ) for # E 9 and X E g(C E U and t E R). Then the differential repre* is defined by sentation d V d ,a = (Vr**)'(X)of Vf. (2.19)

"m)(0=a@/at(c,0).

(dVj-

Proof. Since the Iwasawa decomposition is an analytic diffeomorphism of G onto K x A x N (Proposition 1.4), et(g,C) and u(g, c)=e'(-g'@+e)'*are C"-functions on G x U. Hence the function Ol(C, g)=(V,j.* #) (c) is a C"function on U x G if q3 E 9.So it is sufficient to prove that if V(c, t ) is a complex-valued C"-function on U x Rn and if the function Tt: t;wV(r;, t ) belongs to 8 for each fixed t E Rn, then the function t w T t is an 8valued C"-function on Rn. Let el =(1,0, .., 0), ,.., en=(0,..., 0, 1) and Put

.

222

UNITARYREPRESENTATION OF SU2,R)

Then by Taylor's theorem, we have

ava c , t, h) =2-'h+c,

t

+o w

for some 0 E (0, 1). Since 7P is a C"-function, for any a>O, there exists an M > 0 such that

;1

%C,

s)l4

M for any c E U and s E [t -a, t +a].

Since IF&, t, h)l6 Mahaif Ihl $ a and lim F&, t, h)=O L-0

for any (C, t),

we can apply Lebesgue's theorem and get

s

limllFn(C,f, h)llz=lim h-0

h*O

27

IF&, t, h)lzdC=O.

Hence the function t-?FPLis differentiable with respect to tk and the derivative is equal to the function &4?F(C, t)/af,. In particular (2.19) is proved. Similarly we can prove aa?Ft/atcat,exists and is equal to c-aa?F(c, t)/at,at, and so on. We see that V t has derivatives of any order and is an 8valued C"-function on U. q.e.d.

where

3) If # E 6-belongs to a closed subspace 8' of 8 which is invariant under V.',then dVx'.*# belongs to 8' for all X E g. 4) We can extend the definition of ~ V ~ . ' ( g) X to E X E gc formally by

PRINCIPAL CONTINUOUS SERIES

dVd.' =dV&'+idVd.', Then we have

223

where Z = A + i B e go, A, B E 8.

dJ'Z'. ' f p =(S + p +j)fp+l, dVz'*'f,= ( s - p -j)fp-iy for Z = X l + i Y and z = X l - i Y . Proof. 1) Let ar=exp tXl, then we have

(2.20)

(Vex,tg,l*J#) (C) =e-Jr(5~-'~c)u(at-l, C)2j#(at-1C)

and Ch-

t

ShT

at =exp tXl =[ s h i

chi]'

We want to differentiate (2.20) with respect to t and set t=O. We first observe that the product of the first two factors of the right member cf (2.20) is equal to &J+j ) r ( 5t - 1,C)

(-

C sh- t

+ch-)Xj. t

2

2

(cf. (1.14) and the definition of u). We have

'

To deal with the third factor of the right

member of (2.20), we first calculate

If we set at-'. C = etB(')and write #(ar-l-C)=f(O(t)), then we have d

1

x#(ar-l-~)

We note, thatf'(e(0))

= (D#) (C).

t-0

=f(e(o))e'(o).

We obtain e'(0) from

d ( ~ ~ ' c ) l r - O = e gie'(O)=i@'(O), @(o) which, combined with the result

224

UNITARY REPRESENTATION OF

SL(2,R)

above, gives us 8 ' ( 0 ) =2i3 . Putting these together, we have

(~v,,,~.B)(c)=o.+s~+(c)+2i(-~)+(c)-~~-c-') c+rl

(09)(0,

giving the first formula in 1). If we replace the at above by

and make the corresponding changes all the way through, then we arrive at the second formula of 1). 2) To obtain the results in 2), we apply 1) to =fp and use

Cfm

+

=fp-l(C),

m P t r )

c-YP(c) =fP+l(C), =

- @fP(C).

3) If # E 8- n @', then t-l(Vexp t X j s * # - #) belongs to 8'. Therefore, the strong limit lim t-l( Vex,t x j * - +)=dVxjS8# t.0

+

belongs to 8'. 4) Obvious. LEMMA 1. Every element g in G = S U ( l , 1) or Go=SL(2, R) can be written as follows: g= ucatu+. Proof. Since G is isomorphic to Go, it suffices to prove the lemma for the group Go. Let g be an element in Go, and P be the set of all positive definite real symmetric matrices of order 2. Then Igg belongs to P.So there exists an element p in P such that 'gg=pa. Let u=gp-'. Then we have $uu=p-ltggp-'= 1, i.e., u is an orthogonal matrix. Since detg= 1, we have detp = detu= 1, i.e., p, u E Go= SL(2, R). Since p E P n Go, there exists a v E SO(2) =K such that p = v-latv for some t. Since the elements uv-l and v in K can be written as uv-l = u+ and v = u+, we have g= uv-latv= u+atu+. q.e.d. Remark. Concerning a direct proof for the group G and the uniqueness of the expression for g, see Proposition 5.2.

Theorem 2.1 1 1 1 1) If Re s =- and (j,s) # (- -), then Vjl is an irreducible unitary repre2 27 2 sentation of G.

225

PRINCIPAL CONTINUOUS SERIES

2) Let $3' (resp. 8-) be the closed subspace of 8 spanned by { fp I p r o ) (resp. { fp I p
Proof. 1) By Proposition 2.2, if Res=-, then Vjs a is a unitary represen2 tation of G. Let K be, as before, the maximal compact subgroup of G given

Eefs,#ElI

by K = [ u ~ =

0 5 0 <4z}. The restriction Wj.a = Vj,* I K of

the unitary representation VJt8to K is decomposed into a direct sum of irreducible representations (cf. Ch. I. Theorem 3.1 .) and any irreducible for some p E 212. unitary representation of K is given by xp(ue)= Actually, we have seen in Proposition 2.2, 4) that Vue'*'f,=xn+j(ueYn, wherefn(C)= C-",

n E 2. Since [fn I n E Z } is a complete orthogonal system of functions in .ij= L2(U,dC) (cf. Ch. I. Theorem 3.3 and Theorem 4.1), we have WJ*'=V'*' I K = exn+,. n.8

Let 8' be any non-zero, closed subspace of .ijinvariant under P**. We shall show that 8' contains at least one fn. Since Wj.*I &'= there is an f E 8' (f#O), such that Vuef.'f(C)=

xn+j(ue)f(C)

@

xn+j, nrAc2

=erns+*jef(C).

Since Vuejva f ( c ) = l e * @ l a l - s ( * + j ) (efela)Xff(e-*@c) =e*j@f(e-*@c), we have

f(e-'")

= e'"@f(C)

for almost all C.

Hence we have

f=cfn for some c E C. Hence,f, E 8' n 9.We now note (Proposition 2.6, 4)), that (2.21)

V d *yn +n +j)fn+i, {ddV$*"f,=(s-n-j)f,-i, =( S

belong to 8' (Proposition 2.6, 3)).

and dV&% and dV$% 1) If we assume Res=-

(2.22)

1

2

t

and (j,s)#(-

1 1 -), then we see that

2' 2

(s+n + j ) # 0, (s-n - j ) # 0.

1 andj=O, 2 E 8'. Repeating so again (2.22) is clear. From (2.21) and (2.22) we the argument we get fn*S E a', and so on and we have, finally, fp E 8' for

In fact, if ImszO, (2.22) is clear, while if Ims=O, then s=-

226 all p

UNITARY REPRESENTATION OF E 2.Hence,

1 2) If (j,s)= (?

8’=Q and

Vj.

SL(2,R)

1 1 is irreducible if (j, s) # (- -). 2’ 2

y1),then (2.21) becomes

’* lllfn =(1 +n)fn+i,

If

llz,llafn

= -nfn-l.

This shows, that 8’ n 8- is invariant under dVz1Ia.lIaand d V ~ ~ l ’ *hence ~l’, and dV+ I f a . Since V,,j.”f,= xp+j(ul)fp,Q* invariant under dVXllla*lla is invariant under VVej* ‘. Since @*n Q- is invariant under ~ V X ~ we ~ ~ ~ * ~ ‘ see that Q* is invariant under VexptX1lla,lla= Vat 1fa,112. (This fact is proved by Proposition 6.9 and Proposition 6.11 below). Furthermore and any g E G can be written in the form since Q* is invariant by VUeJ.* g=ulutu+ (Lemma l), we see that Q’ is invariant under VO1l*,lla for every g E G. If Q’is any non-zero closed invariant subspace of @+ (resp. Q-), then, as in l), there is some fn E Q’,n r O (resp. n
q.e.d.

Now we may characterize the representation Vj**from a general stand) the unitary representation of G=SU(l, 1) inpoint. Vj.8 ( s = & + i ~is duced by a one-dimensional unitary representation of the minimal parabolic subgroup r =Z A N of G . Definition 4. Let g be a Lie algebra. Define Wkg(k=1,2, 3, ...) by W1g= rg, s3, w 3= [ * ? 3 , gl. A Lie algebra g is said to be nilpotent, if there is some k E N,such that

.

%=g = {O} Let g be a Lie algebra. A subalgebra Jj of g is called a Curtan subalgebra of g if Jj is nilpotent and the normalizer n(!j)= { X Eg I [ X , Jj] c !j} of lj

is equal to 8. Let G be a connected semi-simple Lie group and g its Lie algebra. If Jj is a Cartan subalgebra of g, then the subgroup H defined by

H = {g E G 1 (Adg)X= X for all XE Ij] is called the Cartan subgroup of G corresponding to Jj. The Cartan subgroup H has Lie algebra lj and is not necessarily connected. Let Ij,!j’ (resp. H,H‘) be two Cartan subalgebras (resp. subgroups) of g (resp. G). Then $(resp. H) is said to be conjugate to J$’(resp. H ’ ) , if there exists an element g in G such that (Adg)E]=!j’ (resp. gHg-l=H’).

227

PRINCIPAL CONTINUOUS SERIES

Proposition 2.7.

Let G=SU(l, 1) and g = l u ( l , 1).

1'

1) Let Xl=z[l 1 0 0] 1 and Xo=-!-[i 2 0 -i

and let

Q1=RXl and Q s = R X o .

Then Q1 and Qs are Cartan subalgebras of g. 2) Let Hi be the Cartan subgroup of G corresponding to rjr(i= 1,2). Then H i = Z A , Ha=K, where Z = { k l}, and A = {at) and K = {u8) are the subgroups of G appearing in the Iwasawa decomposition KAN of G (cf. Proposition 1.2). 3) Every Cartan subalgebra [resp. subgroup] of g[resp. GI is conjugate to either QI or Qa [resp. Hl or Ha].The Cartan subalgebras [resp. subgroups] Q1 and Qa [resp. Hl and H2] are not conjugate. Proof. 1) Since Qi is 1-dimensional, Q, is abelian, hence nilpotent (i= 1,2). By the isomorphism eK(2, R ) ~ l u ( l 1) , (cf. Proposition 1.6), Q, can be identified with the subalgebra { H =

[:

-1131

U E

R} of H(2, R).

~

E SI(2, R). Then, for H E Q1, we have [H, XI=

, so [H, XI E q1 ( H # O ) o x l a = x a l = O o X E Q1, and we

-b X , l have tt(Ql)=rjl.

Hence ljl is a Cartan subalgebra. Similarly,

identified with the subalgebra { H =

[-5: t] I

X is as above and H E Q2 we have [H, XI=

U E R}

can be

of lK(2, R). Then if

VXa1

-2ax11

1

-a(x1a+xa1) ' and we have n(ljn)=

-22aX11 +xla)

so [H, x] E Qa(H#O)axll=O and x12= - x X a l o X eQa,

Qa

Hence Qa is a Cartan subalgebra. 2) By the isomorphism SL(2, R ) z S U ( l , 1) (cf. Proposition 1.6), we consider H I as being contained in SL(2, R) and Q17 in lK(2, R). For g E SL (2, R), we have Qa.

g E HI- (Adg)H= H for all H E Q l

o g H = Hg for all H E Q1 e g = [ "0 Since A = (at=[:I2

"1

u-1

for some a E R - (0).

e-t:]}, the latter condition holds if and only it

g E ZA. For the case of Ha (considered as being in SU(1, I)), we let g E SU(1, 1) and observe, as before, that g E H z o ( A d g ) H = H for all H E rjaagH=

228

UNITARY RHPRESENTATION OF

SL(2,R)

3) If Q and Q' are two Cartan subalgebras of g and H and H' are the wrresponding Cartan subgroups of G, then we have, for g E G,

(Adg)Q =kj'*gHg-l= H' So it suffices to prove the statements in 3) for the Lie algebra g . Identify g with GI(2, R). First we shall show that a Cartan subalgebra Q of g must have dimension 1. Let { H , X , Y ) be the base of g given by

Then we have [H, E]=2E, [H, I;]= -2F, [E, F]=H, so that W g = [ g , g] = g, which shows that g is not nilpotent. Hence dim Q 0, because if Q = {0), then n(Q) = n( {O)) = g # Q. S u p pose dim Q =2. Since dim @Q
and there is some g E SL(2, R ) , such that gXg-l=

-3;

thus we

have gIjg-'= I j l = R X I . (ii) a=ci, b = -ci, c E R , c#O. In this case there is a g E SL(2, R ) , such that gXg-l=

["- G],

SO

gIjg-l= Qa =RXo.

(iii) a=b=O. In this case, there is a g E SL(2, R ) , such that gXg-1=

[:

i],

for H =

so gQg-'= 8' where Jj'=RX2,X2=

['0

-1

"1

i].

But Ij'#n(fj'), because

we have [H, X3]=2X2,so that H E n(Ij'). Hence fj' is not

a Cartan subalgebra, and so neither is Q =g-lJj'g.

229

PRINCIPAL CONTINUOUS SERIES

It is clear that a real matrix of the type appearing in case (i) cannot be conjugate in SL(2, R ) to a real matrix of the type appearing in case (ii), hence fil and 6%are not conjugate. q.e.d.

COROLLARY to Proposition 2.7. Every Cartan subalgebra of g= H(2, R)E h ( l , 1) is a maximal abelian subalgebra consisting of diagonalizable (i.e., semi-simple) matrices. Remark 1. The converse of this Corollary is also true. Remark 2. Every semi-simple element g e G is contained in a conjugate of Hl in G or a conjugate of Ha in G, that is, every semi-simple element of G is contained in a Cartan subgroup of G. The union of all Cartan subgroups of G (=the set of all semi-simple elements of G) is an open and dense subset in G. In fact if g E G is not semi-simple, then the eigenvalues of g are equal, so g E {h E GlTrh = f2). (cf. Proposition 7.8). Proposition 2.8. The set Z?< of all irreducible unitary representations of H&= 1,2) is given as follows: 1 f i =~ *'Z{ u(zat)= xj(z)eiv' Ij = 0 , ; Y ~E R } ,

where j(z) =zaj for z E { f I}.

KL= { I , I j E 2/21, where x,(us) =eve. Proof. 1) We have H2= {us=

Lie''

"I

e-is,l

I O S O < ~ The ~ } .mapf:

B + ~ ~ Z +is Uan~isomorphism from R/4& onto K=H,, and the map g : ?+Z + 4 ~ ? + 4 x Zis an isomorphism from T = R / Z onto Rl4r2, so we have the composition of isomorphisms

TARI f ~~z-H~. It was proved in Ch. I, Theorem 4.2 that f = { ~ ~ ( f + Z ) = eI n~E~Z}. ("~ Thus we obtain I%= {unog-lof-l I n E Z } so we can write But (unOg-lof-l)(us)=un((B/4x) 2)=eSnbfa =

.

+

fi2=

{x,

ljE

2/21.

2) Since Hl is abelian, every irreducible unitary representation of H I is I-dimensional. Since H l = Z x A, we have 1

f i l = 2 x i f = { x , I j = 0 , - } x {at++e*Y*IV E R } , 2 because A s R and hence i f s & ~ R .

q.e.d,

230

UNITARY REPRESENTATION OF

SL(2,R)

Let G =KAN be the Iwasawa decomposition of G = SL(2, R ) z SU(1,l) and let r = Z A N , where Z = { k l } , and U={CECI ICl=l}. Weidentify the three manifolds G/roK/ZxU

by means of the diffeomorphisms grwueZwe‘e= C,

where g=ue at ne. Recall that K = {

[t’’’

e-t8:]

I 05 0 <4a}

and that

the homomorphism uswe‘efrom K onto U has kernel Z . Proposition 2.9. 1) Let of Hl = Z A defined by

11.”

be the irreducible unitary representation

1 Ij*y(zat)=xj(z)e*ut ( j = O , -; 2

Y

ER).

Let Lj.” be the function on r =ZAN defined by N). Then Lj.’ is an irreducible unitary representation of r. 2) We can extend LjsUto a function on G by putting = xr(ue)L f *.(we). Ljs Y(uoatne) (2.23) The extended function LjSy satisfies Lj.y(gh)=Lj.y(g)LjSv(h) for all g e G, h e r . Lj*qhx)=lj*qh)( h E H 1 , X

E

Proof. 1) The first part of this proposition is almost evident. In fact, N is a normal subgroup of r and we have r / N z Z A = Hl. Ifp is the natural map from r to r / N , then L 5 . ~ = l j o. ~p is a unitary representation of r, and it is irreducible because ljv’ is irreducible (in fact 1-dimensional). 2) Let h=zb, where z E 2, b E AN, and g = ueatncthen Lj.u(gh)=Lj*y(uoatnczb) =Ljsv (uezatnpb) = xj(ueZ) Lj**(atnEb) =xj(uo) Xj(z) LjS’(atnc) LjSu(b) =L’*’(ueatnp)LjSu (zb)=Lj*’(g)Lj.’(h).

q.e.d.

Let L,I be the measure on U defined by dp(C)=dy=de/2a (C=e*e). The Jacobian of the transformation CwgC is equal to e-“V*C) (Proposition 2.1). We have, therefore,

PRINCIPAL CONTINUOUS SERIES

23 1

where the left-member is a Radon-Nikodym derivative. The measure g p on U defined by (A is a Bore1 set in V) ( g p ) (A)=p(g-’A) is absolutely continuous with respect to p . Hence p is a quasi-invariant measure on U with respect to the action of G. The induced unitary representation ULj.yon the Hilbert space Q(Lj.y) is defined as follows (cf. Ch. IV, $1): is the set of all functions f on G satisfying The Hilbert space Q(Lj.#) l), 2) and 3): 1) f :G-X is measurable. 2) f(gh) =Ljlu(h)-lf(g) for all g E G and h E r.

3) / ; f ( g ) ] ’ d c c co (where c = g r E G / r = U). From 2) we have If(gh)l= If(g)l for g E G,h E r. It is clear that Q(L’*”) is a Hilbert space with inner product

The induced representation ULjPyon Q(Lj.v)is now defined by setting, w), forfe Q(LJ.

-

( UOL’.yf)(g1)=(~p(g-’g1)/dp(E))+ f(g-lg1)

-e-t(Q-l.C)IZf(g-lgl), where C = g J . We can check easily that ULjsuis a unitary representation of G. For anyf E @(L’*y) we definefi by (2.24) Thenfi satisfies (2.25) In fact we have

f i ( g )=~ ’ W f ( d .

fi(gh) = f i ( g ) for all g E G,h E r. fi(gh)=L ’ ” ( g ~ l f ( d 4 =V * ’ ( g )L’. h) L’, h)-’f(g) p(

y(

f ( g )=fiw So fi is actually a function on G / r = U.Let =L’.W

(2.26) d c ) = f i W =L’*’(glf(g)7 where C = g r E U = G / r . Then the mapping (2.27) A :f t . 4 is an isometry from @(Lj..) onto @=L’(U, dc). In fact, since

1$(c)l2=

232

UNITARY REPRESENTATION OF SL(2,R )

Ifk)tP,we have ll$ll'=[

U ig(C)PdC=/ U If(g)l'~C=IlfII'.

Hence A is an isometry of $j(LjlY)into 8 =La(U,dc). Let $ be an arbitrary element in 8 and put f(g) =Lj.'(g)-l$(c) for = g r . Then it is easily seen that f belongs to $j(L'.y) and $ =A$ Therefore A is an isometry of 8(Lj.') onto 8 =La(U,dc).

Proposition 2.11. 1) Let M be the centralizer of A in K and MI the normalizer of A in K. Then we have M = {+1} =z,M1= {uelO=O, z, 2%327).

2) u,utu*-l= Q-t. 3) Let W =M1/M= { 1, w } (w' = 1, w =u.M( = us.M)) and let r'r.~(h)=z'.~(h")=zj.~(u.hu.-l) ( h E Hl).

PRINCXPAL CONTXNUOUS SERIES

233

Then we have 1'1,

=11.-u

.

Proof. 1) By definition we have M = { U s E K 1 u8aub-l = a for all a E A}, MI= {Us E K 1 usdus-'=A). We calculate the product uratue-' and obtain (2.30) t

=[::h 2 l We now see that, for 0 5 e <4r, we have U s E Meusatus-' =at for all t E R e e f @ = 1e B =0, 2 x and U s E MloFor each t E R , there is some t ' E R such that u,atu-l=at, e e i @ = fl d = o , K, 2 ~3, ~ , hence the first assertion of the proposition is proved. 2) Putting e = K in (2.30), we get u. at u,-l=a-:. 3) The final assertion follows from /'J**(za,) =lj. U(u.zatu.-') =Zj.'(za-t) (by 2)) = Xj(z)e-"' =1'. -'(zat). Theorem 2.2.

If Res-i, then we have VA a g vj. 1-8.

Proof. Since Vj*++'uz UL'.' by Proposition 2.10. It sufiices to prove that u L j v w uLj.-u. Let $ E $j(LjSy)and let f= Bq5 be defined by f(g) =(4) (g) =L'."(U.)$(g)-

Then f belongs to Q(U.-'). In fact we can show that f satisfies the three conditions for a function on G to be in $(Lj.-') as follows: 1) It is clear that f is a measurable mapping from G to C.

234

UNITARY REPRESENTATION OF

SL(2,R)

2) For every g E G , h E H and n E N,we have f(ghn)=L~*’(u.)(b(ghn> =Lj. qu,)Ljqvz)-’ 4(g) =Lj.qu.)Lj.*(h)-’Lj.u(u*)-’Lj.Y(u.)4(g) =l j *quXhux-1)-’f(g)

(by Proposition 2.11)

=Z j p -*( h)-’f(g) =Lj*-”

(hn)-lf(g).

3) Since IU+(u.)l= 1 we have If(g)l= l@(g)land

Moreover, from this it is apparent that B : (b+f is an isometry of 8(Lj.”)onto Q(LJ.-y).F o r f s Q(LJ*-’),let B-’f=(b( E Q(Lj.y)).Then ( B UgLJsyB-lf) (gi)=Lj*’(u.) (UgL’*”$) (gl)

Proposition 2.12. The representation Vj,‘(s E C ) defines a representation of the proper Lorentz group G+(2)if and only if j = O . Proof. Since G+(2)E G/ { rt l } (Proposition 1 4 , Vjs8 defines a representation of G+(2) if and only if Vj.’( - 1) = 1. Note that - 1 =ua.. By Proposition 2.2,4) we have V d :fp Hence we have

=

xP+ j(udfp, where .fp(C) = C - p ( ~ E 2)

Vj.*(- 1)= loX,+j(uax)=1 for all p

EZ

(using x,,(ub)=e‘”‘) e e a j “ (= 1

oj=O,

because j = O or

-.21

235

PRINCIPAL DISCRETE SERIES

53. Irreducible unitary representations 11. PRINCIPAL DISCRETE SERIES 1 Let n E 32.Then

is an irreducible unitary representation of K=Ha. xR can be extended to a (rational) holomorphic representation of K C=

bY

Proposition 3.1. The group G=SU(l, 1) acts on the unit disc D =

{c E C I IcI < 1) by the linear fractional transformations c-gc=- ac+p

where g = [ " E G. /3c+a' B E The action of G on D is transitive. The isotropy subgroup of G at 0 is equal to K. By the canonical mapping f : g K e g 0 , GIK is homeomorphic to D . Proof. Let C + be the upper half-plane and c the Cayley transformation z-i c : ZH-, which transforms C + onto D . We have seen in Proposition Z+l

1.6 that SU(1, 1)= CSL(2, R)C-l, where C = -

A[:

-:I.

We can

check easily that the following diagram commutes: g

D -D

Tc c+-c+

CT

g'

where g'= C-lgC (g E SU(1, 1)).

Hence g D = D (gE G=SU(l, 1)) followsfromg'.C+=C+(g'ESL(2,R)). and the latter can be checked by direct calculation: Let z = x + i y (y>O) and g ' = E

f;]E SL(2, R).Then

Y Im gz =Im- az+b = cz+ d (cx +d)a+ caya The transitivity of G=SU(l, 1) on D follows from the transitivity of

236

UNITARY REPRESENTATION OF

SL(2,R)

SL(2,R) on C+. In fact, the subgroup AN=

acts in simply transitive fashion on C+, as can be seen from the following calculation:

.

1=--

For g =

["

"1

ai+b -ab a-l

+ia'.

E G, we have g0 =,;B and therefore

B a

g*O=O*,9=O*gE K: Let p : G-+G/Kbe the canonical projection. It is continuous and open. Let @ : G-D be the continuous map defined by qj :g w g . 0 . Since G is locally compact and has a countable base, and since D is a locally compact HausdorfTspace, @ is open (we use Baire's category theorem to show this). Sincefis 1 :1 andf=@op-',fis a homeomorphism from G/Konto D.

Proposition 3.2. Let g =

i]

E G,

C E D, and J(g, C)=jC+a. Then

1 ) J is an automorphic factor, i.e., C)=J(gi, a < ) J(ga, C) and

{J(1, c)= 1 J h i ,

hold. In particular, J&', gCF'

=Jk, C) z 0.

2) 1% C)P (1-lgCl')=1-ICl2. 3) Let C =x iy and let dm(C)=h ( x , y) =&dy. Then d,1(4)=U ( x , y ) = ( 1 lCli)-2 h ( x , y) is a measure on D invariant under G.

-

+

J(1, C)= 1 is clear from the definition, and the rest of 1 ) follows easily. 2) We have

PRINCIPAL DISCFLElT SERIES

237

ax’ ay‘ a ( ~ ’y’) , - ax a(x,y)

ax’ ar

ax

aY

1 1) Let n E -2 and

Proposition 3.3.

2

(using the Cauchy-Riemann equations)

nr 1. Then

is a Hilbert space and 1 E sn, 11111= 1. Every bounded measurable function f on D belongs to 9,and satisfies

I l fII4 1 1 1 1 1 . where

II f II,=ess sup If(C)l. LrD

2) For any g E G, we define the operator Tanon 9% by

(Tan@)(C) =J(g,-’, C)-’”@(g-’C)

where g - l = g

on 9 n Proof.

i].Then T n

:g H T a n is a unitary representation of G

1) Since p,is ,an La-spacewith respect to the positive measure

238

UNITARY REPRESENTATION OF SL(2,R)

2n- 1 d m n ( C ) = x ( 1 - IC1s)2"-adm(C)on D ,

of 1 in

Z

n

is a Hilbert space. The norm

is equal to 1 because

(3.1)

=(2n-1)

s:

(l-f)a"-adt=l.

Therefore, the constant 1 belongs to 9 , By . (3.1), every bounded measurable function f on D satisfies

Using Proposition 3.2, l), the latter expression becomes

which by Proposition 3.2, 2) and 3) becomes

= IlW.

Thus, each operator T," is an isometry. Furthermore, we have (Tglgan

0)(C) =J((g1gJ-1L)-2m @((gig&10 =J(ga-', gi-lC)-amJ(gi-l, C)-an@,(ga-lgi-lC) =J(g1-1, C)-an (Toan0)(g1-'C) = T,,"

V9an

@I(C).

In particular, an isometry T," is invertible and hence a unitary operator. If the mapping T n :gHT gnis strongly continuous, then T" is a unitary representation of G on 2". Now we shall prove the strong continuity of T". In virtue of the equality

11 Thgn f- Th"f l l = IIr g n f - f k it suffices to prove strong continuity at g=1, i.e., to prove that for any f E p,, and E >0, there exists a neighborhood W of 1 in G such that the

239

PRINCIPAL DISCRETE SERIES

inequality

IITLlnf-fll< 6

(3.2)

holds for all g E W. Iff =0, then (3.2) is trivial. So we can and shall assume that f +0. Since dm,(C) is a Radon measure on the locally compact space D , the set L ( D ) of all continuous functions on D with compact supports (cf. Appendix D.5).Therefore, there exists a function # is dense in p,, in L(D) satisfying

Ilf- $11 < min(llfll9 43). (3.3) (3.3) implies, in particular, that #+O. Let L be a compact neighborhood of 1 in G. Since J(g-', I)-an is a continuous function of (g, t;) on L x D , there exists a constant M>O such that IJ(g-l, t;)1-an5MMoreover, since J(g-', <)-an and #(g-l t;) are uniformly continuous functions of (g, c) on the compact set L x L-'C, where C is the support of #, there exists a neighborhood W of 1 in G contained in L such that the inequalities (3.4)

(3.5)

IJ(g-l, Wan- 11<(6ll#ll=)-'e

and (3.6) l#(g-lc)- #(C)l<(6M)-le hold for all g E W and r; E C. Then we have for all g E W and C E C, [(To"#) (C)-#(C)l=

IJ(g-', C)-an#(g-lC)-#(C)I

s; I Jk-', t;)l-anl#(g-lc)- d C ) l

+ I J(g-',

Y - 1ll#(C)l

C

< M E + E II#II-=<, i.e., 6M 611#11= (3.7) IIT.J"#- 411-43. By Proposition 3.3, 1) and (3.7), we have (3.8) IIT,"# -#I] < ~ / 3 for all g E W. Since T," is a unitary operator, we get, by (3.3) and (3.8), IIT,nY--fl16 IIT~"f-T,"~ll+IIT~"#-#II+II#-fll <€13+€I3 4- €13= c for all gtz W. Thus we have proved the strong continuity of T".

q.e.d.

22-1

The representation Tn of G on e n = L a ( D ,-(1- Irla)a"-a dm(C)) R

is not irreducible. In what follows, we shall show that

an=[fe pnlf

240

UNITARY REPRESENTATION OF

Sq2,R)

is holomorphic on D } is a closed subspace of pn and that the restriction U" of Tnto anis an irreducible unitary representation of G.

Proposition 3.4. I) Let f be a holomorphic function on D = {C E Cl I ~ l < l )and let Z E D and e>O be such that D.(z)= {Cl IC-zl<e) cD. Then f satisfies

2) For any compact subset B of D, there is a constant C(B) >0, such that the inequality

If01s C(B)llfII holds for all f E 8" and all z E B. 3) For each fixed z E D , the map f t-+f(z) is a bounded linear form on

8". 4) &, is a closed subspace of gn. Proof: 1) Expand f into a power series convergent on D.(z):

-

C

~xc)= am(C-zP m-0

and set fm(C) =(c -z)". Then

Hence

The left member is equal to

1

If(C)I1dm(C),

while the right member

D&)

is L laol*llf~ll,~. Since Ilfoll.2=nt' and ao=f(z),we conclude that

2) Let dist(B, aD) =d >0 and let t =d/2>0. Let

B'= ICE CJdist(B,C)Sc}. Then B' is a compact subset of D . For any z E B, D.(z) c B'. Let

PRINCIPAL DISCRETE SERIES

24 1

Then

3) The third assertion is clear from the second. m-om be a sequence in $n converging to f in g n . Then by 2), [fm} ,,, is convergent to some function #(on D) uniformly on every compact subset B in D. Since # is a uniform limit of a sequence of holomorphic functions, q5 is holomorphic on D. We have 4) Let [fm}

# ( I ) =lim fm(C), m-lim Ilfm-fll=O. m-m

Some subsequence {fm,} of Ifm} converges to f almost everywhere. (This fact is contained in the usual proof of the completeness of Ls(D,mn)= gn.) So f and # are equal almost everywhere, and +=fin gn. Hence # E -pn and f=# E Qn. Therefore, Qn is closed in 9%.

Theorem 3.1. LetnE*Z,n>l.Then 1) Q n + 1 EQn 2) Every closed subspace 4 ' 2 {0} of Qn which is invariant under T", contains 1. 3) Let U;= T," IQn be the restriction of Tonto 8.. Then Unis an irreducible unitary representation of G. ProoJ 1) We have seen that 11 1II = 1< Q) ,n 2 1 (Proposition 3.3, 1)). (Remark: If n < l , then 11111'= a.) Hence 1 E $n and Qn # {O}. 2) LetfE 8.. We can write

-

242

UNITARY REPRESENTATION OF

SL(2,R)

in D. We have

Hence we have (3.9)

+-/re'ne(

If f belongs to

then #=-

uZef)(c)cio =a0 =f(o). eine(U&f)d8 also belongs to 4'. In

's"

2K

0

tnek

8k+l-@k , and 2n k since by the invariance of 8' we have U; f E 8' for each k, each such ek sum is also in 8'. Since 8' is a closed subspace of 8, we conclude that the limit of the sum is also in 8'. But $(C;)=f(O)for all f E D, as was shown in (3.9), hence q5 is a constant and # =f(O) E 8'. If 8' # {O], then there is an f E Q', f#O, and there is some point co E D such that f(Co) it 0. Since G acts transitively on D (Proposition 3.1), there is a g E G such that g-l.O=c0. Hence we have fact, # is the limit of the Riemann sumz e

(Uiekf)

( U , " f )(0)=a-znf(g-1.0)=a-2nf(To) #O, where g-l=

CBa :I*

By the invariance of 8' we have U ; f E Q ' ; by what was said above, ( U ; f ) (0) E 8'; and since (U;f)(0) # 0, we conclude that 1 E 8'. 3) Let 8'2 {0} be a closed invariant subspace of 8%.If @'#$jn7 then the orthogonal complement 8" of 8' is not {O} and is a closed invariant subspace of 8".Hence, by 2), 1 E 8' n @'l= {0], a contradiction. q.e.d. 1 Let n E 32, n2 1 and let Qn (resp. 8-n)be the subspace of g,,of holomorphic (resp. antiholomorphic)functions in 9" Let. u be the map from p,, onto itself given by u :~ ITheJmap u is an isometric antilinear Define the operator T;" on 9% by mapping and we have o(Qn)=&. Tpn=u T ; u and the operator U;" on 8 - n by U - ; = u U;(I=T-;18-,,. 1

Theorem 3.2. Let n E -2, n 2 1. Then .Q-" is a closed subspace of sn, 2 which is invariant by T-". U-"is an irreducible unitary representation of G on Q-,,.

PRINCIPAL DISCRETE SERIES

243

Proof. Via the isometry u : Qn-&,, the first assertion follows from Proposition 3.4, 4), and the second assertion follows from Theorem 3.1,

3). 1 2, Definition.1. The set of irreducible unitary representations { UnI n E 7 In1 2 1) is called the discrete series of G.

Then {O; I p E N } (resp. {@ 1 p E N})is a complete orthonormal system n in 8 n (resp. 8 - n ) >

k

2)

i.

(ue)O;

U&@;=X-n-p

( n 2 1, p

CJi:e = X n + p

€

N)

(~g)?

Proof. Let n 2 1. We have

Letting r s = t this becomes (2n- 1) ,&,,q/l(l

-t)2n-s fPdi

0

=6,,,(2n-

1) B(2n- 1, p + 1)

Hence {O; I p E N } is an orthonormal system. The system is complete. In fact, if f ( C )

amCm E Qn is orthogonal to Opn for all p E N , then

= m=O

am=O for

all m E N , and hence f=O. Replacing C P by Q', we can use the same argument to show that p E N } is a complete orthonormal system in &n. 2) We have (Uugn O p n )(c) =e-"e Opn(e-(@C) = e-t(n+p)@ Opn(C) =X-n-p(ue)

(U,,g-n@

@pn(C)

{el

9

(c) =U(e+@ OPn(e-*@C)) =er(n+p)@ (c) =Xn+p(ue)

@(C)

-

1 1) If n, m E -2 and In[,Imlr 1 and n#m, then Unand 2 Urnare not equivalent. Theorem 3.3.

244

UNITARY REPRESENTATION OF SL(2,R)

2) Un is not equivalent to V'.'. 3) YO.' is not equivalent to V1IrJ. Proof.

1

1) Let I E 72, 12 1. Then the sets of weight functions on K

associated to the representations U', U-'of G are respectively A'= { z - i , A-'= { X I ,

x-i-1,

X-1-a,

~ i + i X, I + S ,

...I , ...I .

1 if n, m a - 2 and In!, lml2 1, then U n z U" implies An=A"; but the 2 description of A', A-' above shows that if An =A", then n =m. 2) From Proposition 2.2,4), we know that the set Aj*' of weight functions on K associated to the representation Vf.' of G is

A'*'= { X p + j I p a 21 . Since Aj-*# A', A-l, we conclude that Un is not equivalent to Vj.*. 3) Finally, Ao.*= { x P I ~ 2 1 E is not equal to A 1 f a - c ={ X p + l l l I P E 21, so Yo*'is not equivalent to V1la.t. q.e.d. The representation Tn can be regarded as an induced representation. We I

shall explain this fact. For n E -2,n2 1, 2 pn (ue)=efne gives an irreducible unitary representation of K. By the isomorphism of

,,7-

K onto U ( l ) = { a ~ Clal=l}, l defined by us= Eel8 ef81a, we identify K with U(1). Then pn is the unitary representation of U(1) defined by pn(u)=aXn . pn can be extended to a holomorphic (rational) representation the complexification U(l)c= C* = { z E C I z#O] by putting pnC(z) =ZZ"

pnC

of

.

Let V" be the unitary representation of G induced by the representation pn of K. The representation space g,, of ' Vn is the set of all functions f satisfying the following three conditions. 1) f : G+C is measurable. 2) f ( g k ) = p 4 k ) f ( g ) , for all g E G, k E K. 2n- 1 3 ) 1 1 f 1 1 2 = y / G , KIf(g>l' dR@)< Q),where dR(C)=(l -Kll)-adm(O is the

245

PRINCIPAL DISCRETE SERIES

invariant measure on G / K = D . Then g',, is a Hilbert space with norm given by 3). The representation Vn is given by (VL7o"f 1 ( g ) =f(go-lg),

fE P ' n

Propsition 3.6. The representations V n and Tn are equivalent: Vn E Tn. Proof. 1) Let A :pn-+g',, be the operator defined by (SO)

(A$) (g)=pnc(J(g, O))-l)#

Note that since J(g, C)= jC+ a, for g =

($ E -Yn)*

[- i], we have J ( g , O ) = n E

C*. We want to show that A# =f thus defined is indeed in PI,. Of the three conditions for elements of 1) is clearly satisfied f o r 5 To check 2) we calculate as follows:

f k k ) = p n c ( J ( g k , O))-l $(gkO)

(by Proposition 3.2) = PnC((J(g, WJ(kO))-') #(go) = P*(W PnC(Jk,0))-' #(go) = P.(k)f(t?)

and find that 2) is satisfied. For 3) we calculate again: 2n-1

l l f l l % = ~ [

Ipnc(Jk, fflK

o>-l)l~ I#W)I' dA(g) (gO=E=C E D )

=lI$lP< *. So condition 3) is verified. Furthermore, the above shows that A : g,-. 9'is . an isometry. We shall now show that A g , , = g t n . Letfs g',, and let

fi(d=pnc(J(g, 0 ) ) f k ) . Thenf l is constant on each coset of G modulo K. In fact, f l k k ) = pnC(J(gk, O)lf(gk)= pnc(J(g, kO))pnC(J(k,O))pn(klf(g). Since M) =0, and since J(ue,0) = e-tbla gives pn( J(k, 0)) = p,,(k)-l, we get fi(gk)=fi(g). Hence fi defines a function # on G / K z D by #(go)=

246

UNITARY REPRESENTATION OF s u 2 , R )

Propsition 3.7. The representation U n(I n I 2 1) defines a representation of the Lorentz group G+(2) if and only if n E Z . Proof. U"defines a representation of G+(2) if and only if Un(-1) = 1. We have - 1 = u2* and for n 2 1 Uu2=QPn = e - i ( n + p ) 2 n U&nq

@ n

P

= e-2nnI

@ n

P

= &("+P)Zn p - = elnut P

@:

Hence U n ( - 1) = 1 e n n ~ Z (for In1 2 1).

111. THE LIMITOF DISCRETE SERIES As we saw in 52, the representation V3v3 is the direct sum of two irreducible representations. In this subsection we shall construct these two irreducible representations on the Hardy space H 2 (0). For each holomorphic function f on the unit disc Definition 2. D and real numbers p > 0 and r (0 S r < l ) , put (3.10) Let Hp = H p (0)be the set of all holomorphic functions f: D fying (3.1 1)

MP

-

C satis-

( r , f > IM

for some constant M and for all r E [0, 1) = { r E RIO 5 r < 1 }. Hp is called a Hardy space. We need only H2. H 2 is a Hilbert space as is shown in the following proposition. Proposition 3.8. The following two conditions (a) and (b) for a holomorphic function m

(3.12)

f ( z )=

1anzn n-0

247

LIMIT OF DISCRETE SERIES OD

are mutually equivalent: (a) f = H 2 , (b)

I a, I < + 00. H 2 is a Hilbert

n-0

space with the norm (3.13) n-0

forfas given in (3.12). Let B be the mapping which maps the functionf OD

in (3.12) to the “boundary function”

4 (ele)= 1andhe.Then B is an n-0

isometry from H 2on to the closed subspace $0 of the Hilbert space $ = L2 (U, dp) spanned by {f-p (&) = dPeIp 2 0). Proof. The orthogonality of the family {elnejneNin $ leads to the equality

n-0

Hence M 2(r,f) is a monotone increasing function of r. We have DD

(3.15)

lim M~(r,f12 = rtl

1 Ia, I ’. n-0

Therefore (a) implies (b). We shall prove the converse. Let A be the mapping which sends the function f in (3.12) to the sequence (a,). Then A is a mapping from H 2 into f2 (N). If we define the norm in HZby (3.13), then A is a norm preserving mapping. cu

Conversely let (a,) be a sequence in f2 (N)and put f(z) =

1a j n . n-0

Then, by Schwarz inequality, we have

Thus the power series UJ” converges on D and the convergence is uniform on any disc lzl S p with p < 1. Hencef(z) is holomorphic on D

248

UNITARY REPRESENTATION OF

sL(2,R)

and belongs to H 2 by (3.15). Therefore (b) implies (a). We have proved that (a) is equivalent to (b). Moreover A is an isometric mapping from H 2 onto the Hilbert space l2 (N). So H 2 itself is a Hilbert space with the norm (3.13). Let C be the mapping which maps (a,) in 1’ (N)to

- a, f-. in $

=

“-0

L2 (U)where I.(&B) = e-me. Then C is an isometry from l2 (N) onto the closed subspace &j0-= 0 Cf-.. Hence the mapping B = C 0 A is an isonZN

metry from H 2onto $;. q.e.d. .and @. for each Previously we have defined two Hilbert space 9 1 1 - n 2, ~ n 2 1. But 9, and 8. are defined for each real number n > 2 2. and 4, with a real parameter n > 1 Propositions 3.3 and 3.4 hold for 9 and no alteration of the earlier proof is needed. The following Proposition shows that the Hilbert space HZis the “limit” of the space $r as t tends to 1 2‘ 1y Proposition 3.9. For each real number t > T let $r be the space of all holomorphic functionsf o n D which belong to the Hilbert space Yr= L2 (D, n - l ( 2 t - 1) (1 - IzI 2 ) 2 r - 1 dm(z)). Then $r is a Hilbert space and contains H2. Moreover we have

llfll zr?g llfllr

(3.16)

forf€H2,

llJlll is the norm in H 2 consists of all elements f in fl 8, for t>l/Z which the limit lim l l f l l r exists and is finite. rll/2 Proof. Let PP( z ) = (r(2t + p ) / T (2t) T ( p + l))l?zP forp E Nand t >

where

1 Then { PPI p E N }is a complete orthonormal system in $r (Proposition -

2‘ 3.5.1)). We remark that

(3.17)

r(P

+ 1)r(2t) -

+ p)

P 2t+p-l

-

P-1 2t+p-2

1 . . . -< 2r

Hence a holomorphic function f (z) = c a n t ” in $r satisfies n-0

1.

249

LIMIT OF DISCRETJ2 SERIES 0)

1

I

luR12=

~lfllz

for all t > 1/2.

R-0

Therefore we have

H Zc at

for all t

> 1/2.

Moreover, iffbelongs to HZ, then the series in the right side of (3.18) is convergent. Hence the series in the left side converges uniformly on the range t 2 1/2. Conversely let f b e a function in n $t for which lim l l f l l r exists and is t>1/2

r11/2

finite. Then the inequality

implies

llfllf 2

(3.19)

-

1

1u,12rzp

= M~( r , n z

for r = ( 2 t ) - l / z .

R-0

llfllt exists and is finite thenfbelongs to a2.We have proved that H 2 consists of all elementsfin r,l/f% for which lim Ilfllt exists and r1112 Hence if lim

r1112

is finite.

q.e.d.

Delinition 3. Let C = e'e be a fixed point on the unit circle U and a be a non negative real number less than 7r/2 (0 5 a < 4 2 ) . Then the angular domain A , (0= {re', I 1 arg (1 I < a} is the domain indicated in Fig. 5. Note that /3 = arg (1 - re'(#-e))=

Fig. 5

250

arg

UNITARY REPRESENTATION OF

SL(2,R)

(F)L

Ocz, where z = rer+'. If a function f(z) has a finite

=

limit c when z approaches to a point c on U along a curve in an angular domain A, (0,then c is called the non-tangential limit off at c. Proposition 3.10. 1 ) Let f be a function in H2and 4 = Bf be the ''boundary function" off(cf. Proposition 3.8.). Then f is represented as the Cauchy integral of 4:

(3.20) 2) f is also represented as the Poisson integral of

for any z

= reie E

4:

D ; and

3) f has a non-tangential limit 4 (&r) at almost every point ( = erron the unit circle U. m

C OD

4 (0= anpfor c = e". Since n-0 n-0 4 belongs to L2(U)(Proposition 3.9), 4 belongs to L1(U).For any fixed Proof. 1) Let f ( z ) =

ad". Then

m

Z E D ,(( - z)-l

= zr"C-'"+"

converges uniformly on U.Hence we have

n-0

The integral in the right side of the above equality is the inner product of 4 and f-,, (e'? = elnrin L2( U ) . Hence the expression on the right side of the last equality is equal to the following:

= n-0

wherefn (0= C-" = e-lnr 2) Similarly the series

u#Jn

=f (z),

LIMIT OF DISCRETE SERIES

25 1

converges uniformly on U for a fixed Z E D . We have

Substract (3.22) from (3.20) we get

for z = refe, C = e" and d( = icdt. 3) Sincefis represented by the Poisson integral (3.21),f(rde) has a nontangential limit 4 ( & I ) for almost every t by a well known theorem of Fatou on the Poisson integral. See Appendix G for the proof of Fatou's theorem. q.e.d. Proposition 3.11. 1) Let j = -T, SEC and f E $ = L2 (U).Put

for g-'

=

6:)

E G = SU(1,I).

Then VJo"is a bounded representaion

of G. In particular, if Re s = 1/2, then Vj.' is a unitary representation of G. 2) Put (3.23)

(m(0= Cf(0

Then T is a unitary operator on (3.24)

8.

f o r f a T satisfies

T OV;' = V-l;. o T

for all gEG.

Hence we have (3.24)'

V-1IZ.r - VlIZ,.

for all SEC.

Proof: 1) The proof of 1) is same as that of Proposition 2.2. 2) Since I 61 = 1, T is a unitary operator on $ = L2 (U).We have the equalities

252

UNITARY REF'RESENTATION OF SL(2,R)

= [(V:/z,' o 2")f ] We have proved (3.24) and (3.24)'.

for all g E G .

(0

Remark. The explicit expression of the operator V;1/2,112 is as follows:

P I W ~ W ~3.~ 12.~ O 1) ILet I f~ H Z ,g-' =

:)

E G = SU(1,l) and put

(3.27) Then U:I2f belongs to H 2 and U1t zis a unitary representation of G. 2) Let B be the mapping which sends f E H 2 to its "boundary function" 4 = Bf (cf. Proposition 3.8). Then we have (3.28)

B o U:I2 = V;1/21/2 o B

for a l l g ~ G .

Put S = T 0 B where T is defined by (3.23). Then we have (3.29)

S

o

UiIz = V:/2.1/z 0 S

for all g E G .

The closed subspace 8; of 8 = L2(V)spanned by { f-,(c) = P ' ( p 2 0} is invariant under V-1/2~1/2. The closed subspace 8- spanned by If-,! p > 0 ) is invariant under V1/z*1/2.We have (3.30)

u1/2

v-1/2.1/2

Proof. 1) Put w = g-l

18;;

y1/2.1/2

.z. Then we have

1 - Iw12 = I/Jz

h ( w )=

1/92

I @-.

+ aI-2 (1 - lzl') + h(z)

and

4 - 4

by Proposition 3.2. Since g E G induces a complex analytic automorphism of the unit disc D, U:/2f is holomorphic on D. Moreover U:/2f belongs to n ,fjt along withf. So we have r>ln

LIMIT OF DISCREIE SERIES

(3.31) lim IIU:/2ffl = lim I 11/2

Ill/2

= lim

111/2

Ilfll:

=

253

n

Ilfl12 < +

03.

When t decreases monotonicaly, then 11 U:12fllIis monotone increasing. Hence lirn 11 U:/2fll,exists and is finite. Therefore U:l2f belongs to H2 by I

I1/2

Proposition 3.9. Moreover we have (3.32)

IIU:/2fII 5 llfll for allfEH2 andgEG.

It is easily verified that (3.33)

U:/2 Uij2 = U:i2 for all g and h in G.

(cf. Proposition 3.3). By (3.32) and (3.33) we have and

llfll = lu:!3 u:/2fll5 IIU:/2fll 5 llfll IIU:/2fll = l l f l l for allf€H2 and gEG.

Hence U:I2 is a unitary operator on H2.The continuity of the mapping U:/2f is proved similarly as in Proposition 3.3. So U1I2is a unitary representation of G. 2) Since the linear fractional transformation

g-

(3.34) induced by g-' EG is a conformal mapping which transforms a circle to a circle and leaves the circle U invariant, the transformation (3.34) maps a diameter of U passing e'e to a circle orthogonal to U. Hence the limit

is a non-tangential limit. So, by Proposition 3.10,3), the non-tangential limit lirnf(g-' rt 1

exists. We have,

re'e) = (Bf)(g-l

do)

254

UNITARY REPRESENTATION OF

SL(2,R)

[(B o U:l2)fl ( 8 6 ) = lim (U:I2f) (refe) r11

= [(V;1/2~1/zo B ) f ]

(89.

(3.27) is proved. Put S = To B. Then S is an isometry from H Z into

8 = Lz (U). By Proposition 3.8, we have (3.35) BH2 = 8; and SH2 = @-. Hence U'I2

V--1/2.1/21@0 -2: V 1 / 2 * 1 / 2 ~ ~ -

by (3.28) and (3.29). Proposition 3.13.

q.e.d. 1) Let u be the complex conjugation on functions

(uf)(z)=fin

Put uH2 = H2 and Q U : / ~Q = U;1/2. Then U-'I2 is a unitary representation of G on F . 2) Let 8' be the closed subspace of 8 = Lz (U) spanned by { f. (89 - .-id? In h 0). Then the mapping B = uBu is an isometry from HZ onto 8'. We have

B o U;1/2 = V:/2,1/2 o B for all g E G

(3.36) (3.37)

u-1'2

v1/2,1/2

and

18'.

Proof. 1) If we introduce an inner product in H2 by (a-ag) = (f,g) for any f and g in H 2 , then ZP is a Hilbert space, and U;'l2 is a unitary operator on F , and U - 1 / 2is a unitary representation of G on H2. 2) First we have the equality (3.38)

for all gEG,

=

O V ; ~ / Z * ~ / ~ UV1/z,1/2

because

(0= (st + a1-1 f(g-1 0 = (/?t+ 6)186 + zI -"(g-'

(u~;1/2.1/2~f)

r ) = ( v y 2 f ) (0.

Multiplying the equality (3.28) by u on both the left and the right, we get (3.36). Since of-,,=I. we have

B (H2) = uBH' Thus B is an isometry from (3.36).

= 08; = @

Cf-. = 8'.

.SO

P onto 8'. Hence (3.37) follows from

255

COMPLEMENTARY SERIES

Remark. H 2 consists of all antiholomorphic functionsf on D for which

M z(r,f) is bounded on the range 0 S r < 1. The operator U;'Iz is expressed as follows :

Theorem 3.3. U1l2 and U - l l 2 are irreducible unitary representations of G on H 2 and F respectively. They are equivalent to two irreducible components of the unique reducible representation V1/2*1/z in the principal continuous series: u1/2E V1/2.1/21$- and lJ-112 E Vl/z.l/z 1 8 ' 9 (3.40) where

6-= Q Cfp, @+ = @ Cf,and& (0=

C-P

= e-rpe.

PSO

P
Proof. In Propositions 3.12 and 3.13, we have proved (3.40). In Theorem 2.1 we have proved that V1/z*l/zl$-and V1/z.1/21$+are irreducible. Definition 4. The set of irreducible unitary representations { UIIz, U - 1 / 2 )is called the limit of the discrete series of G. $4. Irreducible unitary representations

IV.

COMPLEMENTARY SERIES

In 52 we have constructed the representation Vj.*(j=O, 112; s E C ) of G on a= L1(U,dc), where dC=de/2x and

The operator V / * is a unitary operator with respect to the inner product ($9

if and only if s=-

1

@)=I U

$(C)lO(r)dC

+

iv (v E R) (Proposition 2.2). However, if j = O , then 2 for some values of s there exists a positive definite Hermitian form

which is invariant under VUo.*for all g E G. We can prove that if we impose the conditions ($, @)* is Hermitian, ($, 9). is positive definite, ($, (l)*is invariant under V,,O.*for all g e G,

UNITARY REPRESENTATION OF

256

SL (2,R)

then we get 1 &(c, 7)=C(1-Re(~q))'-1 a n d p < l

(cf. V. Bargmann, [l] p. 617). In what follows we start from the kernel O,(C, 0)=(1 Re(Cg))"-' and show that the Hermitian form (4.1) is defined 1 on 8 x 8 and positive definite if and only if 2 <s< 1.

-

Proof.

(Set

2.

e-O)

1 1

2-1

=-B

+=-

(a-y,

x

z)

Remark. It is easily seen, by considering the expression (*), that I 1 diverges for u 5-. 2

Proposition 4.2. 1 Let u > and

and put

257

COMPLEMENTARY SERIES

(4.2)

and by Schwarz's inequality this is S~11#11 lI+lf

0

(1-cosaY-l da

(by Proposition 4.1) = llill 11+11<

1

Q)

if o>--.2

We shall now show that ($, +), is positive definite for 1/2<0<1. (($, +)$is not positive definite if 0 21.) LEMMA 1. Let Tn(x) be the Tschebyscheflpolynomial of the first kind, i.e.,

T, ( x )=cos(n arccos x), T, (cos 8)=cos n8 . Then we have

T ~ +( ~I ) = 2 x T n ( x ) - T n - l (XI, To (x)= 1, Ti (x)=x,

1)

2)

where (2n- l)! != Proof.

{

*

"'

(2n - 1)

+

1) Since cos (n+ l)@cos (n- l)@ = 2 cose cos ne, we get

258

UNITARY REPRESENTATION OF

SL (2,R)

Tn+l(x)+Tn-1 (x)=2xTn( x ) for all X E R . By the definition of Tn,we have To(x)= 1, Tl(x)=x. 2) Let Dn(x) denote the right member of the equality in 2) of the proposition. Then we have Do(x)=1, D1 ( x ) = x , and Dn(x)is a polynomial in x of degree n. If we can show that it satisfies

(*I

Dn+l (x)=2xDn (x)-Dn-l

(XI,

then, since Tn(x)and Dn(x) both satisfy the same recursion formula and To(x)= Do(x), Tl(x)=Dl(x), we can conclude that Tn(x)= Dn(x). To show (*) we calculate as follows: dn-1 x-dn (1 -X~)n-l~~=zn{X(l dn -Xa)n-lly -n-&l - x a)n-iia

dx"

dn-1

=-,-,[{-nx"(n-

dn-1 dn-1 ( - 1 ) (2n-l)-(l-x~)"-a'a=-&3{(-2n+l) dxn-'

l)}(l-xay-y,

(l-x2)n-a/a}

.

These relations show that

+(-

1 ) (2n- 1)-

dn-1

(1-xa)n-aI'

from which (*) follows immediately.

Proposition 4.3. 1)

I Let fp (C)=C-p(p E 2).Then we have, if a>-, 2 (fn,fm)e=

an,,

An(o)=A-n(o), n

&&(a)= 17 L-I

Proof. We have

where Ro(o)=l

Rn(o),

k-o if n > 1. k+o-l

259

COMPLEMENTARY SERIES

=&so I0 .ar

(fn,

fm).

ar

(1 -cos (e - ~

ded@

p - e-*n@+tm* 1

Let

Since cos(-n$)=cos n# we get R n ( ~ ) = ~ - n ( ~ ) and RO(a)=1 is clear from the definition of C, (cf. the proof of Proposition 4.1). If n 2 1, let x =cos#. Then we have

(by Lemma 1)

(Integration by parts n times)

Letting t=-

1-x we see that the latter equals 2

1 1 The definite integral of the last expression is equal to B(a---, n+-). 2 2 Using the value of C, given in Proposition 4.2, we now have Rn(a)=

(- l)"&r(a) 2.+"-l(u - 1) ... (a -n) r ( u - 4) r ( n 2-1r ( a - 4 ) ?r(2n-l)!! r(a+n)

+4)

We have proved 1). The statement in 2) now follows from the formula for ]"(a) given in 1). 1, then Rn(u)$O for at least one n. Hence (#, +)# is Remark. If not positive definite if a h 1.

260

UNITARY REPRESENTATION OF

SL (2,R)

Proposition 4.4. If ) < a < 1, then (#, 4). defined by (4.2) is a positive definite Hermitian form on 8 =Lq(U,dc). Moreover, we have m

(4.3)

C

($3

n:--m

where fn(ete) =cine. Proof. Since Ifn)

nrZ

An(O)

I ($,fn)I2

9

is a complete orthonormal set in 8, we have m

m

Since (#, 9). is continuous on 8 x 8 (Proposition 4.2), we have

m. n---

(Proposition 4.3)

Putting

+ =# we get m

($9

$)n=

C An(o)I(+,fn)12rOn=-m

If (4, #).=O, then (9, fn)=O for all n~ 2, so #=O. Remark. The formula (4.3) is valid for a 2 1. If 0 21, ($, +)#is a Hermitian form on .@ x .@ which is invariant under Yo,.and is not positive definite. In particular, if a is equal to a positive integer I, An(u)=O if and only if In1 &Z. So the kernel N = @ E .@I($, $)0=01 has codimension 21+ 1. In this case, the representation Yoszinduces a finite-dimensional representation on the factor space Q/N. (cf. Theorem 7.5 and Remark after it.) Proposition 4.5.

Let l+i-E -i- E 2 2 :i l-izE

261

COMPLEMENTARY SERIES

be the Iwasawa decomposition of gu,. Then, for g=ut, we have et
-

-cht + shtcos e ,

cos (ar e)=e-tcat")(sht+cht.coso)

,

sin (at .e) =e-t(at*')sine . Proof. By Proposition 1.7,we have for g =at = et(at, r )

-

b3

-laC+BI t

+sh-1'2t

= lch-ete

2 t

=cha-

2

+ s h t' +~ 2ch-2t sh-2t cos e

=cht+shtcos 0.

By Proposition 1.7 we have for g =at -e-t(at,d) -

+ +

(cht cos8 sht i sine) ,

from which follow the second and third equalities in the proposition. Proposition 4.6.

1

Let 7 < O X 1 and let .$&be the completion of 8 with

can be extended uniquely to respect to ($, +)#.Then, for any g E G, VU0p8 a unitary operator Vu@on 8.. The map P : g w V u eis a unitary representation of G on 4.. 1

Dewtion. The set of representations { PIz
(V:lu

C, V,".'+).=(C,

for any $, (o E $3 and g E G.

(o),

Since we have g=u, at u, ($2 Lemma 1) and Vuuro,"= Vuo,~Vuto,e (Proposition 2.2), it suffices to prove (4.4) for g=u, E K and g=ut E A. From the definition it is clear that t(u., 8) =0. Hence we have (

~

(c>=e-""uu-l'

~ u C~ ) 0

,

C)

=$(et
C(u.-'C>

262

UNITARY REPRESENTATION OF

SL (2,R)

- e - ~ t ( a L - l 8), $h[aL-le1. and furthermore (4.6)

-

1 cos(aLe-aL@) = 1-cos ale cos uto -sin ate sin aLw

(Using Proposition 4.5) =e

- ( t ( a L , e)+r(a,,

0))

(1-cos(e-o))

holds. Hence, by (4.5), we have ( Va:.'$h, Vato."+>o

From Proposition 1.5, I), we have t(at-l, at8)+t(at,8)=t(l, O)=O, and from Proposition 1.5, 3), we have d(ut8)/d8= e-L'at'e' (and similarly for 0). Using these relations and (4.6), the above expression becomes

So (4.4) is proved. V ~ may ~ ' now be extended uniquely to an isometry V / of the completion QUy of $ with respect to ( , onto itself. Moreover, since Vvo'"J7v-10.'=Vv-10.'VgO.@= 1, )uy

we get Vv~Vv-l'=Vg-l'Vv'=l

.

Hence, Vvuis a unitary operator on 8.. Since ygo*y,. 0 . = ygg, 0.0 0

3 .

,

263

COMPLEMENTARY SERIES

We can show, as before, that Yo,.is strongly continuous (cf. proofs of Propositions 2.2 and 3.3), and since Q is dense in Qo,' V is also strongly continuous. Hence V.is a unitary representation of G.

Remark. The Hermitian form (#, 9). is invariant under V0*#even if 1, as the above proof shows. We shall now show the irreducibility of V . We always assume & < u < 1.

u>

Theorem 4.1. For any u satisfying Q
11#11.411#11 for all + € @ . Hence if # E Q is a C"-vector for Yo,. then # is also a C"-vector for P and dVu(X)# = dVo.q ( X ) # for all X E g . In particular the functions f p ( C ) = C - p Hence by Proposition 2.6,4) we get

belong to

Qme

for all p E 2.

EfP =(P+ U ) f P + l

for all p E 2,

(4.7) FfP =(P - U l f p - 1

+

+

where E= d P ( X l ) idP( Y) and F= -dV.(X,) idP( Y). Now we prove that a vector f in !& satisfies (4.8) V,,f= Xp(ua)f for all 0 E R and some p E 2-'2 if and only if f = c f p where c is a constant. First, a function satisfying the latter condition satisfies (4.8). Conversely, we assume that f E Q . satisfies (4.8). f can be expanded in terms of the orthonormal basis (& = i n (O)-tfn)nrZ say, (4.9)

f=

C

Cn$n nrz

.

Then operating with V U b e on both sides of (4.9), we get (4.10)

For each n + p , there exists a real number 0 such that X ~ ( U ~ ) # X , , ( U ~ ) . Hence (4.9) and (4.10) show that cn=O if n # p and thus f = ~ ~ ~ ~ = c f , .

264

UNITARY REPRESENTATION OF

SL (2,R)

After these preparation, we prove the irreducibility of V.Let Q' be a invariant under P,and W be the unitary non-zero closed subspace of representation of the compact group K = {ue[OE R} defined by Wu,= Vue[Q'. Then W is decomposed into a direct sum of irreducible representations x,,'s. In particular @' contains a vector f#O satisfying (4.8). Then f is proportional to f p as was proved above. Hence f p belongs to Q'. Then Enf, and Fnfp belong to 8' for all n E N. Since p is an integer and 4
and

Vueufn=Xn(Ue)fn

VUIIy= @ xn nrZ

.

On the other hand, Proposition 2.2.4) shows that

v~I~@ . ~Xn+lll I Y.= nrZ

Hence, P and VIJ'.* are inequivalent. We also have from Proposition 3.5 that, for n 2 1, u"IK= @ X-n-p, U-"IK= @ Xn+p P ~ N

P ~ N

Hence, P is not equivalent to Un(n E 2/2,lnl> 1). (cf. Theorems 6.4 and 6.5). Remark. V' is not equivalent to Vo*++*u Proposition 4.8.

P(3 < u c 1) defines a representation of G+(2). proof. This is clear from the fact that VU2; = 1.

Proposition 4.9. If O= 1, then the Hermitian form (#, +)1 is positive semi-definite. The completion of 8 with respect to (4, +)1 is isomorphic to @//8(0),where @(O)= {#I(#. #)l=O). Moreover dim&= 1. The representation V1 induced by VOJ on Q1 is the (trivial) identity representation I of G. Proof. By Proposition 4.3, Ro( 1) = 1 and In 1) =0 ( for n # 0. Hence we have

-

($9

+)I=

C

( # , f n ) W = ( # , f ~ )

n--O

from which the first statement follows. Since

(+,fa>,

K-FINITE VECTORS

265

(VgO*'$,V17°*14)1=($,$91, + E Q, g e G, holds (cf. proof of Proposition 4.6), Q(0) is invariant under VgO*l,g E G. Let $o =f& Q(0)E Q/Q(O). We have $9

Vvl$o = (Vv0*'h,h)$o. Hence, in order to prove that V1 is the identity representation of G, it suffices to prove that ( V g ' J ~ l ~ , f 0 )for = l ,allgEG.

By $2, Lemma 1, any g e G can be written in the form g=u,atuo, and since V,,O*lfo=fo ,

we have

(v,O*lfo,fo)=(vat OJvqO.'foo,

vu+-1O J f O ) = ( v a , O * ' f ~ o , f O ) (Proposition 4.5)

$5. &finite vectors The present section is in preparation for the classification of irreducible unitary representations. In this section we shall prove that each irreducible unitary representation U of SL(2, R) contains each irreducible unitary representation xm(mE 2-lZ) of its maximal compact subgroup K at most once. Then the representation space 8 of U is decomposed as the direct sum of one-dimensionalsubspaces am(mE M) :a=@ am,where UIQm= mriY x,,,. The elements of the algebraic sum Q K = C Qm of the subspaces Qm m t I

are called K-finite vectors for U.It will be shown that QK is contained in the space 8.. of C"-vectors and is invariant under the differential representation U' of U.The restriction U,(x> of U ' ( X ) to Q K for X E 11(2, R) defines a representation UK of the Lie algebra 11(2, R) on The classification of irreducible unitary representations U of SL(2, R) can be reduced to the classification of Ug.This process is carried out in the next section.

ax.

266

UNITARY REPRESENTATION OF

SL (2,R)

First we calculate the Haar measure on G=SU(l, 1).

Proposition 5.1. The left-invariant Haar integration on G=SU(l, 1) (or Go= SL(2, R)) is given by

for any continuous functionf with compact support, where g =uoatne is the Iwasawa decomposition of g (cf. Proposition 1.3). Proof. Put dg =e’dedtdt. We shall prove that d(gog)=dg for any goE G. By Lemma 1 in $2 (or the following Proposition 5.2), every element go in G can be written as go=u,a,u,. Hence it is sufficient to prove d(g0g) =dg for go= u, and ar. u,ueatnt = u,+oat nt, we get d(u,g) =etd(p e)dtdc=etdedtdE=dg. By a simple calculation we get

Since

+

atneat-’=nett. Put t’=t(a,, 0 ) and €’=€(a,, 0 ) in the notation of Proposition 1.5. Then we have

a,ueatnc=u,, .eat+zclatnt - ua,.eatlatat-lnetatne = U,,.&#+tne-tt.+t

Hence we have by Proposition 2.1, 3), d(a,g)=et+t’d(a,-B)d(t+t’)d(E+e-tE’) -et+t‘e-t’dedtdE=dg.

q.e.d.

In the following, the Haar measure dg of SU(1, 1) (or of SL(2, R))is fixed by the normalization given by (5.1). Proposition 5.2. Any element g in G=SU(l, 1) (or in Go=SL(2, R)) can be written as (5.2)

g=u,atu#, O s p < 4 n 7 Ost, 05$<2n.

If

g=(i

[) E G,

then

a =e*(,++)/2 ch(t/2), B= e*(p-+)/S sh(t/2). (5.3) (5.4) ch(t/2) = lal, sh(t/2)= IBI, et/’= la1 I& In particular if g belongs to G -K,then

+

K-FINITE VECTORS

267

(5.5)

and (9, t, 9) is uniquely determined by g. If g E K, then t=O and p + + is uniquely determined modulo 4r by g. Proof. By the multiplication of matrices, we get ei(r++)lach(t/2), eg(9-+)/2 sh(t/2) u,acu+= (ei(-p++)lash(t/2), e-i(p++)la ch(t/2)).

Hence we have (5.3). There exists a unique tz0 such that sh(t/2)=(,9(. Then ch(t/2)=(1 +181a)t=lal and et/a=lal+lpI. We get (5.4). IfgEG-K,thenp#O. Since lala=l+IBI'r-l,aisalways #O. Wecan find p and 9 satisfying o s p <4r, -2r <$<2T, and the first two equations in (5.5). If -2n <sl, <0, then we can replace u+ by #++a. = - u+. Hence we may assume that 0 5 $ <2n. Assume that an element g in G -K can be written as g = u 9 a t u + = u , ~ a t ~ u,+ ~ where p, t, 9, p' t', $' satisfy the inequalities in (5.2). Then we have t = t' because sh(t/2) = 181= sh(t'/2). By (5.5), we get eK9++)/2 =e i ( p ' + + ' ) I a and e<(q-g)/a =e<(p'-@')15 . (5.6) Dividing the first equality by the second bne, we get e*+=et+'.Since 0 5 9, 9' <2r, we get $ = 9'. Moreover, by (5.6) we have ei91a=etQ'laand p/2 = y'/2 and p = p'. If g = U e E K, then ue=u,++. In this case t -0 and the sum p ++ is uniquely determined modulo 4 r by g. q.e.d. The decomposition (5.2) is called the Cartan decomposition. Proposition 5.3. The Haar integration (5.1) is given by the formula

(5.7)

1

f(g)dg=2~/=/;

/

X

f(katk')shtdkdtdk' ,

where dk is the normalized Haar measure (44-lde of K = {ue1050<4n). Proof. Let g=u,atu+=uea,nt, where o j y , e < 4 r , o j t < + c 0 , 0 j + < 2 ~ , --<s,E<+co. Then (t, $) and (s,E ) can be regarded as two systems of coordinates on the right coset space K\G= { g =Kglg E G) . The relation between (t, 9)

268

UNITARY REPRESENTATION OF

SL (2,R)

and (s, E ) is given as follows: Put

Then we have a =e*+lp ch(t/2) and

p =e-l+rn sh(t/2).

By Propositions 1.3 and 1.6, the parameters s and t in (5.8) are given by e*=la+BIa=cht+shtcos qlt, Im{(a+I) ( a - I ) } = -e-# shtsinqlt . Now we calculate the Jacobian E=e-*

The second factor in the right hand side of the above equality is equal to e-as, because it is the inverse of

The first factor is given by

Hence we get D = -e-# sht and e8dsA dE = sht dqlt A dt. Transforming the Haar integral (5.1) by means of (5.9), we get (5.9)

=IK/:1; =;IK/.1:'

f(kue-'atu+)shtdkdtd+

f(katu,)shtdkdtdqlt

=2r.l K . l o

Jf

(katk')shtdkdtdk'

K

where we use the relation atuI =~ ~ , a ~ u + + ~ . . COROLLARY to Proposition 5.3. The left invariant Haar measure dg

269

K-FINITE VECTORS

given by (5.1) is also right invariant. Hence G=SU(l, 1) is unimodular. Proof. Using the identity u.a-tu.-l=ar and (5.7) we get

/

f(g-l)dg =2 x 1 ff

K r

=2xJ

/+-IK

f(k'-la-tk-l)shtdkdtdk'

O

roo

K J 0

r

r

J

f (k'ark)shtdk'dtdk = K

Hence we get

where f(g) =f(g-l). Definition 1. The vector space of all complex-valued continuous functions with compact supports on a topological space X is denoted by L(X). For each n E 2-'2, the character X n of K = {uelOS8 <4n} is defined by Xn(ud)=einS . (5.10) The subspace of L(G) consisting of all functionsf satisfying

(5.1 1) f(uogu+)=xn(u~lf(g)Xn(u,) for any 8, (O E R and g E G=SU(l, 1) is denoted by A,. Proposition 5.4.

(5.12) and

1) A function f in A, satisfies ~ ( w L ~ s= ) xn(up+(a)f(at>

(5.13) f(at)=f(a-t). 2) The mapping f++fIA of restriction to the subgroup A = {atIt E R) is a linear bijection of A, onto L.(A)= { fL(A)lf(at)=f(a-,)}. ~ The m a p ping f E A , H F E L([l, m)) defined by f(at)=F(cht) is a linear bijection of A, onto L([l, a)). Proof. 1) (5.12) is trivial. Since u.atu.-l=a-t, we get (5.13) from (5.12). 2) By (5.12) and (5.13), the mappingfHflA is a linear injection of A, into L,(A). Let 9 be an element in L,(A). Then we can define a function fonGby

-

f ( u v a 4 = Xn(Up+,) ( O ( 4 The Cartan decomposition g=uputz+ is not unique when g belongs to K. But in this case 9++ is unique modulo 4~ and the function is well

270

UNITARY REPRFSENTATION OF

SL (2,R)

defined on G. It is easily seen that f belongs to A,. Hence the mapping fwfIA is a bijection of A,, onto L,(A). Since the function cht maps the

interval [0,

+ a) homeomorphically onto [l,+ a), the second part of

2) is easily obtained from the first part. Definition 2. In the following, we denote the quantity F(x) appearing in Proposition 5.4,2) by f [ x ] :f(ut)=f[cht]. Proposition 5.5. The set A, is a commutative algebra over C for every n E 2-lZ. The product is defined by the convolution on G. Proof. Let fl and fi be two elements in A,. Sincefi and fa belong to L(G),the convolutionfl*h is defined. Moreover, we see that

n

=d

andfi*h E A,,. To provefi*h=$&,

Up>

(fi*f) (h)~n(u+)

it suffices to show that

(h*f) ( u t ) = ( f * f ) (at)

(5.14)

for all c E R

.

Put (5.15) Then we have

+

a = e * p l a ch(r/2)ch(s/2) e-*sla sh(t/2)sh(s/2),

p =e*pla ch(r/2)sh(s/2)+e-<,lash(t/2)ch(s/2).

By Proposition 5.2, we have

(5.16) and (5.17)

cht'=21ala- 1=chtchs+shtshscos(p e*b'+#')lZ=

dial

+ sh(t/2)sh(s/2) +chrchs+shtshscos(p)lla

=J Z e * ~ 1 ch(r/2)ch(s/2) 2

(1

e-tp'a

27 1

K-FINITE VECTORS

X

+

(ch(t/2)ch(s/2) e-'p sh(t/2)sh(s/2))'" shsdpds. [2-1(1 +chtchs+shtshscosp))

Similarly we have (5.19)

(fi*h) (at)=

X

+

(ch(s/2)ch(t/2) e-'+ sh(s/2)sh(t/2))'^ shsdsd+ {2-'(1+ chscht+ shssht C O S ~ ) )

.

Cornpairing (5.18) with (5.19), we get (5.14) q.e.d. Proposition 5.6. A function f in L(G')belongs to A , if and only iff= where convolution is taken over the group K :

x,,*f*x,,,

Proof. I f f belongs to A,, then f=x,,*f*x,, by (5.20). Conversely if a functionf in L(G)satisfiesf= x.*f*x,,, then n

n

LEMMA1. Let A be a *-subalgebra of the algebra B ( H ) of bounded linear operators on a Hilbert space H, A s and A w be the strong and weak closures of A and R(A) be the smallest von Neumann algebra containing A . Then we have A'= Aw and R(A)=A" = C1+ As.

212

UNITARY REPRESENTATION OF

SL (2,R)

Proof. Since the strong closure of a convex subset of B ( H ) coincides with the weak closure (cf. Dunford-Schwartz [l] p. 477), we have A * = A". Since A""=A" by Ch. I, (2.9), A" is a von Neumann algebra (loc. cit.). From A c R ( A ) , we get A"cR(A) hence A"=R(A). Let D=C1+ A'=Cl + A w . Then D is a weakly closed *-subalgebra of B ( H ) containing 1. Hence D is a von Neumann algebra. Since A" 3 A , we have R(A)= A" =A"* 3 C1 + A * = D 3 A . Hence by definition of R(A), we get R(A)= D. (cf. Dixmier [l], p. 44) q.e.d.

Proposition 5.7. Let U be a unitary representation of a locally compact group G and A be a bounded linear operator on H = H ( U ) . Then A commutes with U, for any g E G if and only if A commutes with U f for every f E L(G). Proof. If AU,= U,A for any g E G, then for any f E L(G), we have AUf =

S.

f ( g ) A U , d g = / f(g)U,Adg= U f A . D

Let V be an element in the set !Jl of all compact neighborhoods of the identity e in G. Then there exists a function f v in L(G) such that fvLO and

1

fv(x)dx= 1 and supp (f v ) c V. For any g in G, put gv(h)=fv(g-l

h). Then s-lim U,,= U,.Hence if A U f = U f A for every YEL(G), then AU, = U,A for every g E G. Proposition 5.8. Let T be a strongly continuous representation of a locally compact unimodular group G by bounded operators on a Banach space H. Such a representation is called a Banach representation. Let k be the set of equivalence classes of (finite-dimensional) irreducible representations of a compact subgroup K of G. For any 2 E k,put &)= TrU2(k)(U2E I), d(I)=deg U2and E(I)=d(I)

Tkmdk.

S K

Then E(1) is the projection on the subspace H(I) of H consisting of all vectors x E H such that the set {T& E K } spans a finite-dimensional subspace H , on which T induces a direct sum of a certain number of copies of U2. Proof. Since x ~ * K=~d(;C)-lx2,E(I) satisfies =E(I). Hence E(I) is a projection onto the subspace E(I)H. If x belongs to H(;C),then there exists

..., V, such that x E 1Vc (direct sum) and to the class 2 E k.The vector x can be written

subspaces Vl,

6-1

TklVt belongs

273

K-FINITE VECTORS

where (tp,(k)) is a unitary matrix. Hence we have P

d(1)

by the orthogonality relations (Ch. I, Theorem 3.2). So we have proved that x E E(2)H and H(R)c E(R)H. Conversely if x belongs to E(R)H, then x = E(R)x satisfies ‘ d(i)

Tkox=

1 t0~(kO)~P,0 3

9.0-1

T m x d k . Hence x belongs to H(1) and E(A)H

where x,.,=d(R)/ K

c H(R).

q.e.d.

Proposition 5.9. The notations are same as in Proposition 5.8. For any 2,let

2E

L(R)= If€ L(G)ld(R)’XI *f* 2 2 =fI .

J.

1) Then H(R) is invariant under T , =

f(g)Tgdgfor anyf E L(R).

2) For any f E L(R),put T,A = T,IH(,I).Then T, =O if and only if T f 2=O. Proof. 1) As is easily seen, E(R)T,=~(R)TF~,,,;, =d(R)T,,;A= T,E(R) forfe L(r7). Hence T,H(R)= T,E(R)H= E(R)T,Hc E(R)H= H(R). 2) Since E(R)T,E(R)= T,, if T I A=0, then T / H = E(R)T,E(R)H=E(A)Tf’H(R)=O q.e.d. and T,=O. Conversely if T,=O then T,l=T,IH(R)=O. Definition 3. If dimH(R)=pd(R), then we shall say that R is contained T.p is called the multiplicity of a in T.

p times in

Theorem 5.1. Let U be an irreducible unitary representation of GSU(1,l) on a Hilbert space H. Then each irreducible unitary representation

274

UNITARY REPRESENTATION OF Xn(us) =e(nu

SL (2,R)

(n E 2-’Z)

of K is contained at most once in U. Proof. Put En=E(Xn)=S K x,OU&. Then the conclusion of the theorem is expressed by dim E , H $ l . The algebra A-n coincides with L(Xn)= {fsL ( G ) l f , * f * ~=f} , . Hence E,H=H(n) is invariant under A n = [UfIH(n) ‘IfEA-n}. Put L = { U , l f s L(G)}. Then by Lemma 1, we have (5.21) Cl+L’=L”.

Put M = { U J g E G}. Then by Proposition 5.7 and Schur’s Lemma (Ch. I, Theorem 2.1), we have L’ = M f= C1. Hence (5.22) L“= M“ =B(H). (5.21) and (5.22) imply c 1 L*= B(H). (5.23) For any f~ A,,,let V(f) be the restriction of U, to the subspace E,H. Every element A in B(EnH) can be extended to a bounded linear operator A. on H by (5.24) A,x= AE,x for x E H.

+

If x belongs to E,H, then by (5.23) there exist nets (or filters) (jJreI and ( u ~ )in ~ L(G) . ~ and C respectively such that Aox=lim (url+ U,,)x. (5.25) I

Put g,= x,, *f,* 2.. Then g, belongs to A-, by Proposition 5.6. By (5.24) and (5.25), we have lim (a,l+ V(gr))x=lim (udl+ Ui,*~,*:)~=lim (uJ +EnUf,E,)x I

=

lim Em(utl + U~,)x=E,Aox=E,AEnx=E,Ax= Ax. I

Hence we have proved that B(EnH)= C1+ A,” = An” (5.26) by Lemma 1. Since the algebra A-, is commutative, so is the algebra A,, Hence we have ,Anc A,’ and A n ’ 2 Am” (5.27)

.

By (5.26) and (5.27), we have

275

K-FINITE VECTORS B(EnH)c A,"C An'cB(EnH).

Hence An"=&' and A,"'",''. The last identity implies that the algebra A," = B(E,H) is commutative. Since B(H) is not commutative q.e.d. if dimHz 2, we can conclude that dimE,HS 1. Definition 4. Let g(G)be the set of C"-functions with compact supports on a Lie group G and let U be a Banach representation of G on H. The subspace Ho of H consisting of all the finite linear combinations of the vectors of the type

-1.

U a-

'

f(g)U,adg

for f E g ( G )

and aE H

is called the Gdrding subspace (for U). Proposition 5.10. The GBrding subspace Ho is contained in the space H , of C"-vectors. Both Hoand H, are dense in H . Proof. Any element X in the Lie algebra g of G is regarded as a left invariant differential operator by the formula

To the element X in g, there corresponds a right invariant differential operator X' defined by

Let f E d ( G ) , X E g and a E H and put f-(g)= U,Ufa and xt=exptX. Then by the left invariance of the Haar measure dg, we have

Wf %xt) -f -(g)) =

1

I3

t-'(f (x-tg-'go) -f (g-'go)) U d g a

+ULsx,fa as t+O. We have proved that f-(g exptX) is differentiable at t = O for any g E G . Since x ~ + ~ = xf-(g ~ x exptX) ~, is a differentiable function of t. And we have

We can repeat the differentiation and prove that for any ,'A ..., X , E g, XI ... X,f- is defined as a continuous H-valued function on G . Hence f - is an H-valued C--function on G . (cf. Ch. 11, $6, Lemma 1. This lemma is stated for a complex-valued function, but is valid for a Banach-valued function without changing the proof.) It remains to prove that Ho is

276

UNITARY REPRESENTATION OF

SL (2,R)

dense in H. Since U is strongly continuous, for any a E H and E >0, there exists a compact neighborhood V of the identity e in G such that IIU&--all<e for a n y g e V. r Let f g o be a C"-function on G satisfying supp(f) c Vand f(g)dg= 1.

Iff

Then we have

IIU1a-alld

L

f(g)ll~&-allds9

Hence we have proved that Ho is dense in H. Since H, =I Ho, H, is dense in H. In the course of the proof, we have proved the following corollary. COROLLARY to Proposition 5.10. For any a E H and E>O, there exists a C"-function f E g ( G ) such that IIU,a-all<&.

Proposition 5.11. Let K be a connected compact Lie group and 2 be the set of equivalence classes of irreducible unitary representation of K. For any Banach representation U of K on H and R E 2,put (5.28)

E ( I )=d(R)/ K

u*imdk,

where x 2 is the character of 1 and d(J)=xl(e)is the degree of R . Then for any C"-vector v E H for U,the series

converges absolutely to v. Proof. Since U is strongly continuous, the set { Ukxlk E K} is compact and hence bounded for each x E H. So by the uniform boundedness theorem (cf. Appendix A), the function p(k) = I I Ukllis bounded on K. Let C be an upper bound of p ( K ) . Then (5.29)

IIE(mlld Cd(R)llXll/ K IX.l(k)l& 5 cdQ)llxll.

Let (Xr)lrldnbe an orthonormal basis of the Lie algebra t of K with respect to an inner product (X,Y) on t invariant under Ad K, and put

K-FINITE VECTORS

277

Then D is the Casimir element of U(1). If v is a C"-vector for U,then for each R E R we have E(R)O'(X)v=/

K &(k)U&'(X)V

(na(k eXp(-tX))-Rz(k))Ukv dk

=Em t-' (-0

=

-1

dk

J K

(Xxl) ( k )Ukvdk

K

.

where X E is~ interpreted as a left invariant a differential operator of order 1 in the last term of the above sequence of equalities. Since the matricial elements of 1 are eigen-functions of the Casimir operator 9 with eigenvalues o ( A ) = ( A , A+26) (Ch. II, Proposition 8.1), the last equality proves that

(5.30)

E(I)U'(L?)V=J K

(Gl)(k)ukV dk

= o(,I)E(R)v

for any C"-vector v and A E k (5.29) and (5.30)imply that lIE(lz)~lI=o(R)-nlIE(OU'(Ja~.)vlI 5 Cd(R) o(R)-"llU'(Jam)~ll. Since d(2) is a polynomial function of the highest weight R (cf. Ch. 11, d ( ~ ) o ( R ) converges -~ if m is sufficiently large. Hence we have EjS), 1+0

E(R)v converges absolutely (normally).

proved that 1.R

Now we prove that its sum is equal to v. Let H* be the dual space of H and p E H*. Then

vl(k)= < U ~ Vp ,>

is a C"-function on K for any v E H,. And we have

278 put

UNITARY REPRESENTATION OF

C E(I)V=

W.

SL (2,a)

Then

2.2

w,(k)=C< UxE(R)v,Q>

=

c

4 2 ) (v, * %l)(k)

l e t

1

is the Fourier series of a complex-valued C"-function v, (Ch. I, Proposition 3.4). Hence its sum w, is equal to v, for any (O E H*. In particular, we have < E ( I ) V (p, = w,(e) = v,(e)= < Y , (0 >

C

=-

1

for any

cp E H*

and

q.e.d.

E(R)v=v. ki?

Remark 1. Put t(k)= Unv.Then we have (E(;Ov)-(k)=d(I)(t*x,)(k). Hence Proposition 5.11 is a generalization to a vector valued C"-function of Ch. 11, Theorem 8.1 (cf. also Ch. I, Proposition 3.4). COROLLARY to Proposition 5.1 1. The result of Proposition 5.1 I still holds when U is a strongly continuous representation of G on a Frkhet space. Proof: Let (P,,),,.~ be a countable family of seminorms defining the topology of H. Then, after replacing the norm with the seminormsp,, the above proof of Proposition 5.11 is valid when H is a Frdchet space.

Proposition 5.12. Let U be a Banach representation of a Lie group G on H, let K be a connected compact Lie subgroup of G and let E(I) be the projection defined by (5.28). Then the space (H, n E(I)H) is dense in H . 1.R Proof. We prove that for any a E H and E > 0, there exist a finite subset F of K and an element alE E(I)H n H, for each I E F such that I a, -a1 I

Ic

.arF

We can assume a#O. By Corollary to Proposition 5.10, there exists a C"-function f E g ( G ) such that <E.

(5.31)

I1 U,a-all< 4 2 .

Let C be a compact subset of G such that C = CK and supp(f) c C. For x,. Put M=supllU,all. Then we any finite subset F of 2, put

xF=C ICF

oco

have O < M < + co . Any function f E s ( G ) is a C"-vector for the right regular representation of G on L1(G). This fact can be proved in a manner similar to the proof of Proposition 2.5. Proposition 5.11 proves that there exists a finite subset F of R such that

K-FINITE VECTORS (5.32) 1lf-f. (5.31) and (5.32) show that

XPI 11

279

<€/2M.

IIU,*ZPa- 4 I 5 I I U,*,p - Ural I + IIU,a-all 5 M IIf*X P -flIl + 1 Iufa-al I <

9

because supp(f * 2). c CK= C. Since Uf*Z,a belongs to the Garding subspace Ho of H, Proposition 5.10 shows that Uf*lpa belongs to (Ha n E(R)H).Hence we have proved the proposition. q.e.d.

1

,
Definition 5. The notations are the same as above. If K is a connected compact subgroup of G, an element in the subspace H K = cE(I)H is 2rr2 called a K-finite vector. Proposition 5.13.

1) Let H(R)=E(R)H and H,(R) =Ha n H(R). Then

H&) is dense in H(;O.

2) If dimH(R)< + m for every R E I?, then H,(R)= H(R) for all R E I? and is contained in H,. H K is invariant under U‘. Proof. 1) Let R and be two elements in I?. Then

H K

E(R)E(~) = K

1

~~~~X~(k)X,(k~)dkdk~

K

by the orthogonality relations of characters (Ch. 11, Proposition 2.2). then E(R)b E H,(R). Let a be an element in H(R). Hence if b E

c

Pfb

Then there exists a sequence ( U ~ ) , , ~ Nin

1H,(R)

such that liman=a by

led

Proposition 5.12. Then we have limE(R)a, =E(R)a=a. Since E(R)a, E n-== H,(R), we have proved that H,Q) is dense in H(2). 2) If dimH(2)< + w , then the dense subspace H,(R)=H, n H(R) coincides with H(R). Hence we have Ha 2H(R). Therefore if dimH(R) < + 00 for every I E I?, we see that H, 2 HK = H(2). Let x be a K-finite vector

c

ieI?

and X be an element of g. Then U’(X)x is K-finite, because U,U’(X)x= U’(AdkX)Ukxand g is finite-dimensional. q.e.d. Definition 6. Suppose that Hg c Ha and let UK(X) be the restriction of U ’ ( X ) to the subspace HK(XE 9). Then UK is a representation of the Lie algebra g on H K . UK may be uniquely extended to a representation

280

UNITARY REPRESENTATION OF

SL (2,R)

of the universal enveloping algebra b of the complexification gc of g. The extended representation of b on HK is also denoted by UK.UK is called the K-jinite diflerential representation of U.

Theorem 5.2. If G = SL(2, R)and K = S0(2), then for every irreducible unitary representation U of G, the space H, of C"-vectors contains the space HKof K-finite vectors. Proof. By Theorem 5.1, dimH(n)5 1 for every n E 2-'2 =15. Hence the assumptions of Proposition 5.13. 2) are satisfied by G=SY2, R) and q.e.d. K= SO(2). Remark 2. As a generalization of Theorem 5.2, the following general theorem holds: Let G be a connected linear semisimple Lie group and K be a maximal compact subgroup of G. Then every irreducible representation 1 E I? of K is contained at most d(2) times in every completely irreducible representation of G (cf. Godement [4]).

$6. Classification of irreducible unitary representations In this section we prove that any irreducible unitary representation

U of Go=SL(2, R) and G=SU(l, 1) is equivalent to one of the representations constructed in 92, 93 and & Our I. method of classification is based on the classiiication of the representations UK of the Lie algebra go= BK(2, R) on the spaces of K-finite vectors. The differential representation U' of a unitary representation U is obtained by differentiating the one parameter group UeIptxof unitary operators. First we give preliminary results on the self-adjoint operators which appear as the infinitesimal generators of one parameter groups of unitary operators. Then we shall show that the classification of unitary representations U of G can be reduced to that of the representations Ug of go. Since the Lie algebra go=81(2, R) has very simple commutation relations (Lie products among the elements of a basis of go), the classification of the representations Ug can be accomplished easily. Comparing these results with the differential representations of the unitary representations constructed in the preceding sections, we get the theorem stated at the beginning of this section.

Definition 1. Let A be a linear operator in a Hilbert space H withdense domain D(A). Let D(A*) be the set of all elements y in H for which there exists an element y* E H such that

CLASSIFICATION OF IRREDUCIBLEUNITARY REPRESENTATIONS

28 1

(Ax,y ) =(x, y*) for every x E D(A).

Then, since D(A) is dense in H, the element y* is uniquely defined by y. Put y* =A*y. Then A* is a linear transformation from D(A*) into H and is called the adjoint of A. Let A and B be two linear operators in H. Then B is called an extension of A, and this relationship is denoted by Ac B, if D ( A ) c D ( B )and Bx= Ax for any x E D(A). A linear operator H is called symmetric if H c H*. A linear operator H is called self-adjoint if H = H*. Propition 6.1. Let %=%(R) be the o-algebra of all Bore1 sets in R and E be a spectral measure on % in a Hilbert space 8. (cf. Difinition 3 in Ch. 111, 93) Put D(H)= {x E QlJ- PdllE(A)Xll’<

(6.1)

-m

+

00

1

Then there exists a unique linear transformation H from D ( H ) into 8 which satisfies (6.2)

( H X ,y)=lend(E(l)x, Y

)

for any x E D ( H ) and y E 8. Moreover the domain D ( H ) is dense in 8 and the operator H i s self-adjoint. ?‘roo$ The domain D ( H ) is dense in 8. In fact, for any x E H and E>O, there exist Q and b in R(a
bl>xlI <E (Ch. 111, Proposition 3.9,7)). And E([a,b])x belongs to D(H), because

IIx-

+

AldllE(1)E([a,b ] ) x l ~ z = j;c~ll~(A)xll’< ~ co a

-0

By Ch. 111, Proposition 3.12, a linear transformation H from D(H) into Since X =A,

8 is defined by (6.2).

(Hx, y ) =(x, Hy) for any x, y E D(H). Hence D(H)c D(H*) and H c H*, thus H is a symmetric operator. Now we prove that H I H * . Let y be an element in D(H*). Then for any a,b E R (a
E([Q,b])H*Y)=(HE([Q,b])z,V )

282

UNITARY REPRESENTATION OF

SL (2,R)

is a bounded linear form on 8. Hence the strong limit F=lim Fu.a: Z H U--*

/I-

rJ++-

~ d ( E ( l ) zy ,) is also a bounded linear form on 8. Therefore

(6.3) 1 ~ ~ z ) l S l l FIIzII<+CO ll for any Z E 8 . In particular, let x ~ be ~the . defining ~ ~ function of the interval [a, b]. Then (6.4)

z= T(Xr.,*&=r ldE(lllY

satisfies E([a,b])z=z and (6.5)

=r

F(z)=/ : i d ( E ( l ) z , y ) =

s:

Ad(E(l)z,y )

k q z , E(l)y)=IIzlla.

a

By (6.3) and (6.5), the element z in (6.4) satisfies llzlls IlFll. Hence

/:

l2~llE(Alyll2= IlzllasIIFIP

+ m , we get ladllE(l)yllasIIFIISC+ CO

for any a, b E R(a< b). Letting a+ - co and b-.

s_

and y E D(H). Since y is an arbitrary element of D(H*), we have proved that D(H*)c D(H). Since

v)=/I;d(E(l)z, v)

(2,H*y) = (Hz,

=S_-d(z,E(Rly)=(z, H r ) for any Z E D ( H )and y e D ( H * ) , we have proved that H*y=Hy and H*cH. q.e.d.

Definition 2. The self-adjoint operator H in Proposition 6.1 is denoted by (6.6)

H=

/--

ldE(R),

and is said to be associated with the spectral measure E.

Remark. Conversely, any self-adjoint operator H can be written as in

CLASSIFICATION OF IRREDUCIBLE UNITARY REPRESENTATIONS

283

(6.6). This fact is the well known spectral representation theorem of J. von Neumann.

Proposition 6.2. Let U be a unitary representation of the additive group R of real numbers on a Hilbert space 8. Then by Stone's theorem (Ch. 111, Theorem 3.2), there exists a spectral measure E on B=B(R) in 8 such that (6.7)

Let H=

1:-

ndE(;c)be the self-adjoint operator associated with E. Then

the following three conditions l), 2), and 3) for an element x E: H are mutually equivalent: 1 ) x E D(H). 2) lim t-l(UC- l ) x exists. c-0

3) The function tHUtx is differentiable everywhere on R . If x satisfies these conditions, then Ucxbelongs to D(H) and satisfies d -( Ucx)= iHUcx= iUcHx. (6.8) dt Proof.

1)*2).

First we observe that lei2c-ll$Inl It1 for any 2 , t E R .

(The length of a chord of a circle is shorter than the corresponding arc.) If x belongs to D(H), then I[(t-l(Ut- 1) - iH)xlls = /:mlt-1(e6tr - 1) - i2I2 d(lE(2)xlla

tends to 0 when t-0, because the function under the integral sign tends to 0 and is smaller than the integrable function 22' when It1 5 1. We have proved that 1 ) implies 2) and (6.8) for t=O. 2) => 1). Conversely, if x E H satisfies the condition 2), then lim Ilt-l(Uc- l)xlla=lim t-0

t-0

1-

It-1(et2c- l)lzdllE(R)xlla

-m

exists. Since the function under the integral sign has the limit P,it follows from Fatou's lemma that Ra is integrable with respect to ~ ~ ~ E ( ~ ) x ~ hence that x belongs to D(H). 1)53). If x satisfies the condition l), then Ucxis a differentiablefunction oft, because d Ucx)= lim s-l( Uc+,- Uc)x= Uc lim s-l( U,- l)x

x(

8-0

.

8-0

284

UNITARY REPRESENTATION OF

SL (2,R)

exists for any t E R. Since 2) is equivalent to 1) and 1) implies (6.8) for t=O, the right hand side is equal to iUtHx. 3)*2). This implication is trivial. Since UtE(A)=E(A)Ut for any A E 23,we have

S;n'dlls(n)u:xIl.

=

/I,

ladl I UtE(2)xlI S

=/~>2dIIE(4xI12.

Hence if x belongs to D(H), then U t x belongs also to D(H) (cf. (6.1)).

Definition 3. The self-adjoint operator H is called the infinitesimal generator of U. Proposition 6.3. Let U t =

R , T=/I:dE(I)

/I.

e"'dE(1) be a unitary representation ot

and xo be any element of D ( T ) . Then the function

x ( t )= Utxois the unique solution of the differential equation

2

---iTx

(6.9)

which satisfies the initial condition (6.10) x(0) =xo. Proof. It has been proved in Proposition 6.2 that x ( t ) = Utxosatisfies (6.9). (6.10) is trivially satisfied because x ( t )= Utxo.Let x l ( t ) be a solution of (6.9) satisfying (6.10). Then

=O

for any t E R .

q.e.d.

Proposition 6.4. Let U and V be two unitary representations of a connected Lie group G on the Hilbert spaces H and L respectively, and let U' and V' be the differential representations of U and V on the spaces H , and L, of C"-vectors for U and V respectively. Then U is equivalent to V if and only if there exists an isometry A of H onto L such that AH- = L, and AU'(X)x= V'(X)Ax (6.1 1)

CLASSIFICATION OF IRREDUCIBLEUNITARY REPRESENTATIONS

285

for any x E H, and X E g (the Lie algebra of G). ProoJ If Uis equivalent to V, then there exists an isometry A of H onto L satisfying A U,= V,A for any g E G. If x belongs to H,, then Ax belongs to L, and satisfies (6.11), because gHV,Ax=AU,x is a C"-function and AU'(X)x=lim A.t-l(Ut- I)x=lim t-l(Vt- 1)Ax= V ' Q A x . t-0

t-0

Conversely assume that there exists an isometry A of H onto L satisfying AH,=L, and (6.11). Put W,=A-'V,A for any g e G. Then W is a unitary representation of G on the Hilbert space H. Because of the assump tion AH, =L,, H, is seen to be the space of C"-vectors for W.Moreover, W'(X)x=A-'V'(X)Ax= U'(X)x €or any x E H, and X E g. Let XE Q and xo E H, and put iT= W'(X) = U'(X). Then both x(t) = Uexptxxo and y ( t )= WexptxXo are solutions of the differential equations (6.9) satisfying the initial condition (6.10). Therefore, by Proposition 6.3, we have x(t)=y(t) for all t E R. We have proved that UexptXX=A-l Je'rptxAX for any X E 0, any t e R and any x E H,. Since G is a connected Lie group, any element g in G can be written in the form g=exp Xl ... exp X,,, for Xi E Q Hence we have proved that U,x =A-1 V,Ax for any g E G and x e H,. Since U, and A-*V,A are unitary operations on H and equal to each other on a dense subspace H,, U,=A-lV,A for any g E G. q.e.d. Our classification of irreducible unitary representations of SL(2, R) is based on the following proposition. Proposition 6.5. Let U and V be two irreducible unitary representations of G=SU(l,l) (or Go=SL(2,R)) on W and L respectively, and let Ug and VK be the K-finite differential representations of U and V (cf. $5 Definition 6). Then U is equivalent to V if and only if there exists an isometry A of H onto L such that AHK =LK and AUK(X)x= VK(X)Axfor any x E HK and X E g . (6.12) Proof: By Theorem 5.2, we have H, 2 HK and L, 2 Lg and the restrictions UK(X)and V K ( X ) of U'(X) and V ' ( X ) to HK and LK respectively are defined. If U is equivalent to V, then there exists an isometry A of H onto L

286

UNITARY REPRESENTATION OF

S L (2,R)

such that AU,= V,A for any g E G. Then AHK=LK, because HK consists of all x E H such that dim(UexlkE K} < CO. Since Ug(X) and V g ( X ) are the restrictions of U‘(X) and V’(X) to Hg and Lg,we have A uK(X)x= VK(X>Ax for any x E HK and X E g by Proposition 6.4. Conversely, assume that there exists an isometry A of H onto L satisfying AHg=& and (6.12). Then again by Proposition 6.3, we get

+

A UexpgxX= VexptxAX for any x E Hg, X E g and t E R. The last equality implies that AU,= U,A for any g E G in the same way as in the proof of Proposition 6.4, because Hg is dense in H. Hence U is equivalent to V. q.e.d.

Proposition 6.5 is valid for any connected semisimple Lie group (Note that Theorem 5.2 is valid for any such group). Moreover, the following proposition was proved by Harish-Chandra [3], I.

Proposition. Let U and V be two irreducible unitary representations of a connected semisimple Lie group G and K be a maximal compact subgroup of G. Then U is equivalent to V if and only if Ug is equivalent to VK* Proof. Since we do not use this proposition later, we give an outline of proof. Let H and L be the representation spaces of U and V respectively. The “only if” part of Theorem can be proved similarly as in the proof of Proposition 6.5. Suppose that there exists a linear isomorphism A of HK onto Lg such that AUg(X)v= Vg(X)Av for all v E HK and X E g. Then for each R E g, we have A H K ( I )=LK(I).Since the representations of K induced on HK(R) and Lg(R) are unitarily equivalent each other, there exists an isometry B, of Hg(R) onto Lg(R) such that B , U k X = vkB,x for every X E Hg(R) and k E K. Since H&) (resp. L&)) is orthogonal to H&) (resp. LK(p))if # p , and since Hg (resp. Lg) is dense in H (resp. L), there exists an isometry 3 of H onto L whose restriction to H&) is equal to B, for each E I?. Let S be a linear isomorphism of HK onto itself defined by Sx=B-IAx (XE Hg). Then S and S-I leave Hg(R) and &(A) invariant respectively. Since dimHg(R) c + co , there exists a linear transformation S* of Hg into itself satisfying (S*x,y)=(x,Sy) for all x and y in Hg.Furthermore S*-1 exists and S* and S*-’ leave Hg(R) invariant. Put T= S*S. Then we claim that TUg(X)T-lx= Ug(X)x for all X E g and x E HK (*) First we notice that if X E g and x , y E Hg, then we have ~

(uK(X)X,Y)= -(x,uK(X)y) .

CLASSIFICATION OF IRREDUCIBLE UNITARY REPRESENTATIONS

287

Similar equality holds for VK.SinceBSVK(X)S-' =A UK(X)A-'B= Vg(X)B, we have for all x , y E HK and X E g (TUg(X)T-'x, y ) = (S*SuK(X)S-'S*-'x, y) =(SUJ?(X)S-1S*-'x, Sy) =(BSUg(X)S-'S*-'x, BSy) = (VK(X)BS*-lx, Ay) = -(BS*-'x, Vg(X)Ay)= - (BS*-'x, A UK(X)Y) = - (S*-'x, B-'A UK(X)Y)= - (S*-'x, SU,(X)y) = -(x, UK(X)Y)= (UK(X>X,Y ) . We have proved (*). Now choose xo E H&o) such that xo# 0 and TXO= cxo for some C E C . Since cIIxolla=(S*Sxo,xo)=IISx~l12>0,c is positive. Let U be the universal enveloping algebra of 9". Then it can be proved that the closure FV of W = UK(lI)xois invariant under U, for any g E G (cf. Prop. 6.9). Hence, by irreducibility, we get P = H . Since dim HK(R) c + co ,we have wn HK(R)=H&) for any R E 2 and W =HK. For each x E HK, there exists an element b E U such that x = U K ( ~ ) X Hence O. we have Tx=TUK(b)xo=U ~ ( b ) T x ~ = by c x (*) and IIAxll'=IISxll'=(Tx, x ) = cllxllp for all x E HK. Hence A is bounded on HK and can be extended to a bounded operator A on H. A satisfies llAxlla=cllxlla for all x E H . Hence c-lIaA= R is an isometry from H onto L. Moreover R satisfies RUK(X)x= VK(X)Rxfor all X E HK and X E g. Now we can prove that U is equivalent to V in a similar way as in the proof of Proposition 6.5. q.e.d.

Definition 4. Let U be a Banach representation of a Lie group G on a Banach space H. Then a vector x in H is called an anaIytic vector for U if the function ,t : gt-+U,x is an analytic function on G. Since U is strongly continuous, a vector x is analytic for U if the function g w c U,x, 'p > is analytic for each 'p in H* (Ch. IV, $5, LEMMA 1). The set of analytic vectors for U is denoted by H,. H , is a linear subspace of H contained in the space H, of C"-vectors for U. Proposition 6.6. The space H , of analytic vectors is invariant under U and U'. Proof. Let go be an element of G and x be an analytic vector. Then since the right translation Roo:g-rggo is analytic, the mapping RgOoR: gwU,,,x is analytic. Hence U,,x belongs to H,. Since the adjoint representation Ad is analytic on G, the mapping gwU,U'(X)x= U'(AdgX) U,x is analytic and U'(X)x belongs to H . for any X E g. q.e.d. Definition 5. For any XE g, we denote the restriction of U ' ( X ) to H . by U-(X). Then U, is a representation of the Lie algebra g on H.. U. can be extended to a representation of the universal enveloping algebra of g. U. is called the analytic diferential representation of U.

288

UNITARY REPRESENTATION OF

SL (2,R)

LEMMA 1. Let D be an elliptic differential operator with analytic coefficients on a real analytic manifold M and g be an analytic function on M. Iff is a C"-solution of the differential equation ( D- a)f= g, (6.13) for a constant a,then f is analytic on M . Moreover iff is a C"-solution of (6.14) (D ='")a 0 for some n E N,then f is analytic on M. Proof. By a classical theorem of S. Bernstein, a C"-solution of (6.13) is analytic. (For the proof of Bernstein's theorem, see F. John [I] p. 144.) The analyticity of a C"-solution of (6.14) is proved by induction on n. If n = 0, then f = 0 is analytic. Assume that n 2 1 and any C"-solution of (6.14) for n - 1 (in place of n) is analytic. Put g = ( D - a)J Then the function g satisfies ( D - a)"-'g = 0. Hence g is analytic by assumption. The original function f satisfies (6.13) and is analytic on A4 by the first q.e.d. half of the lemma.

2. Let G be a Lie group, K be a compact subgroup of G, and LEMMA

U be the universal enveloping algebra of g regarded as the algebra of left invariant differential operators on G. Then there exists an element D of U such that D is an elliptic differential operator on G and (Adk)D=D for every k E K. Proof. Since K i s a compact subgroup of G, there exists an inner product (X, Y) on the Lie algebra g of G which is invariant under AdK. Let X I , ..., X , be an orthonormal basis of g with respect to this inner product and put D=Xi'+ ...+X n 2 . Then (Adk)D= D for every k E K and D is an elliptic differential operator on G.

Proposition 6.7. Let U be an irreducible Banach representation of a Lie group G on a Banach space H, and K be a compact subgroup of G. If there exists an element 20 in (the set of equivalence classes of irreducible unitary representations of K ) satisfying Ocdim H(R0)c+ 00, the space H. of analytic vectors for U is dense in H. Proof. Since H. is invariant under U (Proposition 6.6), the closure I?, is also invariant under U because of the boundedness.of U,.Since U is irreducible, it follows that if H, f {0), then p-=H. Hence it remains to prove that H . # (0}. By Proposition 5.13, we have H, n H(lo)=H(Ao) and H , ~ H ( R ~Let ) . D be a differential operator on G as in Lemma 2.

CLASSIFICATION OF IRREDUCIBLE UNITARY REPRESENTATIONS

289

Since U'(D) commutes with Ukfor any k in K, the subspace H ( h ) is invariant under U'(D). Since dimH(Ro)c+ 00, there exists a basis XI, ..., xn of H(l0) such that for each i there exists at E C and an integer n > O satisfying (U'(D)-at)"xt=O ( l $ i $ n ) (Jordan normal form theorem). For any element q in the dual space H* of H, put A(g) =

'

Then since xrE H(Ao)c H,, we have

for any X in the Lie algebra g of G and

[(D- at)$] ( g )= < Ug( U'(D)- a$%,

q > =O

.

Hence the functionfi is analytic by Lemma 1. So the vector xi is a weakly analytic vector and hence an analytic vector by Ch. IV, $5, Lemma 1. q.e.d. Remark. It can be proved that for any Banach representation U of a Lie group G, H. is dense in H (Nelson's theorem). Cf. Nelson [l], GArding [2] and Warner [l]. Proposition 6.8. (Taylor's formula). If v is an analytic vector for U,then there exists an open neighborhood N of 0 in the Lie algebra g of G such that (6.15)

for any X E N . Proof. If we regard an element X of g as a left-invariant vector field on G, then for any H-valued analytic function f on G, we have

for any m E N. Hence the Taylor expansion of the function ti+f(gexp t X ) around t = O is given by

Sincef is analytic at e (the identity element of G), there exists a star-shaped of g and open neighborhood N of 0 in g with respect to a basis (Xr)lstrn

290

UNITARY REPRESENTATION OF

SL (2,R)

a power series P(xl,...,xn) convergent on N such that n

f(exp

(1XZJ)

...

=P(x~, ,xn)

I=1

for each X = c xtX, in N. Then we have f(exp tx)=P(txl,..., txn)=

- 1 1 -amt m!

m-0

for any t E [0, I]. Since dm

am=[&(exP

tx)lt-0=(Xmf) (e),

we get (6.16)

f(exp X ) =

1- Z1 ( X (e) ~J for)any x

E N.

me0

Now let v be an analytic vector for U.Then applying the formula (6.16) to the function 5 :gHUgv, we get (6.15), because

q.e.d. Proposition 6.9. Let G be a connected Lie group, U, the universal enveloping algebra of the complexification Q“ of the Lie algebra g of G, and let U be a Banach representation of G on a Banach space H. 1) If x is an analytic vector for U,then the closure U,(U)x of the orbit of x is invariant under U. 2) If L is a subspace of H , invariant under U,,then the closure I? of L is invariant under U. ProoJ 1) Let be an element in the dual space H* of H satisfying

< U,(z)x, (D> =O for any z E U. Let a be any element in U,(U)x; then a belongs to H,(Proposition 6.6) and there exists a symmetric open neighborhood N of 0 in g such that

uer, a =

- 1 1 -U,(x)ma m!

m-0

for any XE N.

CLASSIFICATION OF IRREDUCIBLE UNITARY REPRESENTATIONS

29 1

for any XEN. We can assume that the exponential mapping is a homeomorphism of N onto an open set Win G by taking a smaller neighborhood if necessary. Since the real analytic function f(g) = ( Uga,p) on G is equal to 0 on a non-empty open set W, f is identically zero on the connected group G. Hence we have U,a E -for all g E G by the Hahn-Banach theorem. 2) Let x be an element of L and g be in G . To prove that UgxE t,it is sufficient to show that for any E > O there exists an element y in L satisfying IIU,x-yll<s. Put M=IIU,II. Since U, is invertible, we see that M>O. Since x belongs to E, there exists an element z in L such that -=4(2M).

IIX-ZII

By l), we have

-~

u,z E U,(U)ZC

U,(U)LC

L.

Hence there exists an element y in L such that

I I u,z -A I <€12. So we have proved that

II ~ & - Y l l I;IIUP-

UozlI + IIU,Z-YII

<€124- €12=E. q.e.d.

3. Let H be a Banach space,f be an H-valued analytic function LEMMA on a Lie group G , and h be a complex-valued continuous function on a compact subgroup K of G . Then the function p(x) =

1

f (xk)h(k)dk

K

is analytic on G . Proof. Let xo be an element of G and {xl,..., x,) be a system of local coordinates on an open neighborhood N of xo.Since the function ( x , k ) t+ f(xk) is analytic on G x K there exists a finite set (U,),,A of open coordinate neighborhoods in K whose union is equal to K and a set of power series (Pa),cAsuch that

f(Xk)=P,(xi,

..., xn, kl, ..., k p ),x E N , k E U.,

where {kl,..., k p ) is a system of local coordinates on U,.Let 1 =Eva ucA

be a continuous partition of unity subordinate to the covering (U,)r.d. Then

292

UNITARY REPRESENTATION OF

SL (2,R)

is analytic on G.

q.e.d.

Propition 6.10. Let U be a Banach representation of a Lie group G on a Banach space H and K be a compact subgroup of G. Assume that the space H , of analytic vectors for U is dense in H (cf. Proposition 6.7). Then the subspace Arf

is dense in H. Proof. Let x be an element in H and E >0. Then, since H, is dense in H, there exists an analytic vector y E H, satisfying

I Ix -YI I <42. Moreover since the analytic vector y for U is a C"-vector for the restriction UlK of U to K, y can be expanded in the form Y=

c

EOlY

As2

by Proposition 5.1 1. There exists a finite subset I:of R such that

Put z=E(R)y. Then the function Z : gHUgz is analytic on G because I is given by the formula

io= /K 9 k 4 X.l(k)dk (Lemma 3). Hence E(R)y belongs to H , n H(I) for any R E E. We have E ( R )in ~ (He nH(n)) such proved that there exists an element w =

1

A
that Ilx-wll<e.

1 A

q.e.d.

Definition 6. Let K be a compact subgroup of a topological group G. A Banach representation U of G on H is called K-finite if dimH(;O is finite for every R E k An irreducible unitary representation of SL(2, R) is S0(2)-finite by Theorem 5.1. Proposition 6.11. Let G be a Lie group, K a compact subgroup of G and U be a K-finite representation of G on a Banach space H. If the space

CLASSIFICATION OF IRREDUCIBLEUNITARY REPRESENTATIONS

H. of analytic vectors for U is dense in H, then the space HK=

293

C H(R) 162

of K-finite vectors is contained in H , and HK is invariant under U,. Pro05 Since C ( H . nH(R)) is dense in H (Proposition 6.10), we can 1

prove that Ha n H(R) is dense in H(I) for each R E I? in a fashion similar to the proof of Proposition 5.13,l). Since dimH(2) is finite, H , n H(R)= H(A) and H , 3 H(R) for each 17 E I?. We have proved that

H,DC H(R)=HK. A

Since H g is invariant under U ' ( X ) by Proposition 5.13, it is invariant q.e.d. under Ua(X)= U'(X)lHI.

Theorem 6.1. Let U be a unitary representation of G = SL(2, R ) and K be SO(2). Then the following two conditions for U are mutually equivalent. 1) U is irreducible. 2) UK is (algebraically) irreducible. Proof. 1) 3 2 ) . Let L # 0 be a subspace of HK invariant under UK.Then we have (6.17)

In order to prove (6.17), it is sufficient to show that if a=

C

aj (a, E H(n,))

j-1

belongs to L, then each ar belongs to L . Since

Uexp tx,,aj=e

ttnj

aj ,

we have Ua(H)aj=njaj,

where H = -iXo.Since nj#nr ( j z k ) , there exists a polynomial PkE C[z] such that Pk(nj)==jr (1 rj,k s n ) .

Hence we have a / = UK ( P j (H))aE L for all j . (6.17) is proved. Since U is an irreducible unitary representation, U is K-finite (Theorem 5.1), and H. is dense in H (Proposition 6.7), the space HK is containcd in H, by Proposition 6.11. Hence Proposition 6.9 can be applied to L and

294

UNITARY REPRESENTATION OF

SL (2,R)

the closure 1 of L is invariant under U. Since U is irreducible and L # 0, we have E= H. Now we prove that L=Hg. Assume that L#Hg. Then by (6.17) there exists an element Ro in I? such that L n H(Ao)# H(Ro). Put L ( A ) =L n H(A) for each 1 E R. Then there exists a non-zero element x in H(&) satisfying W O ) ,x ) =o.

(6.18)

If R # , ? ~ then , H(R) is orthogonal to H(Ro)and we have (L(R),x)=O.

(6.19)

The equations (6.17), (6.18) and (6.19) prove that (L, x)=O. Since E=H, the last equation implies x=O, a contradiction. We have proved L = HK and Ug is irreducible. 2)=>1). Let L#O be a closed, U-invariant subspace of H and put Lg= ( L n H(A)). Then Lg is invariant under UK, because Lg is the

c

1.f

space of K-finite vectors in L for the representation UJL. Since UK is irreducible, Lg is equal to {0} or HK. Since L = @ LQ) by Ch. I , Theorem 2

3.1, L K = c L(A)is dense in L # {O). Hence Lg# [O} must be equal to 1

Hg. Since L g = H g = c H(A) is dense in H = @ H(A), L is dense in H. 1

1

Since L is closed, we have L =H. The irreducibility of U is proved. q.e.d. By Proposition 6.5 and Theorem 6.1, the classification of irreducible unitary representations U of G=SU(l, 1) can be reduced to the classification of the representations UK of the Lie algebra g of G. Now we start a closer study of irreducible unitary representations U and their infinitesimal representations UK. Proposition 6.12. Let g = Sn(1, 1). Then g consists of all the matrices of the form

’ ) aER,bEC.

(fi b -aiy

(6.20) The three elements (6.21)

form a basis of g. The Lie products between them are given by (6.22)

[Xo, X1]= - Y, [Xo, Y ] = X , ,

[Xl, Yl=Xo.

CLASSIFICATIOrJ OF IRREDUCIBLE UNITARY REPRESENTATIONS

295

The element o0=2-l( - X o a + X l a + Y a ) is the Casimir element of the universal enveloping algebra of g. ProoJ Since the Lie group G=SU(I, 1) consists of all matrices g e GL(2, C) satisfying g*elg=el

and det g = l

(cf. Proposition 1.6), the Lie algebra g=lu(l, 1) consists of all matrices X E M a ( C )such that X*elfelX=O and Tr X=O (cf. Ch. 11, Proposition 5.6.). Since el=

(A

_“l), this condition implies

(6.20). Hence the three elements in (6.21) form a basis of g. (6.22) can be proved by direct calculation. From (6.22) we get the matrix expressions of ad Xo, ad Xl, and ad Y with respect to the basis {XO, Xl, Y}.We see that (6.23)

-B(Xo,Xo)=B(Xl, Xl) =B( Y, Y)=2

and

B(X1, Y )=B( Y, Xo) =B(X0, Xl) =0

Hence 2-l( -Xoa+ Xla + YS) is the Casimir element of enveloping algebra of g. Definition 8. In the following, we let o=Xo’-X12-

Y2.

Proposition 6.13. Let U be an irreducible unitary representation of G = SU(1, 1) on a Hilbert space Hand put O = U‘(o).Then Q is a scalar operator on H,, i.e., there exists a real number q such that sZx=qx

ProoJ

for any

XE

H,.

Since U is a unitary representation, we get

(6.24) (U’(X)u,v) = -(u, U’(X)v) for any u, Y E H, and XE g by differentiating the relation, (Uexptxu,

Uexptxv)=(u,

v)

at t = O . Hence we have (6.25)

(Ou, v) = (u, Ov) for any u, v E H,

,

Thus O c O* and O is symmetric. For any g E G, the automorphism Ad g of g can be extended to an automorphism of the universal enveloping algebra ll of g. The extended automorphism of ll is also denoted by Ad g. Then

296 (6.26)

OF

UNITARY REPRESENTATION

SL (2,R)

(Ad g)w=(gXog-')' -(gXig-')' -(gyg-')' =gag-'="

because [X, 0]=0 for any X E g (Ch. 11, Proposition 5.14). On the other hand, we have (6.27) WgW'(X)Wg-lV= U'(Ad gX)V for any XE g, g E G and v E H,, because g(exp tX)g-'=exp(t AdgX). The equalities (6.26) and (6.27) prove that for any g E G and Y E H,. Let D be the set of all elements x in H such that there exists a sequence ( x ~ ) in~ .H,~ converging to x and for which the limit limQx,=y U,QUa-lv=Sav

n*+m

exists. Then a closed linear operator 0 is defined on D by setting 0 x = y . 0 is the so-called closure of Q. For any element g in G, we have (6.28)

U,D=D and U,Ox=OUax for any X E D .

In fact, let x , ( x ~ and ) y~ have ~ the ~ same meaning as above. Then the following limits exist: lim W,x, = Waxand lim OU,x, =lim UaSaxn= Uay = n*-

n-m

n--

UaBx.Hence Waxbelongs to D and QUax= U,Bx. Let W be the Cayley transform of the closed symmetric operator Q. W is defined as follows. The domain D( W ) of W is given by D(W)= (z=(B+i)xIxE D}

and W is defined by

+

-

W(O i)x= (0 i)x . Since (Ox, ix) = - (ix, a x ) by (6.25), we get (6.29) Il(a+i)xll2= I l D X l l ~ + IlxlJa=II(B-i)xll" Hence W is an isometry of D( W ) into H. By (6.28), we have Waz= Ua(O+i)x =(a+i ) Uax

for any z = (0+i)x E D( W ) , hence UaD(W )=D(W ) for any g E G . Moreover, we have (6.30) WUaz=W(B+i)U,x=(D- i)Uax = Ua(a-i)x= WaWz, for any z E D( W). If ( x ~ ) is~ a. sequence ~ in D( W ) converging to x , then (WX,,),,,~ is convergent because W is an isometry. Hence x belongs to D(W) and

CLASSIFICATION OF IRREDUCIBLE UNITARY REPRESENTATIONS

297

lim Wxn= Wx. This shows that D( W) is a closed subspace of H. Since U is n-oo

an irreducible unitary representation, the closed invariant subspace D( W) is either {0} or H. By (6.29), if (D+i)x=O, then x=O. Since D2D(L?)= Hm# {0}, D(W ) is not equal to {0} but is equal to H. Hence by (6.30) and Ch. I, Theorem 2.1 (Schur's lemma), W = C1 for some complex number c of absolute value 1. If we assume that Wz=z, then by writing z =(a+i)x, we see that (D-i)x = Wz=z =(a+ i)x and x =0, z =0. Hence c # 1. Then for any x E D, z =(0 + i)x satisfies

(a- i)x= Wz= cz=

c(B +i)x .

Hence we get

a x=q x for any X E D where q=i(l +c)(l -c)-l=Zj is a real number. In particular, for any q.e.d. x E H,c D, we have Px=qx.

Remark. Since the image Im W of W is a non-zero closed Ginvariant subspace, Im W is equal to H and W is a unitary operator on H. Hence 0 is a self-adjoint operator. Proposition 6.14. Let U be an irreducible unitary representation of G = S U ( l , 1) and put

H = -iUK(&), E= VK(XI)+I'UK(Y) and F= V K ( - X I ) + ~ U K ( Y ) . Then we have (6.32) [H,E ] = E [H,F]= -F and [E,F]=2H. (6.31)

Put (6.33) p m =q+ ( R +m- 1)(R +m) and urn=q+(R -m+ 1)(R -m) for m e N where q is defined by Q = q l (cf. Proposition 6.13). If X E HR satisfies Hx =Ax, then we have (6.34) HEmx= (2 +m)Ernx, HFrnx=(2 -m)Frnx, (6.35) FEmx=pmx, EFmx= amx and (6.36) IIEm+l~II' = ~ ~ + I ~ ~ IIFm+l~II2 E ~ x I I= ~ ~ m,+ l l l ~ " ' X I I

-

Proof. Since Ug is a representation of the Lie algebra g (Proposition 2.5), (6.32) is a direct consequence of (6.22). Using (6.32), (6.34) is proved by induction on m. Since 2Q= -2Ha+(EF+FE) and [E,F]=-2H,

we have

298

UNITARY REPRESENTATION OF

SL (2,R)

'

(FE+EF)x=2(q+la)x, (FE-EF)x =2Rx . Hence (6.37) (6.38) Replacing x in (6.37) with E*-'x and R with R+m-1, we get the first equality in (6.35). Similarly we get the second equality in (6.35) by replacing x in (6.38) with Fm-lx and R with R -m+ 1. By (6.24), we have (Eu, v)=((U=(Xl)+iU=(Y))u, v ) = - (u, (UK(Xl)- iUg( Y))v)= (u,Fv) for any u, V E Hg,. Hence IIFrn+lxlla =(Fm+lx,Fm+lx)=(F"x, E P + l x ) = o ~ + I ~ ~ F *" ' ~ I I ~ Similarly we have IIEm+l~IIa = pm+lllEmxI12. Definition 9. An eigenvalue of the operator H = -iUK(Xo) is called a K-weight of the representation U. Proposition 6.15. There exists no finite-dimensional unitary representation of G = S U ( l , 1) other than the identity representation. Proof. First we show that the Lie algebra g=Gu(l, 1) is simple, i.e., g has no ideal other than {0} and g. Let a be a non-zero ideal of g. Then a contains an element Z = a X o+ pXl + r Y# 0. If a #O, then a contains a-l [Xo, [Xl, Z]] =XI, hence it contains [Xl, Y] = Xo and [Xl, XO]= Y (cf. (6.22)). So a coincides with g. Similarly if p # 0 or r ;t0 we can conclude that a = g. Let U be a finite-dimensional unitary representation of G = SU(1, 1). Then the kernel ker U' of the Lie algebra homomorphism U' (the differential representation of U ) is either [0} or g. But it is impossible that ker U'= (0). In fact, if we assume that there existed a unitary representation U such that ker U'= {0}, then U' would be an isomorphism of g onto g1= U'(g). The identity U'([X, Y]) = [U'(X), U'(Y)] implies that ad U'(X)= U'oad XoU'-l. Hence the Killing form B, of g is transformed to the Killing form BS1of gl. Namely, Bp(X, Y )=Tr(adXad Y) =Tr(ad U'(X)adU'( Y)) =Be,(U'(X), U'(Y)). Hence the signature of B, is equal to that of BB1.The relations (6.23) show that the signature of B, is ( 2 , l ) . O n the other hand, since U is a

(6.39)

CLASSIFICATION OF IRREDUCIBLE UNITARY REPRESENTATIONS

-

299

unitary repesentation, U'(g) g1 consists of skew-Hermitian matrices. Hence every element X of g1 has purely imaginary eigenvalues. If the degree of U is equal to n, then ad,,X is the restriction of ad,l(,,oX to g,. Thus the eigenvalues of adQ,Xare differences of two eigenvalues of Xand are therefore purely imaginary. In particular we see that the quadratic form B,, is negative definite, i.e., the signature of Bo, is (0, 3). This con/ tradicts the equality (6.39). So our assumption is false and we have proved that ker U'= g for any finite-dimensional unitary representation U of G=SU(l, 1). Then by Ch. 11, Proposition 5.9, we have UeXptH=exptU'(X)= 1 for any t E R and X E g. Hence U,= 1 for any g E G .

q.e.d.

Theorem 6.2. Notation is the same as in Proposition 6.14. Let Ube an irreducible unitary representation of G=SU(l, 1) and M be the set of K-weights of U.Then M is equal to one of the following sets:

1) 2, 2)

1

+ 2,

3) Mn+= {n+plp E N )

4) M n - = { - ( n + p ) I p E N )

( n ~ 2 - ' N n>O), ,

There exists an orthonormal basis satisfying

(6.40)

Hym =mqJrn,

Fp,

= o,-'(q

(nE 2-'N, n>O),

((pm)mrM

Epm = wm+l(q

5 ) (0} .

of the representation space H

+ rn(m+ 1))'~m+l,

+m(m- l))tcpm-l

for any m E M ,

where q is determined by Q=ql and loml= 1. In case 5), U is the identity representation I . Proof. Since K z U(1) is a compact subgroup of G, the restriction UIK of U to K is decomposed into a direct sum of irreducible unitary representations 1% : ue-etne (n E 2-lZ). Each irreducible unitary representation zn is contained in UIK at most once (Theorem 5.1). Hence we have UIK= @ X n . nrM

If a non-zero vector x E H satisfies Uuex=etnexfor all e E R , then Hx= nx. Since H # {0} by the definition of irreducible unitary representation (Definition 4 in Ch. I, $2), the set M is not empty and there exists an element A E M and a unit vector x E H such that Hx = Ax. There are four possibilities for the vanishing of Emx and Fmx. A) Emx#O and Fmx#O for all m E N . B) Ekx=O for some k s N and Fmx#O for all m e N . C ) E*x#O for all m e N and Fkx=O for some k e N .

300

UNITARY REPRESENTATION OF

SL (2,R)

D) E%=O for some kE N and Fnx=O for some n E N . In any case, the subspace L of HK spanned by {Emx,F " x } , , ~ is invariant under UK and must be equal to HK because UK is irreducible by Theorem 6.1. Since H K is dense in H, the non-vanishing vectors in {Emx, F m ~ } m ,form N an orthogonal basis of H. Hence we have M = [2+ml Emx#O} u [R-mlFmx#O}. Since 22 is an integer, we divide the case A) above into two subcases A.l) and A.2) according to whether 22 is even or odd. The case A.l) Since R is an integer and Emx#O and Fmx#O for all m E N, we have M = { J f m l m E N } = Z (the set of all integers). Since p,#O and am#O for all m E N in the case A), we can replace x with E2x if A 0 and assume that I =0. Put IIErnxll=rm and IIFmxll= tm. Then by (6.36), we have m

(6:41)

rm=

npn+and

n-1

m

tm=

nun+,

n-1

where pa+ and a,,+ are the positive square roots of pn>0 and in (6.33). Since 2=0 in our case, pm=q m(m - 1) = urn (cf. (6.33)) and r, = t , . (6.42)

un>0

given

+

The condition that rm>O for all m E N implies that q+m(m- 1)>0 for all m E N. The latter condition is equivalent to (6.43) q>o. Now we introduce an orthonormal basis ( P , ) ~ ~ Zin the following way. mX, (mE N , m l l ) , (6.44) ( D O = ~ O X~, m = r m n - l ~ m E+m=tm-l~-mFmx where 71m may be an arbitrarily chosen complex number of absolute value 1. By Proposition 6.14 and (6.42), the basis (p), satisfies the equations (6.40). The case A.2) 1 In this case the set M of K-weights is equal to - 2. Hence we can 2 1 assume that R =-. The equalities (6.41) remain unchanged. But the values 2 of p,,, and urnare given by

+

(6.45) p m = ( q - 71) + m 3

and a m = ( q - + ) + ( m - l ) a

Since Emx#O and Fmx#O for all mE N , p m > O and a,>O Hence we have

for any mL1. for all m E N.

CLASSIFICATION OF IRREDUCIBLE UNITARY REPRESENTATIONS

301

(6.46)

in this case. We obtain an orthonormal basis (pm)mtYby means of the following definitions: (6.47)

pi=viX, vt+n=rn-'vi+nE"x,

vi-n=tn-lvi-nFnX

(n E N , nr 1).

The relations (6.40) still hold in this case. The case B) There exists an integer k>O such that Ekx=O. Then Ehx=O for all h z k . Let h + l be the smallest integer for which this occurs. Then x, Ex, ..., Ehx are not equal to 0 and p l > O , ..., p h > O , p h + l = O by (6.36). Hence FmEhx#O for all m E N.So we can replace x with Ehx and assume that Ex =0. We have pl =0 and hence (6.48) q=-1(A+l) by (6.33). The value of urnis given by (6.33) and (6.48). urn=m(m - 1 -21) The condition that u,>O for every rnz 1 is equivalent to the condition (6.49) 1 O is an integer or a half-integer. Then the set M of K-weights is equal to { -n--glp E N). An orthonormal basis (pm)rnsY is defined in the following way: (6.50)

v-n=v-nX,

p-(n+p)=tp-lv-(n+p,FPX.

The equations (6.40) still hold with the following convention: The equation for E p n containing the undefined vector p-(n-l) should be interpreted as E p n=0. This convention is justified because the coefficient of is equal to 0 by (6.48). In this case, we have (6.51)

q=n(l-n)

.

The case C) The case C) can be treated in the same way as the case B). We can start with x satisfying Fx=O. Then we have u1 =0, q= -1(A - 1) and prn= m(21+ m - 1) >0. Hence (6.52) 1>0. Put ~ = n .Then (6.51) still holds. The set M of K-weights is equal to {n+plp E N) . An orthonormal basis (pm)mnrM is defined by (6.53)

( ~ n = p n ~p n, + p = r p - l p n + p E P X

-

302

UNITARY REPRESENTATION OF

SL (2,R)

The equations (6.40) remain valid with the convention that the equation for Fp,, is interpreted as Fp,, =0. The case D). In this case, the subspace V spanned by the vectors {Emx( O $ m i k - l), Fmx (0 5 m 5 n - 1)) is a finite-dimensional subspace of H invariant under UK.Since VKis irreducible (Theorem 6.1), V must be equal to HK. Since HR= V is dense in H and a finite-dimensional space V is closed, we have V = H. Hence U is a finite-dimensional, irreducible, unitary representation of G=SU(l, 1). So U is the one-dimensional identity representation by Proposition 6.15. The cases Al), A2), B), C) and D) correspond to the cases 1), 2), 4), q.e.d. 3) and 5) in the theorem respectively.

In the course of the proof, we have proved the following corollary. COROLLARY to Theorem 6,2. The eigen-valueq of thecasimir operator $2 satisfies the following conditions in each of the correspondingly num-

bered five possible cases 1) to 5) in Theorem 6.2: 1) q>o, 2) q>+, 3) q=n(l-n), 4) q = n ( l - n ) ,

5) q=o.

An important consequence of Theorem 6.2 is the following fact: An irreducible unitary representation U of G = SU( 1, 1) is uniquely determined up to equivalence by the set M = M(U) of K-weights and the eigen-value q =q( U)of the Casimir operator SZ. In fact, we have the following theorem. Theorem 6.3. Let U and V be two irreducible unitary representations of G = SU( 1, 1). Then U is equivalent to V if and only if M( U )=M( V) and q(U)=q(V) where M ( U ) and M ( V ) are the sets of K-weights, and q ( U ) and q(V) are the eigenvalues of the Casimir operator SZ for U and V respectively. Proof. It is clear that if U is equivalent to V, then M( V )= M( V) and q(U)=q(V) (as to the second equality, cf. Proposition 6.4). Conversely, assume that M ( U )=M( V) = M and q( U )=q( V )=q. Then by Theorem of H ( U ) and of 6.2, there exist orthonormal bases (p,JmrMand (v'~),,,~~ H ( V ) satisfying the relations (6.40) with the family of constants ~ ~ any ~ family . ( c ~ ) of ~ .complex ~ numbers Cm of absolute and ( o ' ~ )For value 1, an isometry A of H ( U ) onto H ( V ) is defined by

-

(6.54) Apm=cmpm' (m E M ) Since A satisfies A UR(- iXo)pm=mcmpm'= VK(- iXo)Apm AU=(X~+iY)p,=om+icm+i(m(m+ 1)+q)'pm+i'

CLASSIFICATION OF IRREDUCIBLEUNITARY REPRESENTATIONS

Vg(Xi + i Y ) A p m = w m + l ’ c m ( m ( m +

and

A Uk( -XI + i Y ) p r n = w m - ’ c m - l ( m ( m VK(

303

1)+q)+vi+l’ 1)

+ q)*pm-<

-XI + i Y ) A y m = wm’-lCnr(m(m - 1) +q)+ym-i’,

the operator A satisfies (6.55)

AUK(X)vm= Vg(X)Aym

for any X E a and m E

if and only if (6.56)

Cm+l/Cm=~m+l’/wm+l

for every m E M

.

Hence if we fix an element m o E M and choose a complex number cmo with the absolute value 1 and define the other cm’s by (6.56) and the operator A by (6.54), then A satisfies (6.55). Therefore U is equivalent to V by Proposition 6.5. q.e.d. In the following Theorem 6.4 we give M and q for the irreducible unitary representations constructed in $2, $3, and &I.

Theorem 6.4. The sets M of K-weights and the eigenvalue q of the Casimir operators Sa for the irreducible unitary representations of $2, $3, and $4are given by the following table:

*t

s(1 -s)

n(1 -n) n(1 -n) u(l - u ) 0 Proof. Let n be an element in $2.Then n is a weight of U if and only if there exists a vector x#O in the representation space H of U such that

UUex= etnex. 1) V’.’. Since an orthonormal basis { fn(<) =C-” In E Z } of &? satisfies t6.57) VuBJ*S fn =et(n+f)@f n , the set M of weights is equal t o j + Z (cf. Theorem 2.1). By Proposition 2.6, we have (6.58)

Hence

Efn= { Vg’.’(Xl)+iV~’.‘(y)~fn=(n+j+s)fn+l Ffn = { - Vg’s’(X1) iVg’J( Y ) )fn =(n + j - s) f n - 1

+

.

304

UNITARY REPRESENTATION OF

(6.59)

SL (2,R)

2Qfn=(-2H2+EF+FE)fn = {-2(~~+j)~+(n+j-~)(n+j-l+~)+(n+j+~)(n+j+l-~)}fn =2s(l - S ) f .

2) PI+. By (6.58), the set of weights for V+*+IQ+ is ++N, because 8' is spanned by {fnln E N ) Similarly, since 8-is spanned by { f - J n E N, n z l } , the set of weights for ?".+1@is equal to - 4 - N . By (6.59), D=dY'**(W)=+l. 3) U".Let n E +Z (n2 1) and let Unbe the discrete series representation of G on the Hilbert space Qn or Q-,,described in $3. Then {@,"(C)= {r(2n+p)/(r(2n)r(p+l))}*CpJpE N ) is an orthonormal basis of Qn and {@lp E N} is one of Q-n. By Proposition 3.5,

.

Uu8nQjn =e-t(n+p)rQj n,

u,,#-ne=eNn+pM-n@P .

Hence the set M of weights is equal to - n - N for U n and n+ N for U-". By the definition of Un,we have Uexptgln@pn(C) = { r ( 2 n+ p ) / ( r ( 2 n ) r ( p+ l))} +( - sh2-ltc +~h2-lt)-'~-p x (ch2-ltt;- sh2-lt)p and V,"(X,)@pn =2-'J(2n + p ) ( p+ l)@p+P-2-'J(2n + p - 1)pOp-l.

*

Similarly

U",( Y p p n = -i2-'J(2n +p)(p + 1)Op+P-2-'iJ(2n + p - l)pQjp-ln. Hence we have (6.60)

EQjP"

=J ( h + p ) ( p

+ 1)

@P+ln 9

Fopn= J ( h+ p - I)p Qp-1"

-

We have

+

+ +

2QQp"= { -2(n +p)' (2n +p)(p 1) (2n + p - 1)p)OPn =2n( 1-n)Op" .

For the representation U-n=dJn~, put q = K - P n . Then we have U,-~(Xl)K-p* =2-1J(2n +p)(p + 1)K-p-ln -2-'J(2n + p - l)pK--p+ln Y)K-," =i2-1J(2n +p)(p + 1)K-p-1n +i2-'J(2n + p - 1)pK-p-ln

u,-q and (6.61)

Therefore

EB-," = -J(2n+ p - 1)p K-p+l. FK-p" = -J ( b+p)(p + I) V-p-1.

CLASSIFICATION OF IRREDUCIBLE UNITARY REPRESENTATIONS

+

+

305

+

252Y-," = { - 2(n + p y (2n + p - 1)p (2n + p ) ( p I)) K-," =2n(l -n)!F-p".

P.Since VgU=V:*' on HK which is spanned by (fn(C)=C-nln E 23, the results in 1) hold in this case. Hence the set of &weights is equal to 2 and Q=u(l -u)l. 5) I. If l i s the identity representation gi-1, then M = (0) and q=O. q.e.d.

4)

Theorem 6.5. (Bargman) Any irreducible unitary representation U of G = S U ( l , 1) (or of SL(2, R ) ) is equivalent to one and only one of the following representations. 1) 2) 3) 4) 5) 6)

Vj*';;j=O,8, s = & + i v , v ~ ifj=O, O v > O i f j - 4 (cf. $2), V***l.jj+ (cf. Theorem 2.1), V'*'I@-, un;n E gz, In1 e 1 (cf. 43), v-;+ < u < 1 (cf. @I), Z (The identity representation).

Proox Let U be an irreducible unitary representations of G =SU(1,l). Then by Theorem 6.2, the set M of K-weights of U coincides with one of the following sets.

*

i) z, ii) + Z , iii) M,,+ = {n+pip E N } (nE )N,n>O), iv) M n - = ( - ( n + p ) I p ~ N ( ) n ~ i N , n > O ) ,V) (O}. i) If M = Z , then q > O by Corollary to Theorem 6.2. We divide the case i) into two subcases La) and i.b) according to whether qz&or O f . In this case there exists a unique real number Y >0 such that q= 4 +v2. Put s = + + i v . Then M(U)=M(V+**)and q(U)=q(V+.*). Hence U is equivalent to V+s0by Theorem 6.3. iii) M =M,+, q =n( 1 -n) (n E 2-lZ, n >0). We divide this case iii) into two subscases iii.a) and iii.b) according to whether n = 8 or n > &. iii.a) M=M++,4 = f . In this case M(U)=M(V*.*&j+)and q(U)=q(V+**IQ+). Hence Uis equiva-

+

306

UNITARY REPRESENTATION OF

SL (2,R)

lent to V*.*1$+by Theorem 6.3. iii.b) M=Mn+, q=n(l -n) ( n ~ 2 - ' 2 ,n>)). In this case, M(U)=M(U-n) and q(U)=q(U-^) (cf. Proposition 6.10 and Corollary to Theorem 6.1). Hence U is equivalent to U-". iv.a) M = M , - , q = ) . In this case U is equivalent to V**+ I$-. iv.b) M=M-,-,

-7.

q=n(l -n) (.E 2-'2, n < y

In this case U is equivalent to Un. v) M = {O}, q=o. In this case, U is the identity representation as was proved in Theorem 6.2. Since the pair (M(U), q(U)) takes different values for the different representations enumerated in Theorem 6.4, no two of them are equivalent to q.e.d. each other by Theorem 6.3.

$7 Thecharacters The character of an irreducible unitary representation of the Euclidean motion group M(2) was defined as a distribution on the group. (Ch. IV, $3.) Similarly, the character of an irreducible unitary representation U of G = SU(1, 1) is defined as a distribution on G.Namely, for any complexvalued C"-function f on G with compact support, the operator n

(7.1) belongs to the trace class, and the mapping fHTr U, is a distribution on G. We shall calculate the characters of the representations of G which belong to the principal continuous series, the complementary series, and the discrete series. It turns out that the characters of these representations are functions on G. Proposition 7.1. Let 9 ( G ) be the space of all complex-valued C"functions with compact supports on a Lie group G, U be a Banach representation of G on a Banach space H and V be the uniform closure of the subspace ( UfI f € a ( G ) ) in B= B(H) (the Banach algebra of bounded operators on H). 1) Put f,(A)= U,A and r,(A) = A U,-l

for g B G and A E V. Then I and r are Banach representations of G on V.

307

THE CHARACTERS

The operator U, (f E 9 ( G ) ) is a C"-vector for the representations I and r. Let L and R be the left and right regular representations of G and I', r', L', R' be the differential representations of I, r, L,R respectively. Then we have

l'(Ddr'(D4 U,= U L * C D ~ ) R , C D ~ > ,

(7.2)

for every f in d ( G ) and for all D1and Da in the universal enveloping algebra U(g) of the Lie algebra g of G. 2) Let K be a compact subgroup of G, let A be an element of the dual R of K, and let d(A) and x1 be the degree and the character of 1. Let E(R) be the projection defined by

sg

E(R)=d(A)

(7.3)

xA(k-l)U&

and let Et(A), &(A) be the similarly defined projections for the representations I and r. Then we have B ( I ) ( A )=E(A)A and Er(R)(A)=AE(A'),

(7.4)

where 1' is the class contragredient to A. Proof. 1)The operators I, and r, leave the subspace Vinvariant because W

(7.5)

1

f =

f(g1)U,,,dg1=

ULgf

and

1 0

r a ( u f ) = / f(gl)Ug1o-'dgl= u R g / . Since ~ ~ l , ( A ) l 1 5 ]IIAll, / U g ~I,~ is a bounded operator on V. It is clear that I and r are representations of G on V. Now we prove that I is strongly continuous. Take an element A in V and a positive number E . Then there exists a functionfin S ( G ) such that

(7.6) IlU,-4I<~ * Let N = N-1 be a compact neighborhood of the identity e in G and put M = sup 11 Ugll.Then for every element g in N, we have arN

ll~g(A)-~g(~f)ll 5 IIugll ~ ~ 6Mf ~. ~ f (7.7) Since the support C off is compact, f is uniformly continuous on G. Hence, if the neighborhood N of e is sufficiently small, we have

(7.8)

[f(g-'g1)- f(gl)l < S

for all g E N and gl E G.

For any g in N , the function p(gl)=f(g-'gl)-f(gl) vanishes outside of the compact set NC. Put D = sup ]~Ugl~l. Then we have 91cNC

~

~

308

UNITARY REPRESENTATION OF

SL (2,R)

for every g in N. By (7.6), (7.7) and (7.9), we have

5 IIUuA- U,U,ll+ l l ~ u ~ , - ~ A l +I1u,- All

ill,(A)-All

$(M+l)E+BE.

Hence 1 is strongly continuous at e. Hence, by the inequality

l l ~ , o ~ ~ ~ - ~ 5, ~II U, ~ ll~ llll,-1,o(~)--~Il l and by the fact that G is locally-compact, 1 is strongly continuous on G. Similarly r is strongly continuous. A C"-function f in g ( G ) is a C"-vector for the regular representations L and R.Hence by (7.5) we see that U, is a C"-vector for 1 and r and that U, satisfies (7.10) I'(X)U,= ULl(H), and r'(X)U,= URlta,, for every X in g. Hence we have (7.2). 2) For each element A in V, we have E(l"

= d ( q xJC-l)b(A)dk K

= d(,?)s x1(k-l)UkAdk=E(,?)A and K

SK

Ev(A)(A)=d(A)

xA(k-')A Uk-ldk

=d(;l)

~r(k-')AUxdk=AE(I').

S R

Proposition 7.2. Let G = SL(2, R),K = S0(2), and U be a Banach representation of G. Then the series

(7.1 1)

2

IIE(n)U.rE(m)ll

n.rn.2-12

converges for every f in 9 ( G ) . Proof. Let n be a half-integer, i.e. n E 2-lZ. Then x,, : uBHeinois an irreducible unitary representation of K and the dual k of K is identified with 2-'2 naturally. Let V be the uniform closure of the subspace [Ufl f E g ( G ) } and W be the Banach representation of G x G on V defined by W(,,,,(A)=U,AU,-' If n and m belong to 2-'Z=k, then

for A E V.

THE CHARACTERS

309

is the projection corresponding to the irreducible representation x,,&,,, of K x K. By Proposition 7.1, we get (7.12) F(n, m)(A)=E(n)AE(-m) . Iff belongs to g(Q, then the operator U, is a C"-vector for W.Hence the series

converges absolutely (normally) by Proposition 5.11. The equality (7.12) and the last stated result imply that the series (7.11) converges. q.e.d.

Definition 1. Let G be a a-compact Lie group, g be the Lie algebra of G, and U=U(g) be the universal enveloping algebra of g. Then for any DEU,

P d f 1 = SUPl(Df >(4l arG 9

fE9 ( G )

be is a seminorm on 9 ( G ) . Let C be a compact subset of G and the subspace of g ( G ) consisting of all functions whose supports are contained in C. Then s c ( G ) is a FrCchet space when supplied with the family of seminorms {pDIDE Ll} . Since G is 0-compact, there exists a of compact subsets of G whose union is equal countable family (QnrN to G. We give to g ( G ) the structure of topological vector space defined as the strict inductive limit of the spaces 9 c n ( G ) (cf. Ch. 111, $4, Definition 6). This topology of S ( G ) is independent of the choice of (C"). A continuous linear form on S ( G ) is called a distribution on G. Put f"(x)=f(g-'xg) for each g E G and f E 9 ( G ) . A distribution T on G is said to be central if (fa, T)= (f,T) for every g E G and f E d ( G ) .

Theorem 7.1. Let G = SL(2,R), K = S0(2), and U be a unitary representation of G on a Hilbert space H. If there exists a constant a such that (7.13) dim E(n)Hg a for all n E 2-'2= R , then, for every f in s ( G ) , the operator U f is of trace class and the linear form Ow:f w T r U f is a distribution on G. The distribution Ow is central. Proof. By the orthogonality relations, we get

310

UNITARY REPRESENTATION OF

= Gn,mE(n)

SL (2,R)

-

Hence E(n)H is orthogonal to E(m)H if n f m. Collecting the orthonormal bases (ea)x,rCn) of all non-zero E(n)H, we get an orthonormal basis (ek)ktI of H where I= u I@). Hence we have by Proposition 7.2, n

5

c c

dim E(n)H dim E(m)HIIE(n)U,E(rn)ll

n,rnr!l-lZ

sus

iIE(n)U,E(m)li-=

+ co

for all

SZ(G).

n.me2-12

Hence U, is of trace class by Lemma 1 in Ch. IV, $3. Let Xo be the element of the Lie algebra t of K such that exp 0x0 and put Do=Xo*. Since

Us=

I(ue)E(n)U, =efnbE(n)UI, we have

(7.15) I'(Do)E(n)U,= -n'E(n)U,. In this proof, contrary to our usual convention, we regard an element DO in the enveloping algebra U as a right invariant differential operator on G.Then we have (7.16) L'(DY= Df for any f E g ( G ) . Since t is commutative, Ec(n) commutes with Z'(D0). Hence by (7.15), (7.16) and by Proposition 7.1, we have (7.17)

and

E(n)UDo/=E(n)l'(DO)U/=--n'E(n)U~.

31 1

THE CHARACTERS

1

(7.19)

II~(W311SNII~Do311 3

nd-w

n-l+l)<+ m .

where N = M ( ma-1Z

n+O

By the first equality in (7.14) and by (7.19), we have

(7.20)

IT^ u315

C C nca-lZ

s

I(E(WA,

a)i

XrZCn)

1 d i m ~ ~ ~ ) ~ I I E ( W 3 cI I s IIE(n)ufII a nra-tz

nrl-12

SaNII U D ,. ~ Let C be a compact set in G and (fn) be a sequence in s ( G ) converging to 0 such that eachf, vanishes outside of C. Then, since (1 U,ll is bounded on C, we see that

I l u ~ ~ nSl I I(Dfn)(g)lIIU~ll&-iO s f 3

as n - i m for every D in U. Therefore by (7.20), we have

, ~aNll UD,,D~,II -i0 ITr U D J 5 as n-i co for every D E U. We have proved that Ow :f H T r U3 is a continuous linear form on 9 c ( G ) . Since C is an arbitrary compact set, @' is continuous on the strict inductive limit s ( G ) (Ch. 111, Proposition 4.2). Hence Ou is a distribution on G. If ((D&N is an orthonormal basis of H , then orthonormal basis of H for every g E G.

Since

Uf = f(g-'xg)U,dX J a = U,UfU,--t,

=

(Ug-1pn)nrN

is another

JQ

f(X)U,,,-ldX

by Lemma 1 in Ch. IV, $3, we have Tr U p = T r U3 for all g E G and f E g ( G ) . Hence the distribution Ow :fw Tr U3 is central.

q.e.d.

Definition 2. The distribution Ow :f H T r U3 is called the character of the representation U. Example. If U is a finite-dimensional continuous representation of G, then the (distributional) character Ow of U coincides with the usual r This can be seen from character of U, namely, the function xu : g ~ T U,. the equality Tr U3=

312

UNITARY REPRESENTATION OF

S L (2,R)

Now we proceed to the explicit calculation of the characters of irreducible unitary representations of G=SU(l, 1). First we treat the principal continuous series representations VJ.' (j=O, 4,s E C) which are not unitary unless Re s=*. We use a modified realization Wj.' of VJ.*.The realization W j .* is more convenient for the calculation of the character than the original realization Vj.*. For each s E C,there exists a Banach representation W' of G=SU(l, 1) on the Hilbert space La(K), defined by (7.21) ( Wglf)(u8) =e-*tQ-l* of(ug-1.0) . For each j E {0,41, we define a projection operator E, on H = La(K) by the formula (7.22) Clearly we have

(E,f)(u)=2-' If(4+(-l)a'f(-U)l

-

+

(7.23)

Eo E*= 1, EoE, =EiEo =O , EoH= {fe H l f ( - u ) = f ( u ) } , and E+H= (feHlf(-u)=-f(u)} .

Proposition 7.3. The equation (7.21) defines a Banach representation W * of G = S U ( l , 1) on H=La(K). For each j E (0, h), E, is a projection operator and commutes with every Woa.Hence Wg8induces a Banach representation Wj*' of G on the invariant closed subspace E,H of H. Let A , be a linear mapping from La(U)onto E,H defined by

(7.24) ( A f ) ( u e )=e-*J8f(e"). Then A j is an isometry of La(U)onto EjH and satisfies (7.25) A j o Vgf*'=Wgj$ao A , for every g E G and every j and s. Proof. It is clear that E, is actually an orthogonal projection. Since us== -1 belongs to the center Z of G,we see that guo+l. = ul.gu0 =u,~+alatco,omcg.~) .

Hence we have (7.26) g - ( 8 + 2 x ) r g . 8 + 2 ~mod 4x, t(g, ~ + 2 ~ ) = t ( gB ), . Therefore, E,oWO*=Wg'oE, forall g e G . Moreover we have u(g, 0) =ef(-W+@)lZ=u (g, 0 +2 4 (7.27) by (7.26). By (7.26) and (7.27), we have

THE CHARACTERS

313

-In the following, dg denotes the Haar measure of G-SU(1, 1) described in Proposition 5.1, i.e., 1

if g =usacne . dg =-etdedtdt 4a Moreover, du, da, and dn denote the measures (47r)-ldS,dt, and d f , respectively, where u =U s , a =at and n =ne.

(7.28)

hposition 7.4.

For every function f i n g ( G ) , the operator w3.E/of(g)wg.dg

is an integral operator on H=La(K) with the integral kernel L3(u, v) =

(7.29) Proof.

/

f(uarnV-')ea8dtdlt. AxN

It is easily seen that

(7.30)

t ( p s - ' , S ) =t

if g = vatne

is the Iwasawa decomposition of g. By a direct calculation, we have (7.31) a,-'ntat =ns-tc . Put u=ur. Then by (7.30)and (7.31), for every f in 9 ( G ) and for F in

N- La(K),we have (W3aF)(u) =[ol(g-')(Wg-'.F)(u)dg =

f(g-')e-"'"."'F(ug.~~g

I.

=

f(usg-')e-at(gu,-l*# ) F ( U ~ > ~ S

314

UNITARY REPRESENTATION OF

SL (2,R)

q.e.d.

where Lf(u,v) is given by (7.29).

Proposition 7.5. (7.32)

For every f in S ( G ) , we have

{IKxAxN

Tr (Wf'Ia) =y

+(-

f(uunu-l)e*cdudadn

1.

f(u(-u)~u-l)e*cdudadn

1)aj/ KXAXN

Proof. Put Xn(uB)=e*ne for each n E 2-lZ. Then xn belongs to EoHif and only if n E Z . We see that (7.33) TrWfjJ=Tr(EjW,*) for j = O , 4 and S E C , because

C

Tr Wf0**= (Wf'Xn,

Xn)=

C (Wj*EoXn,EoXn) nra-iz

nrl

=Tr (EoWf *Eo) =Tr (EoW fa ) ,

*.

and similarly Tr Wf * =Tr (E, W f'). By Proposition 7.4, we see that (EjWf'F)(u) =2-1 {( Wf'F)(u)+(- l)'j( Wf*F)(-U)} =2-'{/

f(uanv-l)e*tF(v)dvdadn KxAxN

+(-I)"/

f(u( -a)uv- l)e*'F(v)dvdadn KXAXN

I.

Hence EjW f* is an integral operator on H=L'(K) with the integral kernel (7.34)

MfjJ(Z4, v)=2-'

f(uamv-l)e*tdtuh {JAXN

+(-

1)"s

AXN

f(u(-o,)nlrl)e*tdtdn)

.

Since MI is a C"-function on K x K, the trace of the operator EjWf * is given by (7.35)

Tr(EjWf*)=

u)du Mfj@*(u, J K

THE CHARACTWS

315

(Ch. IV, Proposition 3.5). By (7.33), (7.34) and (7.35), we get (7.32). q.e.d. Now we shall prove that the right hand side of (7.32) is equal to the integral

IQf

(g)@j.’(g)dg

for some function @ I * a on G. This fact implies that the distribution f~ Tr ( W / J ) (cf. Theorem 7.1) is equal to the function W J . To prove this, we need some integral formulas on G. Definition 3. Let X be a locally compact Hausdorff space. Then the space of all complex-valued continuous functions on X with compact supports is denoted by L(X).

LEMMA 1. Let G be a locally compact group and H be a closed subgroup of G. For any functionf in L(G), put

f”(E)=/

f(gWh9

€ I

where dh is a left Haar measure on H and g = g H . 1) Then the mapping f H f is a linear mapping from L(G)onto L(G/H). 2) If GIH has a G-invariant measure d g f 0 , then we have O

(7.36)

for every f in L(G), where dg is a suitably normalized left Haar measure on G. Proof. 1) It is clear t h e y belongs to L(G/H). Hence it is sufficient to prove that for any F i n L(G/H),there exists a function f in L(G) such that f” =F. Let C be the support of F and V be a compact neighborhood of e in G. Then there exists a finite number of elements ( g f ) l s f s nin G n

such that the image of the set C1=u g t V by the canonical mapping r : %=1

g H g = g H contains C. Put C O = X - ~ ( n C el. ) Then COis a compact set in G satisfying n(CO)=C. Let U be an open neighborhood of Co whose closure 8 is compact. Since the underlying space of a topological group G is completely regular, it follows that for each point x in Co,there exists a continuous, non-negative function f z such that f z ( x ) = 1 and f z = O on G - U. Since fs >0 on a certain neighborhood U, of x and the compact set Co is covered by a finite number of Uz’s, there exists a function foin L(G) such thatfoZO on G andfo>O on Co.Put

316

UNITARY REPRESENTATION OF

SL (2,R)

Then f belongs to L(G) and satisfiesf " =F. 2) Since the linear form

is a positive Radon measure on G and satisfies O(L,f)=O(f) for all g E G. Hence O is a left-invariant Haar integration on G.

Proposition 7.6. Let G=SU(l, 1) and G=KAN be the Iwasawa decomposition. Then: 1) The factor space G/A is diffeomorphic to K x N by the mapping p-l, where ( ~ ( un)=unA, , u E K, n E N. If we identify K x N with G/A by the mapping p, then the G-invariant measure dg on GIA is equal to the 1 product Haar measure du dn =-dt?d[ on K x N. Namely we have 4n (7.37) for all F i n L(G/A). 2) Put H = A M = A { f l } . Then the factor space G/H is diffeomorphic to (KIM) x N and G-invariant measure dg on GIH is given by

(7.38)

Iff

lRF(g)dg=

1

F(un)dudn

=./KxNF(un)du dn .

Proof. 1) Every element g in G may be written uniquely as g =udatnF= usacneat-'at=uen,toat. Hence K x N is diffeomorphic to GIA by the mapping p : (u, n)wunA. Since both G and A are unimodular, the factor space GIA has a G-invariant measure dg. Since the Haar measure dg OD! G is given by dg =e'du dt dn, we have by Lemma 1,2)

=J

=I

KxAxN

f(uatn)etdu dt dn

f(un,tcar)ecdudtd(

KxAxN

THE CHARACTERS

/

=

317

f (uneat)dudt d t KxAxN

=/Rxfo

(twudn

for all f in L(G). Since {f”lfEL(G)} is equal to L(G/H), by Lemma 1, l), (7.37) is proved for all F i n L(G/H). 2) By the result of l), we see that G/H is diffeomorphic to ( K / M ) x N . (7.38) is proved in a way similar to (7.37). Since a function f on KIM is identified with a function f o on K which is constant on each coset of M, the last equality in (7.38) follows from

q.e.d. Proposition 7.7. If t # 0 and m = f1, then

for all f E L(HN). Proofi The element ht =atm satisfies nEhtnE-l=hrhc-lnchtn-t=htncs-t-l,c. We have e-‘ - 1>O or O. Putting r ) =(e-I1)t, we get l e - s - l ~f(nchtne-l)dt=le-’~ l l r-_f(htn(,-t-lddt -OD

q.e.d. Definition 4. An element g in G=SU(l, 1) (or SL(2, R ) ) is said to be regular if the two eigenvalues of g are distinct. The set of all regular elements in G is denoted G‘. Let B be a subset of G. Then the set B n G‘ is denoted by B‘. Proposition 7.8. Let G=SU(l, 1) (or SL(2, R)), and put

GI=UgHg-’, G I = U g K g - ’ , Ga= U g ( + N ) g - ’ ad3

ad?

UfQ

318

UNITARY REPRESENTATION OF

and Grl= Gt n G' (i= 1,2,3). Then Gal= in G. We have G=G;U G;U G ~ , (7.40)

SL (2,R)

+, and G1' and GI' are open sets

(7.41) G' = G1' u GI' (disjoint union) . G' is a dense open set in G and the Haar measure of its complement G - G is equal to 0. Pro05 Since SU(1, 1)z SL(2, R), it suffices to prove the proposition for G =SL(2, R). Let R and 2-l be the eigenvalues of g E G. Then we have (7.42)

Gi'=(gEGIAER},

(7.43) and

G s ' = { g E G I 1R1=1, R # & l }

Gs'=+. (7.44) In fact, if 17 >0 for an element g in G', then there exists t E R such that A=e"*. Since R # P , the Jordan normal form of g is equal to at. Hence there exists an element x E G satisfying g=xatx-l. Similarly if R <0, g = x ( - a t ) x l . Hence we have (7.42). If 111 = 1 A # & 1, then there exists a vector el # 0 in C' such that gel =etelaeland 1 =erela.Put C1=ea. Then gea=e-tefsez. Since A # P , el and ez are linearly independent and xl= 2-'(el ep) and xa = (2i)-l(el-ea) are non-zero real vectors. Since gx, = x1 cos 19/2-x3 sin 1912 and gxa =xl sin 1912-I-xlcosff 12, we have x-lgx =uR for the matrix x = (xl, xa). Multiplying x by a scalar if necessary, we can assume that x E G =SL(2, R). Hence we have (7.43). Since the eigenvalues of an element in Gs are both equal to + 1 or to -1, we have Gs'=+ By (7.42) and (7.43), we have G1' n Gal=+. If an eigenvalue R of g E G' is not real, then I is also an eigenvalue of g and must be equal to 1-l. Hence Inla = 12 =R R - l = 1. We have proved that G' = GI' u Ga'. Since the characteristic equation of g has the form ta-(Tr g)t + 1 =0,

+

1 is real if and only if (Tr g)a2 4. Put g =

(I: 5;).

Then we have

+

Gi' = { g E G I (a >4} , G a ' = { g ~ G I ( a + d ) s < 4 } , and G-G'= {gE G J a + d = f 2 ) .

Hence G1' and GI' are both open sets in G and G' is dense in G. G - G' consists of two analytic surfaces in the 3-dimensional manifold G. Therefore, G - G is a null set. If g belongs to G-G, then R = +. 1. Hence the Jordan normal form of g is equal to Consequently we have (7.40).

and g belongs to Gs.

319

THE CHARACTERS

Proposition7.9. PutH=AM= {+a,It.sR} andH+= {+atlt>O}.Then the mapping :G / H x H++G1' defined by +(gH, h) =ghg-' is an analytic bijection of GIH x H + onto G;. Proof. Assume that gu,g-l=g'ar~g'-l and put g'-lg=gl. Then we have glatgl-l=ae~.Taking the eigenvalues, we get {e,", e-c'a} = {et'", e-t'''}. Hence t is equal to either t' or -t'. If we assume that t>O and

+

t'>O, t must be equal to t'. If gl=

(: I;)satisfies glar=atgl, then we

have (eC-l)b=(et-l)c=O and b=c=O because t>O. In this case gl belongs to H and gH=g'H. Hence we have proved that (0 is a bijection from G / H x H + onto G1'. is also an analytic mapping. q.e.d.

+

Remark 1. Put H ' = H n G'. Then +, defined by +(gH, h)=ghg-l, is a 2 to 1 mapping from GIHx H' onto G1'. The proof of this result is clear from the proof of Proposition 7.9 and the fact that u.atu;l=a-t. Remark 2. In the following, we shall see that the Jacobian of the m a p ping does not vanish on G / H x H + . Hence is an analytic diffeomorphism of GIHx H + onto G1'. The calculation of the Jacobian of (0 is the essential part of the integral formula in Proposition 7.11 which will play a key r61e in determination of Moreover, since the method of calculation is the character Tr( W/**). used frequently in Lie group theory, we give a fairly detailed explanation.

+

+

Definition 5. Let M be a C"-manifold and 9-= T ( M )be the algebra of real valued C"-functions on M. A tangent vector of M at p e M is a linear form X on fl satisfying for every f and g in 9. The set of all tangent vectors of M at p forms a vector space T p ( M )which is called the tangent space of M at p. If p is a C"-mapping of M into another C"-manifold N , then there exists a linear ~ p ( M )into T,(,, (N)defined by mapping ( d ( ~of) T

-

[(diD)Pmf)=X(fOp) (d(0)p is called the diferential of (O at p. Let G be a Lie group and L , : x ~ g x and R , : x ~ x g

be left and right translations. We use the following notation: (dL,),X=g*X and (dRp),X=X*g for X E T,(G).

320

UNITARY REPRESENTATION OF

SL (2,R)

We identify the tangent space of G at the identity element e with the Lie algebra g. LEMMA 2. If X E T,(G) and g, h E G, then we have 1) g * (h* X ) = (gh)* X , 2) (X*g)*h= (gh), 3) g * (X*h)=( g*X)*h. Proof. These identities are proved directly using the relations L, o Lh= q.e.d. Lgh, Rh R,= R g h and L, Rh=Rh 0Lg.

x*

0

0

LEMMA3. Let X E Q = T , ( G ) and g E G . Then we have g * X * g - ' = (Adg)X. Put o(g)=g-l. Then we have (du),(g*X)= --X*g-l and (du),(X*g)= --g-l*X. Proof. Let a p : x w g x g - l . Then we have Adg=(du,),. Since ao=Lgo Ra-', we have (Adg)X=g*X*g-'. Put 8(t)=exp fX.Then we have (d8)&f/df)=x . Put 0' = o 8. Then we have 8'(t) =exp (- fX) and (dO'),(d/dt)= -X. Hence we have 0

(dU),x=(do de),(d/dr)=(de'),(d/dt)= --x. Since u o L, =R,-1o U, we have (do),(g * x> = (do dL,),X= (dR,-lo = (-x)*g-'= -X*g-'. 0

Similarly we have (du),(X*g ) = -g-l* X .

LEMMA4. Let H be a Lie subgroup of G and m: G x H+G be the mapping defined by m(g, h) =gh. Then we have (drn)(,,h,(X,Y ) = X * h + g * y for every X E Ta(G) and Y E T h ( H ) . Proof. When the element h in H is fixed, G is identified with G x {h} by the analytic injection ih :gi+(g, h). Similarly, for any fixed g E G, H is identified with [ g } x H by an analytic injection j , : hH(g, h). We see that (dih),X= (X, 0) and (c$,)~Y= (0, Y) for all X E T,(G) and Y E Th(H).On the other hand we have (m ih)(g)= m(g, h) =gh = Rh(g) and (nz oj,)(h) =L,(h), i.e. moih=Rh and m o j a = L , . 0

Therefore we have ( d m h .dX7 Y)= (hh, h)((dih)vX+ (dj,)h y) =(dRh),X+(dL,)hY=x*h+g* Y.

THE CHARACTERS

321

LEMMA 5. Let H be a Lie subgroup of G and p be a mapping of G x H into G defined by lo(& h) =gk-' * Then we have (dp)(g,h)(g *

x,h * Y)=ghg-' * (A&) ((Adh-'-l)X+

Y)

9

for every XEg and YE tj (the Lie algebra of H). Proof. Put p(g, h)=g. Then the mapping p is decomposed as m (mx (u op)): 0

p :(8,h)

- -

m x (u *PI

(gh, g-'1 Hence by Lemmas 3 and 4, we have

m

ghg-'

(P=

-

(d(O)(g,h)(g*X, h* Y>=(dm)(gh,g-l>(h d(0 oP))(o&*X~ h* r ) =(d~)(gh,l-l)(g*X*h+gh*y, -X*g-') =g*X*hg-'+gh* Y *g-'-gh*X*g-l =ghg-l* (gh-1*X*(gh-')-I -g *X*g-l +g * Y *g-1) q.e.d. =ghg-l*(Adg) {(Adh-1- 1)X+ Y}.

Proposition 7.10. Let G=SL(2, R) (or SU(1,l)) and @ be the mapping +(gH, h)=ghg-I. Then 4 is of maximal rank at every point of G / H x H'. 4 is an analytic diffeomorphism of G / H x H + onto GI'. More precisely, let (Xo, Xl, Xa) be the basis of g defined in Proposition 1.1, and put p(g)= gH=g and (dp),(X)=f ( X E T,(G)). Then we have (7.45)

Proof.

--

(&)(;.*d(g*Xo)A( g * x s )A (*at * X I ) )

Put Y=Xa-2Xo. Then (Xl, X2,Y)is a basis of Q. We have

-

Xs]=Xa and [XI,yl= Y (cf. (1.2)). Hence we have AdatX2=etXs and AdatY=e-'Y. By the relation (CI o (p x Id) =(O and by Lemma 5, we have [XI,

(d+)(i,h)((g*XO)A\)A (h*Xl)) = -2-'(d(O)
322

UNITARY REPRESENTATION OF

SL (2,R)

where h= f a t . Hence we see that det(d+)(-,,*,,,fO if t # O . Hence + is of maximal rank at every point on GIH x H'. Therefore, is an analytic q.e.d. diffeomorphism of G / H x H + onto G1'by Proposition 7.9.

+

Definition 6. Let tz" be the S-module of C"-vector fields on a C"manifold M (N = C"(M)).Then an alternating N-multilinear mapping x ... x tz" (k-times) into fl is called a k-form on M. The value of wq of o at a point q E M is the R-multilinear mapping of T,(M) x ... x Tq(M)into R defined by %((Xl),, .**, ( X k ) q ) = O ( x l , *.., X d ( q ) * Let 0 be a C"-mapping of another manifold N into M. Then a k-form + * o = o o + on N i s defined by (0

+)P(Xl,

* *.

,Xk)= O@(P)((d+)P-&,

..a,

(d+)Jk)

*

In the following, the Haar measure dh on the group H = A M = ( k u c l t E R) is normalized by requiring that

Proposition 7.11. If the support C of a functionfin L(G) is contained in GI'= ugH'g-', then we have vrQ

(7.46)

1

f(g)dg =/rec a sh-&dt

f ( ~ a m u - ~ )dn du

KxN

f(u( -ar)nu-l)du dn .

+/rec%h+t/ KxN

Proof. Let Xo, Xl, X z be the basis of the Lie algebra g of G = SU(1,l) defined in Proposition 1.1, and w0, 01, ozbe the 1-forms on G defined by ot(g*X,)=6r,. Then o1 is left invariant, i.e. O ~ L ~ for = Oevery ~ g E G. Hence B = o0A o1A is a left invariant 3-form on G. On the other hand o = ( 4 x ) - l e ~ d 1 9 ~ d tis ~ dalso ~ a left invariant 3-form on G (Proposition 5.1). Hence there exists a constant c such that D=co. Now we determine the value of c. Since the function I9 :ueatne H I9 satisfies O(u,) = 8, O(a,)= e(ne)-0, we see that

323

THE CHARACTERS

Similarly we have

Hence we have

So the Haar integration on G is given by (7.47) Similarly, let 6' (i=O, 2) be the G-invariant 1-form on G / H defined by Br(g* x,)= Then a = 00A 02 is a G-invariant 2-form on GIH. On the other hand, the manifold GIH is identified with KIMx N and p=duA dn = (2~)-lde~ d ist a G-invariant 2-form on G/H (Proposition 7.6). Hence we see in like manner as above that a =27cp and

f(usnc)d8dE.

=(47c)-'] KxN

The 1-form o1regarded as a 1-form on H satisfies (7.49)

f(h)dh=/;-

=I

f(at)dt+/m-ca f(-at)dt

f0l.

If

Finally, Proposition 7.10 shows that (7.50)

Q 0 (0 = -4( Sh-$)laA

WI

Using the formulas (7.47)-(7.50), Remark 1 and the transformation formula for multiple integrals, we have the following equalities for every function f in L(G) whose support lies in G1':

/

f (g)dg= (47c)-'/

fQ = (47~)-l Ql'

-2 1 H+

f (ghg-')dg

(sh+)'dh/ QIH

324

UNITARY REPRESENTATION OF

SL (2,R)

By Proposition 7.7, the last integral is equal to +-

f(uamu-')du dn sh-TdhS,.Nf(u(-a,)n~-l)dudn t

+I:ecla

q.e.d. Proposition 7.11 is a partial result. It is valid for the functions whose supports lie in Gl= ugHg-l. The complete integral formula is obtained by taking the integral over G, = u gK$l into consideration. Such a formula is given in Proposition 7.13. Proposition 7.12. The mapping r :(a, n)wanK is a diffeomorphism of A x N onto G* = G/K. If we identify A x N with G* = G/K, the product measure dadn gives a G-invariant measure on G* = G/K. Namely, we have (7.51)

where g*=gK. The mapping 9 :(g*, u)wgug-l is a diffeomorphism of GIK x K' onto GI' = u gK'g-' . wQ

Proof. Taking the inverse in the Iwasawa decomposition G = KAN, we see that the mapping (n,a, u)wnau is a diffeomorphism of N x A x K onto G. Since acnpac-l=n,tp,A N = N A and the mapping (a, n, u)wanu is a diffeomorphism of A x N x K onto G. Therefore the mapping is a diffeomorphism of A x N onto G/K. For eachfin L(G), put

/Kf(gu)du=f*(g*)

*

Since G and K are unimodular, there exists a G-invariant measure dg* on G* = G/K such that

The last integral is equal to

325

THE CHARACTERS

The last equality is derived from the relation neat=atnc-:F.

')

If g=(-" E G=SU(I, 1) satisfies gu,g-'=u,, we have e*@l)ly=eWaa Be and e-rerap=ecpls,8.Since l a ) s = l+],9l'#O, we have 1 9 ~ mod4x 9 and (ete-1)i3=0. If us+ -t- 1, then p=O and g belongs to K. Hence we have proved that if p(g*, ue)= (o(gl*,u,) for ue, u, E K', then ua = u, and g*= gl*. Namely, 'p is a bijection of G / K x K' onto Ga'. 'p is induced by the C"-mapping 'po :(g, u)++gug-' of G x K' onto GI1. Since the projection q :gwg* =gK is a C"-mapping and po= 9 (q x Id), (0 is a C-"mapping of GIKx K' onto Ga'. As will be seen in the proof of the next proposition, the Jacobian of the 0

mapping 9 at (g*, ur) is equal to 4(sin+)'.

e ego, 2K m o d k and sin-#O. 2

Since us E K' is regular,

Therefore, by the inverse mapping

theorem, the inverse mapping 9-l is also a C"-mapping. Proposition 7.13. we have

q.e.d.

For every Haar-integrable functionfon G =SU(1, I),

proof. Let 7' ( i = l , 2) be the G-invariant 1-form on GIK satisfying vI((g*Xj)*)=6u, where (g*Xj)*=(dq)(g*Xj). Then ~ = V I A is V ~a Ginvariant 2-form on GIK and satisfies

/G/xf(g*)dg*

=JB,a f

The 1-form o0 defined in the proof of Proposition 7.11, gives the Haar integration on K:

Let p: G / K x K+G, be the mapping (g*, u)ngug-'. Then by Lemma 5 and (6.22) we see that (dp))

* (Xl A Xa A XO). Hence we have

326

UNITARY REPRESENTATION OF

SL (2,R)

(7.53) Therefore, for any function f in L(G) whose support is contained in G%', we have Q

a'

QllCXK

Here we use the fact that the mapping (O is a diffeomorphism of GIKx K' onto G2'. Since G-G is a null set and G'=G1'u Gt' is a disjoint union (Proposition 7.8), we have

for any f in L1(G). Moreover, the restrictionflG'' is integrable on GI' (i= 1,2): f l G; E L'(Gt') (i= 1,2). Since (7.54) holds for every function f i n L(Ga'), (7.54) holds for all f in L1(Ga'). By Proposition 7.11, (7.46) holds for all f in L1(Gl'). Hence q.e.d. (7.52) holds for all f in L'(G).

Proposition 7.14. (7.55)

Let m = f1 and f be a function in L(G) satisfying

f(ugu-')= f ( g ) for all

UE

K and g s G .

Then the Radon transform Ff(mar)=et1.S f(macn)cin N

satisfies the functional equation I;&zz-O

=FArnat) .

Proof. By Propositions 7.6 and 7.7 and by the assumption (7.55), we have (7.56)

Ff(mar)=etla f(ma&€n

:IsN S N

I

= 2 sh -

f(nmatn-l)dn

THE CHARACTERS

327

f (unmucn-lu-l)dudn

=2/sh+// KXN

=2lsh$-1/

f(gmatg-')dg. UlH

Put v =u,. Then v normalizes H. More precisely, we have

(7.57) v(mar)v-l= m-c (m= zk 1) (cf. Proposition 2.11). Hence the mapping s :gH-vgv-lH defined diffeomorphism of G / H onto itself. Put

is a well

for each p in L (G). Then by Lemma 1 we have

On the other hand, by (7.57) we have (D".(g)=/

Sa

p(vghv-1)dh = H

p(vgv-'vhv-')dh

=/Hp(vgv-lh)dh.

Hence (7.59)

p~'=ID'oS.

Since the mapping p ~ p maps * L(G) onro L ( G / H ) (Lemma I), (7.58) and (7.59) imply that dg=d(sg). Then, dg =d(sg)=d@-') . (7.60)

By (7.59, (7.56), (7.57) and (7.60), we have

Theorem 7.2.

For each j

B

{0, +} and s E C, put

328

UNITARY REPRESENTATION OF

SL (2,R)

where k = O or 4. Then the character of the principal continuous series representation W' of G=SU(l, 1) (or SL(2, R)) is equal to the function W J . Namely we have

for every function f in .g(G). Remark. Since G - G' is a null set, the values of the function @ * a on G-G' can be taken arbitrarily. On the other hand, since G' is open in G, the values of @js* on G are uniquely determined by (7.62) (cf. Lemma 8 below). Similar remarks are applied to Theorems 7.3 and 7.4 below. Proof. Since W a gWjJ (Proposition 7.3), it suffices to prove (7.62) for the representation Wj** instead of Vj.' By the definition of 0 j v r and by the integral formula (7.52) (or (7.46)), we see that

Ql'

Putting f o ( g )=

f(ugu-')du,

the latter sum of integrals is equal to

d K

+(-l)sj[e8'dt/

KxN f(u(--ar)nu-l)dudn]

.

Comparing this with the expression (7.32) of Tr( W/J) (Proposition 7.5), we get

J.

f(g)Wa(g)dg=Tr ( W / J )

.

q.e.d.

329

THE CHARACTERS

Now we compute the character of the complementary series representation P(4
Theorem 7.3. Let 4
Tr V,. =Tr V f

for any f E : ( G ) . Hence the character of the complementary series representation V is equal to the function

if g E G-G1'

.

ProoJ Let Q=L'(U) be the representation space of Then the representation space 8. of V' is the completion of 8 with respect to the positive Hermitian form (+, +)u on 8 (cf. $4). Since (4, +)< is a bounded strictly positive Hermitian form on 8 (Propositions 4.2 and 4.4), there exists a bounded strictly positive Hermitian operator A. on 8 such that (#, +).=(A. 9, +) for all 4 and in 8. Put &=Ae-*. Let (9n)n.N be an orthonormal basis of 8. Then ( B u p , ) n .is~ an orthonormal basis of Q., because 8 is dense in &. Hence for everyfin we have

+

:(a,

0

C = C(AuVfo*uAe-f9n, Ao-'vn)

Tr Vj' =

(V,'&n7

&pJ.

n-0

n-0

=Tr(A.+ V fO-'A,-*) =Tr V fOo8

because the restriction of VgUto .fjis equal to V:,.. We have proved (7.63). By Theorem 7.2 and by (7.63), the character of P is equal to the funcq.e.d. tion W in (7.64). The character of a representation in the discrete series is computed in quite a different way. The method is in some sense similar to the computation of the characters of finite-dimensional representations. The idea is that we take the sum of the diagonal elements of a representation matrix, but there arises a problem of convergence. However, this difficulty is overcome by introducing a convergence factor and using Theorem 7.1. We begin by calculating the diagonal element (UnOpn,0,"). We recall that every element g in G-K can be written uniquely in the

330

UNITARY REPRESENTATION OF

SL (2,R)

form (7.65) g=upatuI,0 < ~ < 4 a , O < t , 0<9<2x. (Proposition 5.2). The element g in (7.65) is denoted by g(y, 1, 4). In the following Xo, Xly and Xa denote the elements of the Lie algebra g of G = S U ( l , 1) defined previously (cf. (6.21)). They are regarded as left invariant vector fields on G.

Proposition 7.15. As left invariant differential operators on G -K,the elements Xo,Xl, Y = Xa-Xo and the Casimir operator o =X O-~X12- Y' have the following expressions in the coordinates (9,r, @) defined by (7.65):

a +cos (0- a -cth t sin $--a Xl =-sin9 sht ay at a4 YE---C O S ~a +sin +-+cth a shr a p

Proof.

t cos 9-a 34

at

Let g = g ( y , t, g) E G-K. Then

where i(9+I)

a =e a c h -

(7.66)

2

ch-

u) t s + e sh- sh2 2 2 2

t

f(cp++)

t s sh- ch2 2

t(a-cs)

p =e a c h - 2 sh-2 +e

f

.

Hence, by Proposition 5.2, we get t

s

sh2(t(s)/2)= I,qa= (chT sh- 2

+sh-2t ch-2S cos+ S

= c h L ch s-ch2-+2-l sh t shs cos 2 2 Differentiating at s= 0, we get t'(0) =cos # . (7.67)

.

THE CHARACTERS

33 1

Again by Proposition 5.2, a= &-(p(S)+#(J)

)/a

ch(t(s)/2) @=&v(*)-#(J))/2 sh(t(s)/2) Differentiating four equations in (7.66) and (7.68) at s=O, we get (p'(O)+ g'(0) and ~ ' ( 0) +'(O), hence also ~'(0)and +'(O) : (7.68)

-

+'(O)

(7.69)

= -cth t sin

~ ' ( 0 =sin ) +/sh t .

+

By (7.67) and (7.69), we get

a

a

+ +'(O)- a+a sin+ a + cos +--cth a =-t sin +-a . sht a9 at a+

+

Xl = V ' m - aP t'(O)%

Put u,ucusexp sY=g(&), tds), +I@)). Since ~ , ~ a - ' u =exp ~ ~ sY, , ~ ~yl,tl and +1 can be expressed by means of the functions 9, t and +. Put u,utu,a, =g(&,

+), t(s, $1, +(s,

9)). Then we see that

&)=p(s,

+-+),

Therefore pI'(O), tl'(0) and

&'(O)

can be obtained from p'(O), t'(0) and +'(O)

by replacing

+

by

i7

G--. 2 Hence we have

+

cos Vl'(0) = --sht ' tl'(0) = sin , +1'(0)=cth t cos (I

+

and cos+ a a +cth t cos +-a . y = -sin sht ay at a+ The expression for o is easily obtained from those for XO,Xl and Y. q.e.d.

-+

+--

Proposition 7.16. Let p E 2-lZ and s E C. If a real analytic function f on G satisfies a) f ( w u d =XP(U,+#)f (g>9 O I =s( 1- s ) f , and b) c) f(l)=1 , then f must be equal to the function defined by (7.70)

f ( u , u C u # ) = ~ ( c h ~ ~ ' * F ( s-p, s + p ,1;

332

UNITARY REPWENTATION OF

SL (2,R)

where F(a, b, c; z) is the hypergeometric function. Proof. By the assumption a), f is completely determined by its restrictionflA to the subgroup A. By the assumption b) and because of the expression for o given in Proposition 7.15, flA satisfies the differential equation (7.71)

{-$+%-$+[s(l -s)+-

I1

2P' (ch t- 1) f ( a t )=O sh t

.

Let us introduce a new variable x related to t by the equation t x =ths-

2

and put f(at)=(l-~')'F(~). Since d d =4 3 1 -x)and dt dx da da d -dt a - x(1- x y dt y +24(1 - x ) (1-3x)- dx '

the function F(x) satisfies the differential equation (7.72)

daF

x( 1- x b

dF +[1-(2s+ l)x]d, +(p'-s')F= 0 .

This is the hypergeometric equation for which a = s + p , b=s-p and c = l (in the standard notation). As is well known, the unique solution of (7.72) which is continuous on the interval [0, 1) and takes the value 1 at x=O (the assumption c)) must be equal to the hypergeometric function F(s+p, s-p, 1; x). [Another solution Fl of (7.72) which is linearly independent of F(s+p, s-p, 1;x ) has the form

-

FI(x)= C C ~ ~ ~ + F ( S + ~ ,1;x)lOgx. .S-P, k=O

Hence Fl is not continuous at x=O.] Therefore the real analytic function q.e.d.

f on G satisfying a), b) and c) is given by (7.70).

Remark. The complex number s is not uniquely determined by b). However, the function (7.70) is uniquely determined because (7.70) is invariant under the replacement of s by 1-s. The last fact is due to an identity for hypergeometric functions: F(u, byC; z)=(l-~)C-"-~FF(c-b, C-U, C; z).

In fact, if we puts'= 1-s, then s(s-l)=s'(l-s').

333

THE CHARACTERS

Proposition 7.17. Let Un(n E 2-'Z, In1 2 1) be the discrete series representation of 53 and put

+

Opn(C) = [I'(2n+p)/r(h)r(p 111' C p

for p E N

.

Put

(7.73) upn(g)=(UgnKpn,Kpn) for each n E 2-lZ(lnlz 1)

and p E N. Then the function upn is equal to

and uP-n(g)=upn(g) for any n 2 1

Proof.

.

Since cDPn satisfies

(7.75)

Uu,nOpn =X-n-p(Us)Opn

(Proposition 3.9, it is a K-finite vector and hence an analytic vector for U" by Proposition 6.11. Therefore the matrix element upnis a real analytic function on G. Moreover it satisfies the three conditions in Proposition 7.16: First we assume that n 2: 1. a) By (7.75), upn satisfies ~P"(U,B+)

=Xn+p(UI+))Ui(g)

*

b) For any X E g, we have

=(ugnU~'(x)@p(o Op").

Hence @UP*

=n(l -n)up"

because Q = Un'(o)=n(l -n) (Theorem 6.4). c) is trivially satisfied. Therefore upn is given by (7.74) if nll. Since Ug-"=uUgnu, where u : f ~isfthe conjugation (cf. 53), we have Up-n(g)

=(uUg"u a e p n , =UP%)

U @ p 9=

(UgnOpn, cDp")

q.e.d.

*

The hypergeometnc function G,(u, c; z)=F(a+m, -m, c ; z )

(mB N)

334

UNITARY REPRESENTATION OF

SL (2,R)

is a polynomial of degree m which is called the Jacobi polynomial. The Jacobi polynomials (G,(a, c; z ) ) , , ~form an orthogonal system on the . need the followinterval [0, 11 with the weight function xC-'(l- x ) ~ - ~We ing relations for G,.

LEMMA6. Let (7.76) Gpa(~)=F(u+p,-p, 1 ; X) for a> 1 and p E N. Then we have (7.77)

1:

(l-x)u-lG,a(x)G,a(x)dx=O

if p z q

and (7.78)

[( 1 -x)u-aGpu(x)2dx=

(a- l)-l

for all p E N.

0

Proof. Gpaand Gqasatisfy the hypergeometric equations (7.79) [x(l - ~ ) ~ G ~ ~ ' ] ' + +a)(l p ( p- x ) ~ - ~ =GO ~ and ~ (7.80) [ ~ (- lx ) ~ G ~ ~+q(q+a)(l ']' =O . Integrating (7.79) x Gqm-(7.80) x Gpa from 0 to 1, we get [~(~+n)--q(q+41[ ( ~ - x ) u - l ~ p u ( x ) ~ q u ( ~ ) ~ =

- [ ~ (- lx ) ~ ( G ~ ~ ' G ~ ~ - (x)]; G ~ =O ~ G. ~ ~ ' )

Hence if p # q, we get (7.77). The hypergeometric function ab z a(a+ l)b(b 1) z2 F(a, b, c ; ~ ) = l + - - - + - +... c l! c(c+l) 2! is equal to a polynomial of degree S p if b= -p. Gpuis a polynomial of exactly degree p, because the coefficient of x p in Gpa(x)is equal to

+

c p= (- 1 )pa(a+ 1) ... (a+P-lJ,o

P!

Hence the monomial xn is a linear combination of the polynomials (Gqa)osq s n . Therefore, by the orthogonality relation (7.77), we see that (7.81)

1:

(1-x)'-lGpu(x)~(x)dx=O

for every polynomialfof degree Q-1. By the definition of GPU(x)=F(a+p,-p, 1 ; x), we have

.

(7.82) Gpu(0)=1 By (7.81), (7.82), and integration by parts, we have

335

THE CHARACTERS

1:

(1 -x)a-*Gpa(x)adx= [-(a-

1)-l(1 -x)"-lGP5(x)]:

(1 - ~ ) ~ - ~ G ~ ~ ( x ) G ~ ~ ' ( x ) d x = ( a - l ) - ~ .

+2(a-l)-l[ 0

q.e.d. Proposition 7.18. The matrix element upn(g)= (UgnKpn,Kpn) (cf.(7.73)) of the discrete series representation Un is a square integrable function on G and has the norm

lo.

(7.83)

I~,"(g)ladg=4~(21~l--1)-'

for every n E 2-lZ (In12 1) and p E N. Proof. The Haar measure dg is expressed by dg =2n sh t dkdtdk' for g = kark' (Proposition 5.3). Hence by Proposition 7.17, we have

Sr

lupn(g)ladg=2n

I(ch t/2)-51n1F(21nl + p , -p, 1; (th t/2)*)l%htdt .

Putting x=(th t/2)%, we get dr=x-*(l--x)-ldx, (ch t/2)a=(l-x)-1, sh t=2x*(l-x)-'. Hence the above integral is equal to 4x1: (1-x)al"~-~Gpa'n~(x)adx=4n(2~~l-1)-1

by Lemma 6.

q.e.d.

The above proposition determines the part of the Plancherel measure corresponding to the discrete series representations. (4~)-~(2(nl1) is called the formal degree of Un.The formal degree of a square integrable representation plays a role analogous to that played by the degree of a finite-dimensional representation of a compact group. The following lemma gives the generating function of the Jacobi polynomials Gpa(z)=F(a+p, -p, 1;z). LEMMA^. I f I w l < l a n d - l < z c l ,

-

(7.84)

1

WPF(U +p,

thenwehave

+ + 1)]1-5

-p, 1;Z)=~ - 1 [ 2 - 1 ( ~r

P-0

where R = [1-2w(1-22) + wa]+. R is a univalent analytic function in D = (w E Cl Iwl< 1) which takes positive values on the positive real axis in D. We give an outline of the proof (cf. ErdClyi [l]). Introducing a new variable u = t( 1-tz) (1-t)-l in Euler's formula

336

UNITARY REPRESENTATION OF

F(u-b, b, 1; z)=[r(b)r(l-b)]-l[

SL (2,R)

t*-1(1-r)-*(l-rz)*-mdr, 0

we get 2-'(1-u+U)=

(7.85)

1-22, U-ldu=(l-f)-ldf

and

F(u-b, b, 1 ;z)= [r(b)r(l -b)]-1/rub-1[2-1(l -u+ U)]'-"U-'du,

where U=[1 +2u(l-2z)+ua]*. The integral (7.85) is the Mellin transform of the function [2-'(1 -u+ U)]l-aU-l.Putting u=eS, the Mellin transform becomes the Fourier-Laplace transform and we get the inversion formula

(7.86) [2-'(1-~+ U)]'-'U-'=-.

21Cl

s""-

r(b)r(l - b ) F ( ~ - b , b, 1 ;z)u-*&,

p < m

where 9, is a suitably chosen positive real number. The integrand in (7.86) is a meromorphic function of b and has poles only at b= - p @=O, 1, 2, ...) in the left half-plane S= { b = x + y i l x < p } . The integral (7.86) is equal to the sum of the residues at the poles in S. This can be proved by taking a suitable contour C in 3 and letting C tend to infinity. Since Resr(b)=(-l)p/p!, the right side of (7.86) is equal to *--p

c m

(- l)PF(a+p, -p, 1 ;z)uP .

P-0

Putting u= -w, we get (7.84).

q.e.d.

LEMMA8. If (p is a continuous function on G satisfying

for every f e *I(@= {fe S(G)lsuppfc G'], then p(g)=O for every g E G'.

Proof. Let go be a point in G . By the continuity of p, for each there exists a compact neighborhood W of goin G' such that Ipk)--lo(go)l<e

r>O

for all g c W.

There exists a functionfin S1(G) which vanishes outside of Wand satisfies

fro

and

1.

f(g)dg=l.

Hence we have

Iv(go)l=I

s.

(p(g0)-

(p(g))f(g)dglS E. Since c can be taken arbitrarily small, we get p(go)=O. q.e.d.

Now we proceed to the calculation of the character of the representation U" in the discrete series. We use Abel's method of summation for

337

THE CHARACTERS

the divergent series C(UgnYpn,V p n ) and prove that the sum taken by P Abelian summation is equal to the character, i.e., the distributional sum of the series. Before stating the theorem, we recall results on the representation of the conjugate classes in G'. Every element g in GI)= u goH'g0-' may be written gocG

&?=go(-l)S'acgo-'

for a certain goE G, t E R, and j~ {0, &}. The values of j and It1 are uniquely determined by g (Proposition 7.9). Every element g in Gal= u goK'go-lmay be written VO'G

g =goUego-'

for a certain go E G and 19E [0, 4n). I9 is uniquely determined by g.

Theorem 7.4. Let U" (n E 2-'2, In[2 1) be a representation of G = SU(1, 1) in the discrete series (cf. $3). Then the character of Unis equal to the function e-(lnl-t)ltl lecrs-e-tia

I

(-l)tn9

if g = g o ( - l ) s ~ a c g ~GI', -l~

if g E G-G'

,

where j = 0 or 4 and e(n) =sign n. Namely, we have

s.

Tr U p =

(7.88)

f(g)@"(g)dg for all f E S ( G ) .

Proof. We introduce the convergence factor r (0 c r < 1) in the summa-

tion of

i

(

~

~

n

K,"). ~ ~ nPut ,

P-0

@,"(g)=

-

CrP(Ug"Kp*,vPn). P-0

Then the series converges uniformly on G for each fixed r E (0, l), because lupn(g)l5 1 for all g E G. Hence the function 8," is continuous on G. Since up-"(g)= up"(g),we have (7.89)

-

@r-"(g)=@rn(g)

Hence it is sufficient to calculate 8," for n2 1. Assume that n~ 1. Then by Proposition 7.17 and Lemma 7, we have

338

UNITARY REPRESENTATION OF

+ +

SL (2,R)

t

-In

=e-tncp+*~~-1(2-1(~ w I) )l-an P - j- )

where w=re-t(p++)and R=[1-2

,

t ( I-2th2~)w+wa]'.

Hence a finite limit exists for lim @,"(g) for every g=u,atu+ E G-K. r-1-0

Since R =R(w) takes real values on the positive real axis, we see that lim R ( r ) = ( 4 t h aty ) * =21th$/ and 7-14

By (7.89), we have

and for all n E 2-'Z, In)2 1. On the other hand, on the subgroup K,we have

and us+ f1. Hence

exists and is finite. By (7.89), we have

Hence lim @,"(g)=Oln(g)

r-1-0

exists and is finite for every g E G'. Moreover, we have

.

&"(g)=@(g) if g E H' u K' To prove eln=8" on G', it is sufficient to show that elnis invariant under inner automorphisms. This fact will be demonstrated at the end

(7.90)

339

THE CHARACTERS

of the proof. Since the operator Ufnfor eachf in 9 ( G ) is of trace class, the series

converges absolutely. Therefore, by Abel's theorem on power series, we get m

(7.91)

Tr U p = P-0

m

(UlnVPn,lPpn)= r-1-0 lirn

rP(UfnFpn, VPn) P-0

for every f s g(G'). Since

c OD

s r"llfll Ilu,"ll s [4n(214-l)-'I'

llfll(1

+

00

P-0

by Proposition 7.18, the right hand side of (7.91) is equal to n

n

Thus we have proved that r

(7.92)

Tr U," = lim

r-1-0

J f(g)@r"(g)dg Q

for every f E g ( G ) . Put R = R(r, 1 ) = (1 -2( 1-2 th$)r+

ra)+and R1= lim R(r, t) r-1-0

Then we have 0 6Ri- R= (Ri'-Ra) (RI R)-' (7.93)

+

I ;6(1 -r)Rl-'

Since R1-1=2-1(1 +e-t)ll-e-cl-l R*=R-{O}, we see that

. is bounded on every compact set in

lim R=Rl r-1-0

340

UNITARY REPRESENTATION OF

SL (2,R)

uniformly on every compact set in R - {O} . Hence 8," converges uniformly to eln on every compact set in G -K. On the other hand eraconverges to elnuniformly on every compact subset of K' because 1-r IQ~"(~o)-@i"(~a)l = I -re*tel 11 -psi * Therefore, 9," converges to elRuniformly on every compact set in G . Hence Qln is a continuous function on G . Moreover, by (7.92), we have

Tr U3"=S6'g)Q1"(g)&

(7.94)

for every f in g l ( G ) = { f ~ g ( G ) I s u p p f c G ' } . (7.94) proves that the distribution fwTr U," coincides with the function Elln on the open set G' of regular elements. For each x E G, put f.C(g)=f(x-lgx). Then we have j-/(g)@l"(xgx-')&= =Tr U;= =Tr U3"=

f"(g)@l"(g)&

/ff

1

f(g)Ol"(g)dg.

ff

By Lemma 8, we get (7.95) Ql"(xgx-l)=Ol"(g) for all g E G' and x E G

.

By (7.90) and (7.95), we have Qln(g)=@n(g) for all g E G' . Since Il-e**B111--re*~o~-1~4if O < r < l , we see that

(sin~~j@,"(u,)-e.(u,)I 64(1-r) and (7.96)

lim (sin~~(Q,"(ua)-Bn(u.))= 0

r-1-0

uniformly on K. Let ro be a fixed real number in the interval (0, 1). Then R(r0, t ) = 2 l-ro for all t E R. Hence for every r in the inter-

Val [ro,l), we have by (7.93)

THE CHARACTERS

341

We see that

uniformly on every compact set oft. Therefore, (7.97)

lim r-1-0

(sh

fa,)-@( k a t ) )=0

uniformly on every compact subset of the Cartan subgroup H = M A . By the relations (7.96) and (7.97) and the integral formula (7.52) (Proposition 7.13), we have (7.98)

lim /Gf(g)Qrn(g)dg=

r-1-0

1

f(g)@"(g)dg

for allfin d ( G ) . (7.92) and (7.98) prove the desired relation (7.88). q.e.d. Theorems 7.2, 7.3 and 7.4 show that the characters of representations in the principal continuous, discrete, and complementary series are in fact functions which are real analytic on each of the connected components Gl' and G2'of G' and are integrable on every compact set. This is a special case of theorem due to Harish-Chandra [6], [lo] stating that a central eigendistribution (in particular a character of an irreducible unitary representation) on a connected semisimple Lie group is a locally summable function which is analytic on each component of G . At the end of this section, we give a remarkable relation between the character Of*"+'of the non-unitary representation VJ,"+jin the continuous series (92) and the character 8" of the discrete series representation. In this relation, the character of a finite-dimensional representation appears.

Proposition 7.19. Let Vnbe the vector space of all homogeneous polynomials of degree n in two variables zl, z2. Let F" be the representation of G=SU(l, 1) on V, defined by (7.99)

(F"(g)9)(4=(OM*

(cf. Ch. XI, $1). Then the character En of F" is given as follows. If A and A-' are the eigenvalues of a regular element g E G , then An+l--R-(n+l)

En&)=

2-2-1

-

ProoJ Actually (7.99) defines a representation Fn of the complex Lie group GC=SL(2, C). If g is regular, then 2 # P , and there exists an element in G' such that

342

UNITARY REPRESENTATION OF

SL (2,R)

Hence we have €"lg)=TrF'(g)=TrFn(() R

O

- (In+I-R-(n+l) 1t (1-2-9 as was shown in the proof of Ch. 11, Proposition 2.5.

Theorem 7.5. For each TI E N,n r 1, and j we have (7.100) Proof.

+

E

q.e.d.

{O,g) and for all g E G ,

+

@j.n+j(g)= (a(n+J-l) (g) @"+j(g) @-"-j(g)

.

By Theorems 7.2 and 7.4, we have

and

Hence we have (7.100) for all g E G1'. On the other component GI'of G , we have OJ*"+'(goUago-') =0 ,

and (@+j

(n+j-# +@-n-')(gousgo-') = -sinsin (0/2)

0

Hence (7.100) is valid for all g E G,'.

'

q.e.d.

Remark. Theorem 7.5 reflects the following relations among the representations, VJsn+j,Facn+j-')and Un+J@U-"-j. By Proposition 2.6,4), we see that dVzf*n+jf-,-a,=dVjj.n+jf*=0

and the subspace Q(j, n+j) of Q=La(U) spanned by IfPlpzn or pS-n-2jl is invariant under V f ~ " +The j . subspace Q ( j ,n + j ) has codimension 2(n + j)-1 and the representation induced by Vfvn+j on the quotient space

343

INVERSION FORMULA

Q/@(j,n+j) is equivalent to composed thus :

The subspace Q(j,n+j) is de-

FZcn+j-l).

W, n+j) =Q+Ci, n + A @ @-(j, n + A , where Q+(j,n+j) and &(j, n+j) are spanned by {fplp>,n}and {f,lpS -n-2j} respectively. Roughly speaking, the representations induced by Vjtn+fon Q+(j,n + j ) and on &(j, n+j) are equivalent to U-"-j and to Un+jrespectively. More precisely, we get Ur("+J) after the completion of a dense subspace with respect to a suitable inner product. These circumstances are reflected in the relation (7.100). Another explanation is also possible. By Theorem 7.2, we see that (7.101) @.J=@jJ-J. (if s = + + i l , (7.101) is another expression of the result of Theorem 2.2: V j J z VJ-'). By (7.101)7(7.100) can be rewritten as a j . 1 - n - f - z(n+j-l) + @n+j + 0 - n - j (7.102) -t This relation can be interpreted as follows. The representation VjJ-"-j leaves invariant the (2n +2j- 1)-dimensional subspace Q(j, n j ) l of 8 and induces a representation equivalent to F2(n+j-1) on Q(j, n + j ) l . The representation induced by VjJ-"-j on the quotient space Q/@(j,n + j ) l is equivalent to Un+J@U-"-jafter extension to a suitable completion of a dense subspace.

+

$8. Inversion formula As for the groups SU(2), R" and M(2) treated in the preceding chapters, we can develop the theory of the Fourier transform on the group G = SU(1, 1). Let 6 be the dual of G. This means that 6 is the set of all equivalence classes of irreducible unitary representations of G . We can choose a representation U Afor each class 2 in 6. Then the Fourier transformf of a function f in L1(G)is defined by the formula n

Thus f is an operator-valued function on 6. We can expect that the original functionf may be recovered from its Fourier transform f,at least for a function f in s ( G ) . Similar inversion formulae for the groups SU(2), Rn and M(2) suggest that there exists a measure p on 6 such that

344

UNITARY REPRESENTATION OF

SL (2,R)

In fact such a measure exists and is given as follows. Put fcj, s )=

1

f ( g )V , - i W g = V f

a

and

f ( n ) = / f(g)Un,-ldg= Ufv n , where f"(g)=f(g-l). Then we have +f(g)= Tr (f(0,i (8.3)

&Io

+1

+iv) Vgo*++ru)~ th

K V ~ V

+

(2n -1) {Tr( f ( n )U,") Tr (f(-n) U,-")} nta-12 nS1-

for every f e g ( G ) . Hence the support of the Plancherel measure p is equal to the union of the principal continuous series and the discrete series. The complementary series do not contribute to the inversion formula for the functionsf in s ( G ) or L2(G).Putting g = 1 in (8.3), we get +(8.4) f(l)=&I0 Oo.*+fu(f)v thxvdv

where W'(f) = Tr( Vf

=

f(g-')W'(g)dg

and

/a

-s.

W f )=T W f 1v

"

-

f(g-l)@n(g)&

The functions W 8and 8" are the characters of W aand U n respectively (cf. $7). The general formula (8.3) is obtained from the special case (8.4) by replacingfwith L,-lf. Hence we study mainly the formula (8.4). In the study of harmonic analysis on the group G = S U ( l , l), the integral formula in Proposition 7.13 plays an essential role. This integral formula gives a decomposition of the Haar integration on the group G into two stages; one is the integral (the mean) over each conjugacy class consisting of semisimple ( =diagonalizable) elements; the other is Haar

345

INVERSION FORMULA

integration over each of the Cartan subgroups H and K with the weight functions 2(sht/2)' and 4(sin 6/2)* respectively. In virtue of this integral formula, the Fourier transform {W'cf), P c f ) } off E g ( G ) is obtained by first taking the means Ff and Ff on the hyperbolic and elliptic conjugacy classes, and then carrying out the Fourier transforms on the abelian Lie groups Hand K.We shall see later that the derivative F f ' ( 6 ) of the mean FfKgives the value f(1) off at the identity element 1, and that consideration of the jump of Ff=(6) at 6 =O gives a relation between Ff and FfK.Then these two facts and harmonic analysis on A E R and K s T give the inversion formula on G. The results and the proofs of this section are due to Harish-Chandra [2], [8]. First we give two Lemmas which will be decisive in the proofs of the above stated results.

f(6e', 6e-t)(eC-e-')dt,

g(6)=6

(820).

J O

Then g is a C-function on R - {0} with compact support and satisfies lim-(O)= dg 0-0 de

-2 f(0,O).

In particular dgld6 can be extended to a continuous function on R. Proof. Since f belongs to g ( R a ) , g is a C"-function on R - (0), and we have

ss

f (Be', 6e-')(ec-e-')dr

g'(e)=

{e%+e-'fv} ( W ,ee-')(e*-e-')dt

+\:B

=J +J

e-tf dt

( e ' f + eeslf,)dr-j 0

0

(6fv-6fs-6e-2t

fv)df for any 6 2 0,

0

lo rn

where fs = aflax and fy =af/ay. Since 1f(x, y)]g ]Ifl]-

and

we have

(8.6)

limj e-C f(Bet, Be-')dt =f(O, 0) 0-0

0

by Lebesgue's dominated convergence theorem. Similarly

e-Vr = 1,

346

UNITARY REPRESENTATION OF

SL (2,R)

Since the support off is compact, there exists a T>O such thatf(x, y)=O unless Max(lx1, Iyl)s T. Then f,(Be', Be-')=O ( t 2 0 ) unless lelet$ T. Hence

Therefore we have

J

f,(eet, ee-')dt=O,

lim 8 0-0

0

because lim 0 log 1191=0. Similarly 0-0

rm

lim 0 8-0

J

f=(Bec,Be-')dt = O . 0

Put x=Be' and y=Be-'. Then we have

and

Since +/;$(xf)dt=

B-l[(xf)(Se',Be-')];= -f(B,

0) ,

we have

Summing up these results we get Lemma 1 .

LEMMA2. The function g in Lemma 1 satisfies

Proof. Let O<e$T and Bef=x. Then

q.e.d.

INVERSION FORMULA

Since If(x, Px-')lS Ilfll-,

(8.8)

347

we get

lim 01; etfdt =/:~(x, 0)dx =/"f(x,

R*+O

o)&.

By (8.6) and (8.8), we get the first equality in (8.7). If 8<0,then e/retfit

=/->(x, 0

o)&

~x-')dx-/->(x, 0

as 8- -0.

q.e.d.

Delinition 1. We denote by go(@ the set of all functionsf in g(G) satisfyingf(x) =f(x-') =f(uxu-') for every x E G and for every u E K.

Proposition 8.1. For each functionfin g o ( G ) ,put (E

= f1)

,

where dg and dg* are G-invariant measures on GIH and G / K (cf. Proposition 7.6 and 7.12). Then the function F f K is a C"-function on R-2aZ and its graph has jumps at e =0 and 8 =28. The jumps are given by (8.9) F f K ( + O ) - F f K ( - O ) = ~ F f ( l )and (8.10) FfK(2r+ 0) -FfK(2z-0) = -A F ~-(1 ) . ProoJ Since K is an Abelian subgroup and a null set in G, we have

by Lemma 1 in $7.By Proposition 5.3, the last integral is equal to (8.11)

.

2n/~f(a,u,~~-~)shtdf

Since

and f E g o ( G ) has the compact support, there exists a constant M>O such that f(atueat-l)=O if Itl>M and e # 2 d .

348

UNITARY REPRESBNTATION OF

SL (2,R)

Hence the integral (8.11) is convergent and FIK(R) is a Cm-functionon If 0 = 2 n s (n E Z), then the integral (8.11) R - ~ ~ Tfor Z eachf in go(@. is divergent unlessf((- 1)")=0. By (8.12) and Lemma 2, we have

Ffg(+0) = lim sin (e/2)H(6) =2-1 lim 0 H(0) e-+o

8-+O

Similarly we have

FlK(e+ 2 ~ ) =-F,lK(e)

(8.13)

wherefi(x) =f(-x), we have (8.10).

Proposition 8.2.

For each f E .go(G), put d G f ( 6 ) = ~ F 7 ( 0 ) , 0 E R-2nZ.

Then Gf(R) is a C--function on R -2 x 2 and can be extended to a continuous function on R. If the extended function is also denoted by GI, we have Gf(0) = -rf( 1) and G,(27~)=sf( -1) . Proof: By (8.11), (8.12) and Lemma 1, we have d -2nf(l)=lim-(~H(~)) e-o de

d d8

=2 lim-(sin (@/2)H(@)) =2 lim GI(@ e-o

.

By (8.13) and the above result at 8=0, we have limG,(O)=zf(-l). 3-9. Hence lirn GI(@ exists and is finite for each n E 2, and G, can be extended e+'.mt

INVERSION FORMULA

to a continuous function on R.

349

q.e.d.

The derivative dG f / d e=daFfy e ) / d e 2is discontinuous at e = 2nx (n E Z ) , but it has finite limits when e tends to 2 n x 4 0 . Hence GI is a piecewise C1-function. To prove this fact, we need the notion of "radial part" of the Casimir operator. Although this notion may be defined in a general setting, we restrict ourselves to give the necessary minimum. (For the general definition and properties of the radial part of a differential operator, see Helgason [6].) Definition 2. Let w=XOa-Xl'- Y' be the Casimir operator on G = SU(1, 1). For each function 4 in g ( K ' ) , put f(gusg-')=@(ue).Then f belongs to C"(G,'). Let (of)' be the restriction of wf to K'. Then the is a linear transformation in C"(K') and mapping d ( w ) : $-(wf)supp (d(w)$) c supp (+). Hence d(w) is a differential operator on K' which is called the radial part of o on K'. For each f in g ( G , ' ) , put (8.14)

f(gusg-')dg*.

F(B)=/

GlR

Then F belongs to s ( K ' ) . Since 0 is invariant under the left and right translations La and Rg, we have i3lR

Proposition 8.3. Put a(@)= 2sin(B/2) and L= da/dea.Then we have d ( w ) =6-' L 6-4-', (8.16) where 6 in (8.16) implies the operator f~ 6f , and 0

(8.17)

0

F . f K ( ~ ) = ( ~ + 4 - 1 ) ~ f ~ ( ~ )for all f e g o ( @ and 6

E 2aR.

The second derivative daFf(@)/dB1 has finite limits as B-+2nxfOfor each n E 2.In particular, Gf(B)=(Ffg)'(B)is a piecewise C1-function on R. Proof. Since o =Xol X12 Y', A(@) is a second order differential oper-

-

-

ator on K' whose highest term is equal to Xoa=da/de'=L,i.e. (8.18)

d ( w ) =L

+lower order terms .

350

UNITARY REPRESENTATION OF

SL (2,R)

Let d ~ ( G 2 'be ) the set of all real-valued C"-functions on Gt' with compact supports. Then we have (8.19) (of,&) =(f,4) for everyf andji in d ~ ( G s ' )where , the inner product is given by (fi,A)=

lGs,fi(g)A(g)dg. ~f fi satisfies fi(gu,gg-l)=fi(u,) for every g E G and

U ~ E K 'we , have by (8.14), (8.15)and Proposition 7.13

inner automorphisms, we and products are defined by ($1,

$l(us)$a(us)6(B)2d8. The mapping fH F is a surjective

+a)=('Ir)-'J 0

mapping from d ~ ( G s 'onto ) d a ( K ' ) . (The proof is entirely similar to that of Lemma 1,l) in 97.) Hence&) is a symmetric operator. Since 6-10L06is also a symmetric operator whose highest term is equal to L, d(o)-6-10 Lo6 is a symmetric operator of order S 1. Since there exists no symmetric differential operator of order 1, d(0)-6-~oL06 must be a function c. Operating with d(0)-6-~oLo6 on the constant 1, we get c=d(o)1-6-'L(6)=(01)-+4-'=4-'

.

we have Hence we have d(0)=6-~0L06+4-'. For each f in go(@,

LK

Fe,K(e) =2 - 1 ~ ) (of)(gusg-l)dg*

+

=2-1s(e) ( A ( ~ ) F (e)) =2-1(~6 4 - 9 ) ~ ( e )

By Proposition 8.1, both F,g(B) and F-,"(e) have finite limits when e-+ 2 n r f 0 (n E Z), hence the second derivative (Ffg)"has the same property by (8.17). Thus (Fig)"is a piecewise continuous function and G, =(Fjg)' is a piecewise C1-function on R. Definition 3. For each function f in d ( G ) =

cf : G-CI f is a C"-func-

351

INVERSION FORMULA

tion with compact support},put

As we shall see later, the different parts of the inversion formula correspond to f + and f - . Hence it is convenient to study f + and f - separately. These two parts are analogous to Fourier cosine and sine series which correspond to even and odd functions respectively.

+

Put s =t i;l and

Proposition 8.4.

p*(t) =~,,(a) =e ~ / ~ / ~ ~ f * ( u t n . ) ~ .

Then for every f E go(@, we have (8.20) W'( f - ) =@'**(f+) =o , Do**(f + )=

(8.21) (8.22)

G***(f-) =

1:. S

p+(t)e*Wt and (p-(r)e"tdt

.

-0I

Proof.

By Proposition 7.5, we have

{l m

o j - s ( f )=2-1

e"tF,(ut)dt+ (- I)s~[

-m

e"CF,(--a,)dt)

-_

.

for every f s go(G), where j E {0,+} and s=++i17. Since f*(-ut)= ff(ut) and F l ( f u t ) = fF,(ut), we see that (8.20), (8.21) and (8.22) hold. Definition 4.

For each n E 2-'2 and n r 1 , and for each f E S ( G ) , put D"( f

Proposition 8.5.

(8.23) (8.24)

)=en( f ) +e-yf ) .

For each f E s o ( G ) , P ( f - ) = O if n~ 2 and Dn(f+)=O if n s f . 2 - 2 .

Put

F*(O)=

1

QlK

f*(gusg-')dg*

.

Then for every f s SO(@ and n E 2,n 2 1, we have

352

UNITARY REPRBSBNTATION OF

-2n-l

SL (2,R)

1:.

sin (n-+) e sin+c?F+(B)d@

and (8.26)

On++($) =2

e-n5p-(t)dt J

O

s:.

-27~-1

sin no sin +8F-(B)dB .

By Theorem 7.4, the distribution D" is equal to a function e-(n-+)ltllsh+tl-l(- l ) 4 n j if g=go(- l)afutgo-lE GI' , D"(g)= -sin(n-+)O(sin+O)-' if g=goueg0-l E Ga',

Proof.

lo

if g E G - G .

Hence we have by Proposition 7.13 D"(f>=2fme-(n-+)tsh +t

+ 2(-1)sn[e-(n-h)t

Hf(gatg-l)dg

sh+rdt/

f(g(-at)g-1)dg

QfR

By Propositions 7.6 and 7.7 and by the assumption that f(uxu-l)=f(x), we have

We treat the case 1 ) n E Z and the case 2) n E + Z - Z = {nlnE $2 and n 4 Z } separately. 1) n E Z . Then (- l)sn= 1 and f,(g(-a)g-l) = &f,(gatg-l). Hence for Dncf+) (or Dn(S-)), the first integral in (8.27) is equal to (or cancels) the second one.

INVERSION FORMULA

353

+

Since sin (n-3)e sin +e =+[ -cos no cos (n- 1)8] has the period 27~and f*(gUb+lxg-l)=f*(-gusg-l) = ff*(gubg-’) , the third integral in (8.27) is equal to

(8.28)

Hence we have proved (8.23) and (8.25). 2) n E 3 2 - 2 . In this case (-l)2n=-l and a(e)=sin(n.-$)ex Hence we can prove (8.24) and sin38 satisfies a(8+2n)=--a(B). (8.26) similarly.

Proposition 8.6. Put G+=GI, for each f E 9 ( G ) and

1:.

G+(O)cos(n+3)8dB,

c~=zc-’

g , = .-1[

-.

G-(8) cos n8 d8 for each n E N,

and put

Then we have

(8.29) and m

(8.30)

G-(O)=

C

6ngn

.

n=O

Proof. By (8.28), G-has the period 2n and G+has the period 4n. Hence A ( @= ) G+(B)cos38 has the period 27~.Since both Ga and A are continuous and piecewise C1-functions (Proposition 8.3), they can be expanded in Fourier series which are everywhere convergent (Ch. I. Theorem 5.1). Since G- is continuous at 8 =0 and is an even function, we have G-(0) = m

C &gn. Similarly we have

n-0

(8.31) where

354

UNITARY REPRESENTATION OF

SL (2,R)

an=x-l[G+(@) cos 319cos no do .

Since B(0) = G+(B)sin+e is an odd function with the period 28, we have m

0 =B(0)=

(8.32)

1bn ,

where

n-1

bn=r - 1 1 G+(e) sin 30 sin no de . -x

Since

an--bn=cn

-

and an+bn=cn-1 for all n2 1, we get

C

G+(~)=+co+

n=l

m

Cn=+co+

C

Cn

9

n=O

by taking the difference and sum (8.31)+(8.32). Since the series Ccn m

ca

is convergent, we conclude that co=0 and G+(O)= C cn = C cn . n-1

n-0

Proposition 8.7. For each functionfin ~ o ( G )we , have

Proof. Put Ht=F,&:lag.Then Propositions 8.2, 8.6 and 8.1 and integration by parts show that

Similarly we get (8.34) by integrating (8.30) by parts.

q.e.d.

355

INVERSION FORMULA

LEMMA 3. For any 1)

/:@+)-'sin

3)

/::thysin

A E R,we have

At dt =2n th xA ,

Atdt= 2r cthA r ,

t

t d t cthye<"dt = 2x-(cthxR), d2 1) Put I = S_(sh+)-'

Proof.

(the integral of a distribution) sin At dt and f(z) =etAg/sh(z/2). Then

f is meromorphic on the whole z-plane and has its simple poles at z=2nri (n E 2).We integratef(z) along the contour C in Fig. 6. 2 Ti

-

ir

"

c

Fig. .6

Since sh(z/2) is an odd function, we have lim

R-+m, r*O

(I:', + /;)f(x)dx

Since f(z +2 4 = -e-aslf(z), we get ( / r + l d +/ - R + l d ) lim R*+-* r-0

R+aA

= iZ .

f(z)dz= ie-'x'Z.

-r+ht

The inequality If(&R+ jy)l 5 2 ezxl2l(eRfa-e-R"1-' , 2 n 2 y h 0 , implies that

s.

R+lrt

+ .

-0

and -R+lni

when R-, co Since z =0 is a simple pole of f(z), the Laurent expansion around 0 has the form

356

UNITARY REPRESENTATION OF

f(z)=

SL (2,R)

>+ao+a1z+ .... Q

Hence the integral on the semicircle around 0 is equal to a/: id8= -da= -xiRes f ( z ) = -2wi. r-0

Similarly, the integral on the semicircle around 2xi is equal to -ni Resf(z) =2 ~ i e - ~ ~ l . .-Zd

Summing up the above results and using Cauchy's theorem,

we get

+

i(1 e-~r~)l=2ni(l-e-3*~) and t=2nthwA.

"(

2) Since a1 (sh+)-'sin

I f ) = t (sh+)-'

cos At and t

over R,we can differentiate 1) under the integral sign and get

1:-

t(sh t/2)-l cos At dt = 27cd(th nA)/dA.

Since t(sh t/2)-' is an even function, we get 2).

1:-

3) Put J=

erlccth t/2 dt and g(z)=e*llcth (42).

The poles of g(z) are same as those of f(z) in 1). We integrate g(z) along the same contour C as in 1). In this case g(z 2 4 =e-l=lg(z), Res g(z) = 2 , Res g(2)=2e-ax2

+

S-0

a-lrt

.

Hence we have (l-e-an2)J=2ai(l +e-z*2) and J=2x cth nR . 4) We can differentiate 3) under the integral sign and get 4). Proposition 8.8.

For any f in g o ( G ) , we have

357

INVERSION FORMULA

I::+*.

(8.36) ProoJ

f)R cth X R d~

++(I(

Put and @ L 2 ( f ) = @ i * + + t 2 ( f - ) .

@+l(f)=@O~++~~(f+)

Then by Proposition 8.4, we have

1-

~ f= )

pa(t)e(a'dt

.

-01

Since (p+(t)=F,,(at) belongs to s ( R ) , we get S_p.'(t)e(lldt=

-iAQal(f)

by the integration by parts. Put

+,(A)

=

1-

t-lp*'(t)e$Jtdt.

-OD

Since p+ is an even function, p*' is an odd function and t - I p+'(t) belongs to g ( R ) . Hence belongs to 9 ( R ) (Ch. 111, Theorem 1.2) and can be differentiated under the integral sign. We have

+,

#*'(A)

=i

1:-

p*'(t)eWt .

Since t - l ~ + ' ( t belongs ) to y ( R ) and t(sh$)-l

belongs to L1(R),we

can apply the Parseval equality for the Fourier transform on R (Ch. 111, Theorem 1.3), and get d t-'p+'(t) t(sh t/2)-'dt= ++(A)z(thnR)dR

/Im

jm -_

=-

1:-

j:-

++'(A)thddA = -

R th ZR ~ D + ~ ( f ) d 2 .

Similarly, we get t-'p-'(t)t cth t/2 dt = -

Proposition 8.9. For any f E go(@, we have

and

358

UNITARY REPRESENTATION OF

277f-(1)=

(8.38) Proof.

fb

SL (2,R)

m

@-'(f)IcthxRdR+ CnD"+'(f-). n-1

By Proposition 8.5 and integration by parts, we get

r-

m+l

+

1:

v+'(t)(sh t/2)-ldt

=2lrf,(l)-+f-

-m

@ + ~ ( Jth) dR d . ~

The last equality follows from Propositions 8.7 and 8.8 and the fact that q+'(t)(sh t/2)-' is an even function. Since V j .*+%a z Vj* (Theorem 2.2), we have O*^(f)=cD*-^(f). Hence the last expression in the above sequence of equalities is equal to 277J+(l)-/m@+2(f)R 0 thxRdI. We have proved (8.37).

s:

Put D * ( f - ) = 2

p-(t)dt. Then we have

INVERSION FORMULA

{

m

=2 lim (m++)(o+(O)-2n-'E m-+"

359

n r sinnOH-(e)de} n-1

-r

+jLp-'(t)cth-dtt 2

Dewtion 5. For each functionf in L1(G),the Fourier transform f of f is defined as the operator-valued function on = {(j,3 + iR) I R >0, j E {O, &I1 u {nln E 2-'Z, In1 2 1) given by

e0

f ( j , &+in)=

f(g)V,-if.*+*Adg=vjvj,++*2 So

and

f(n)=/

f(g)U,-lndg= UIvn,

1 where the Haar measure dg is normalized as dg=-eecdedtdf. 47r

Theorem 8.1. (The inversion formula for d ( G ) ) . For each function f i n g ( G ) , we have (8.39)

++a)Vgo,++*l)l thddr2

f ( g ) = z1/ " Tr(f(0,

+

Tr(f(+,+ 0

1 +-4nnr%-le E (2n-1)

+ il) V,*.*+*')A

cth n l dR

{Tr(f(n) V,")+Tr(f(-n)

V,-")} .

n2l

Proof. First we assume thatfbelongs to go(@, i.e.,fsatisfiesf(um-1) = f ( x ) =f(x-l) for all x E G and for all u E K. Then by Proposition 8.4, we have Tr(f(O,+

+ 9 ) ) =O0*+ + c l ( f=) O0* (f+) *+*l

=@+^(f).

Similarly

+

Tr(f(&, & iR)) =@ - l ( f ) By Proposition 8.5, we have Tr(f(n))+Tr(f(-n))=

.

K(f+) D

if n E Z ,

(f-) if nE3.Z-Z

360

UNITARY REPRESENTATION OF

SL (2,R)

Hence by adding (8.37) to (8.38), we get

(8.40)

f(1) =

$1:

Tr(f(O,3 +il))R th nR d2

1

1

+& n r t e . n p i

(2-

1)(Tr(f(4)+Tr(f(--n))

for all f e ~ o ( G ) . Now we turn to the general case and put f"(g)=J

K f(kgk-')clk.

Then we have Vh-lj.JV,f.*Vhj.*&.

V,OhJ= /K

Since V,j**is of trace class (Theorem 7.1),

-

(V,j**V&*fP,

=Tr

Vlj.Jfp)

V,jTJ

p--m

for every k E K and the series is absolutely convergent. Hence we have

=IK

Tr(VfVj.*)dk=Tr(Vjvji~)=Tr(f(j,s)).

Similarly, we have Now put f"(g)=f(g-l).

Trf(n)=Trf(n) . Since W(*a+) =P( +a,) ,

by Theorem 7.2, we have W(g-') =W*(g) and

By Theorem 7.4, we see that

36 1

INVERSION FORMULA

Hence we have Tr((f")W) =Tr(f(-n))

*

Put

h =.)(P +P") Then by the above results, we have Tr h(j, s) =Trfu, s) (8.41)

.

and Tr h(n)+Tr h( -n) =Trf(n) +Trf( -n)

(8.42)

.

Since h belongs to so(G), (8.40) is valid for h. Hence by (8.41) and (8.42), (8.40) is also valid for$ We have now proved (8.39) for every f in 3(@. Put (L,f)(x)=f(g-lx). Then we have

(L, f )"(j, s) = f(j,s) Vg-lf*'and (L, f )^(n) =f ( n ) V,-l . Therefore, we obtain (8.39) by replacingfin (8.40) by L,-$ q.e.d.

Tbeorem 8.2. (Parseval equality). For every f in s ( G ) , we have

where I] Proof.

[I, means the Hilbert-Schmidt norm. Put a= f * f *

where

f*k) =f(g-').

for each f ES(G)

Then u(1)=JG I f(g)lVg ,

Tr 40,s) =Tr(f(j, s ) f ( j ,s)*) = IIfG, s)ll2 for s = 2 4 + i 2

and Trd(n) =Tr(f(n)f(n)*) = Ilf(n)llla .

Hence equation (8.40) applied to u gives us Theorem 8.2.

q.e.d.

362

UNITARY REPRESENTATION OF

SL (2,R)

99 Harmonic analysis of zonal functions Definition 1. A function f on G=SU(l, 1) is said to be two-sided Kinvariant or a zonal function, if it satisfies f(kgk') =f ( g ) for each g E G and k, k' E K. The set of all complex valued zonal functions on G is denoted by A. In this section, we study Fourier transforms of zonal functions. In particular the images of the spaces of C"-functions with compact supports and of rapidly decreasing functions are determined. Let n be a half-integer (i.e. n E $2)and put xn(ue)=elne

The set of all C-valued continuous functions on G with compact supports is denoted by YG). Proposition 9.1. Let U be a Banach representation of G=SU(l, 1) on a Hilbert space H such that u k is a unitary operator for each k E K and fn andf, be two vectors in H satisfying (9.1) Utft=xi(kYi for each k E K (i=n, m)

.

Then the operator

Uf =/Gf(g)uGd.g for a functionf in L (G) n A satisfies

.

(Uff n ,f,) =0 unless (n,m)=(0,O)

Proof.

Sincef is two sided K-invariant, we have (uffn,fin) =

1

f(k-lgK-1) (u a f n fm)dg 9

Proposition 9.2. If the restriction to K of a unitary representation U of G does not contain the identity representation xo, then U,=0 for every function f in L(G) n A. In particular, f(+,s)=O for every S E C and f(n)=O for every nE+Z,InlBl. Proof: The representation space H of U has an orthonormal basis

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

363

Cfr)rcr satisfying (9.1). Hence the first part of Proposition 9.2 is a direct consequence of Proposition 9.1. The last half is clear from the first half and Theorem 6.2. Proposition 9.3.

Put

fn(c)=cn

for ~ E (lCl=l). Z Then we have

( V , o ~ ~ ~=,0f unless ~ ) (n,m)= (0,O)for every f E L(G) nA and s E %. Proof. Clear from Proposition 9.1. By Proposition 9.2, the support of the Fourier transform f of a zonal functionfis contained in the set { Vo.++321120}. By Proposition 9.3, the operator valued Fourier transform f ( 0 , s) of f E L(G) n A is reduced to a C-valued function ( Vlo~*fo,fo)which will be denoted by {(s). The inversion formula for a function f in g ( G ) n A can be written in terms of the scalar valued Fourier transformf(s). In fact we have the following proposition.

el=

Proposition 9.4.

Iff belongs to s ( G )n A, then we have

+f ( g ) = k j f(++il)#(g, ++il)AthxldR for each gEG

Proposition 9.5. The function #(g, s) defined by (9.2) satisfies (9.4) #(kgk', s)=#(g, s) for every k,k' E K and g E G, P

364

UNITARY REPRESENTATION OF

SL (2,R)

Delinition 2. The function $(g, s ) is called a zonal spherical function. Proof. Sincefo satisfies (9.1) for i=O, 4 satisfies (9.4). By the definition of Vg0.*,we have

(vgO,* f)(C)=e-*t(o-'.C)f(g-'.C) * Hence +(g, s)=

e-*t(g-l.c)dC

(~>ssfo,fo)=/ l7

q.e.d. Proposition 9.6. I f f belongs to L1(G)n A, then the Fourier transform f(s) is given by (9.6)

f ( s )=

[Ff(t)e(*-'%ft for Re

s=

1 3

-m

in terms of the Radon transform Ff ( t )=et"

L-

f(atne)& .

Proof. By Proposition 9.4, we have

n

n

q.e.d. Dewtion 3. The transformation :f H f l defined by (9.3) is called the spherical transform on G. By Proposition 9.6, the spherical transform f-f on the integrable zonal functions is decomposed into the Radon transform f - F f and the

365

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

ordinary Fourier transform on the real line (PA). This decomposition enables us to determine the precise image of g ( G ) nA.

LEMMA1. 1) If v E C ( R ) is an odd function, then there exists an even function E C"(R) such that

+

.

v(t)=t+(t) for every t E R 2) Iff E C"(R) is an even function, then the function,

+

g(x)= f(&) is a C"-function on [0, a). 3) The function k(x) = [log(x J k T ) ] * belongs to C"([1, 4) Iff E C"(R) is an even function, then the function h(x)=f(log ( x + J m ) ) belongs to C"([l, a)). Proof. 1) It is sufficient to put

+

+

03)).

+

+

2) It is clear that g is a C"-function on the open interval (0, a). Hence it is sufl6cient to show that g is differentiable infinitely many times at x=O from the right. Put x = P . Then we have (-$>.,(x)

=

zyf(f),

(&

for every k E N and x >0.

Sincef is an even COD-function, the right hand side of the above equation is also an even C"-function. Hence by l), the limit

exists and is finite. Hence g is differentiable k-times at x=O from the right. Since k is an arbitrary integer 20,g belongs to C"([O. 03)).

+

t

3) Put f(t)=xf. Then f is an analytic function on a neighborhood of t = O . Put

Then I(x)=f(t) and

Hence I belongs to @"([1, + a)) by an argument similar to that in 2). Hence k ( x ) = [ l o g ( x + , / ~ ) ] 2 = ( x 2 - 1 ) I ( x ) fbelongs to C"([1, a)).

+

4) Since h(x)=f(t)

=g(t ') = (g k)(x), 0

366

UNITARY REPRESENTATION OF

h belongs to C"([l,

+ a)).

SL (2,R) q.e.d.

Definition 4. Let C+"(R) be the set of complex-valued, even C"-functions on R and 9 + ( R )be the subset of C+"(R) consisting of functionswith compact supports. For each function f in C+"(R), we put f [ x ]=f(cht) for x =cht . By Lemma 1, the mapping f(t)t+f[x] is a bijection from C+"(R) onto C"([l, w)) and maps 9 + ( R ) onto 9+([1, a)).

+

+

Proposition 9.7. If the functionf belongs to C"(G)n A, then the function d t )=f(at) belongs to C+"(R). The mapping f ~p is a bijection from C"(G) n A onto C+"(R) and maps g ( G ) n A onto g + ( R ) . Proof. Since the Iwasawa decompositiongnusatncis a diffeomorphism (Proposition 1.4), (@,t, c) can be take as coordinates on G. Hence p(t)= f(at)is a C"-function on R. p is an even function by Proposition 2.11, 2). Since every element g in G can be written as g=u,at,u, (proposition 5.2), a function f in A is uniquely determined by its restrictionflA to A. Thus the mappingf-p is injective. Let p be a function in C+"(R). Put (9.7)

Then f is a well defined function in C"(G) n A. In fact, the expression g= u,atu, (O
In the following, we identify f E C ( G ) nA with (p(t),, and put f[x]= f(aJ where x =cht.

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

Proposition 9.8.

367

The Radon transform R :f HFl(f)=etfa/Im f(utnt)dE

(9.9)

is a bijection from g(G) n A onto g + ( R ) . The mapping S inverse to R is given by

Proof. Since F,(t)is an even function by Proposition 7.14, F,(f) belongs to g + ( R ) .Put x=cht. Then by (9.7) and (9.8), we have

(9.11)

FZ[x]=ecia

Differentiating the above equation with respect to x, we get

Ilcnce we have

We have proved that SRf= f for each f E d ( G ) n A

.

Now let F(f) be an element in . g + ( R ) . Then F[x] belongs to d([1, + w)) by Lemma 1 and the functionf[x] defined by (9.10) also belongs to Cm ([I, a)). Hence the function f(t)=f[chf] belongs to . g + ( R ) and f can be extended to a function in s ( G ) n A by Proposition 9.7. The latter function is also denoted byf. Then we have

+

Therefore we have

RSF= F for each F E g + ( R ) .

q.e.d.

COROLLARY to Proposition 9.8. The Radon transform R maps the set { f c g ( G ) n A l f[x]=O if x>a} onto { F ~ g ( [ l+, a ) ) [ F [ x ] = O if x>u).

Proof.

The corollary is clear from (9.11) and (9.10).

368

UNITARY REPRESENTATION OF

SL (2,R)

By Propositions 9.6 and 9.8, we obtain the following analogue of the Payley-Wiener theorem for the spherical transform. Theorem 9.1. Let E be the set of all C-valued functions p(s) of a complex variable s satisfying the following three conditions: 1) p is an entire function. 2) For each m E N,smp(s)is of exponential type, i.e. there exists a constant C, and a>O such that (9.12)

l(1 -tIsl')"p(s)l

s C,,,eolBeal for all

s 6C

3) p satisfies the functional equation (9.13)

p(1 -s)=

p(s) for all s E C .

Then the spherical transform

Y : f 4 M=/S(g,4.-:

s)&

maps 9 ( G ) n A onto E bijectively. For a fixed constant a> 0, the spherical transform p=fof f in g ( G ) n A satisfies (9.12) if and only iff satisfies (9.14) f(ac)=O for Itl>a. Proof. Iff belongs to 9 ( G )n A and satisfies (9.14), then the Radon transform F&) off belongs to g + ( R ) and satisfies F,(t)=O

for Itl>a

by Proposition 9.8 and its corollary. Sincef ( s ) (more precisely (2r)ff(i(3-s)) is the Fourier-Laplace transform of I;, E g+(R)(cf. Proposition 9.6), thePayley-Wiener theorem for the real line (Ch. 111, Theorems 4.1 and 4.2) proves that f ( s ) is an entire function of exponential type and satisfies (9.12) for each m E N. Since F,(t) is an even function, we have f(+-i1) =

1

-_~ , ( t ) e =~[?(t)e-"Cdt=f(+ t

+i l ) .

Hencef satisfies (9.13) on the line Res= 1/2. Since the both sides of (9.13) are entire functions of s. f satisfies (9.13) for all s E C. We have proved that fl mapps g ( G ) nA into E. 9-is injective by Propositions 9.8 and 9.6 (or by Proposition 9.4) Now we shall prove that Y is surjective. Let p be a function in E satisfying (9.12) for a fixed a>O. Then by the Payley-Wiener theorem (Ch. 111, Theorem 4.1) there exists a function F in g , , ( R ) such that l?(l)=cp(++i;O for all 1 E R. Since cp

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

369

satisfies the functional equation (9.13), F is an even function of t. Hence by Proposition 9.8 and its corollary, there exists a function f ~ g ( G ) nA satisfying (9.14) and Ff=F. Thus we have proved that f= 9. q.e.d. The next aim of this section is to establish the uniqueness of the inverse spherical transform, which will be done in Theorem 9.2. For this purpose we give some elementary properties of the zonal sphericalfunction $(g, s). Most of these properties are valid for any non-compact semisimple Lie group G and a maximal compact subgroup K of G. Proposition 9.9. The zonal spherical function $(g, s) =(Vgo.yo, yo)has the following properties: 1) For each fixed g E G, $(g, s) is an entire function of s.

2) $(g, s) =$(g, 1 -s )

for every s E C and g E G,

3) $ k , s) = $ k 9 3)

for every s E C and g E G.

4) $(g-l, s)= $(g, s )

for Re s= 1/2 and g EG.

Proof. 1) From the proof of Proposition 9.5, we see that (9.15)

$(g, ~ ) = / ~ e - ~ ~ ( dk. g-',~)

Hence $(g, s) is an entire function of s. 2) If Re s= 112, then there exists a unitary operator A on .fj=L4(U,dc) such that A V;J-' = V$'A for every g E G. (Theorem 2.2) Since V$'Afo=AV$l-#fo=Ah

for every k E K, Afo must be a scalar multiple of fo.Namely there exists a complex number c such that Afo=cJb, Icl=l.

Hence we have #(g, 1- ~ ) = ( V ~ ' - ' f o , f o ) = ( V ~Ah, ' Ah)

= ( v ; ~ y o , f o ) = ~ (s) g,

for Re s= 1/2. Since $(g,s) and $(g,l -s) are entire functions of s. $(g, 1 --s)=$(g, s ) for all s E C .

370

UNITARY REPRESENTATION OF

SL (2,R)

3) The expression (9.15) of #(g, s) gives 3) immediately. 4) Since Yo*'is a unitary representation when Res- 112, we see that

fW1,s)=(vv-~O.Jfo,fo)=(fo, V v O

~ ' h ) = ~ .

Proposition 9.10. Let s=o+i2 be a complex number in D = Is E C104 Re s$ I}. Then Proof.

ldg, s>ld ldg, dl d 1 for every g E G By (9.15) we have

1

I#(g, s>ls

.

e-*t(g-l.k)dk=+(g, a)

K

If 1/2 c a < 1, then there exists the complementary series representation V'. By Propositions 4.3 and 4.4, we have m

( Vg"fO,fO).=

1

=

~ n ( u ) (Vgo.*fo,fn)Cfn,fO)

(vo~*"f~,f~) = @(g,

0)

n=-m

Since V' is a unitary representation, +(g,0 ) is positive definite and satisfies (9.16) l $ ( g , ~ ) l s d 1 , ~ ) = 1forevery g e G (Ch. 111, Proposition 3.1). By the functional equation $(g, 1-s) = #(g, s), (9.16) is valid for O
f off

Proposition 9.11. 1) g ( G ) is dense in L1(G). 2) g ( G ) n A is dense in Ll(G) nA. Proof. 1) Smce the space L(G) of continuous functions with compact supports is dense in L1(G), it is sufficient to show that for each g E L(G) and E >0 there exists a functionf in g ( G ) such that (9.17) Ilf-gIll<E There exists a function h in a ( G ) such that Ilg-h*gll1-

*

(This part is proved in a way similar to that in the proof of Lemma 3,

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

371

Ch. 111, $1.) Thenf=h*g belongs to 9 ( G ) and (9.17) is proved. 2) If g belongs to L1(G)n A and E >O, there exists a function f E 9 ( G ) satisfying (9.17). Put (9.18)

f"(x)=

11 K

f(kxk')dkdk'.

K

Then we have f" E 9 ( G )n A and

=Ilf-gll1
*

Proposition 9.12. For everyfe Ll(G) nA, the spherical transform f of f has the following properties;

IfMl r:Ilf

1) (9.19)

f(1-s)

(9.20)

111

=f(s)

for every s E D = {slog Re s g lI} 2) f ( s ) is analytic on D = [slOtRes< l} and continuous on D. Proof. 1) (9.19) is a direct consequence of Proposition 9.10. (9.20) is follows from Proposition 9.9, 2). 2) By Proposition 9.1 1, there exists a sequence (fn)nrN in 9 ( G ) nA converging to f in L1(G). Then, by (9.19), the sequence (fn)nrN converges to f uniformly in D. Hence, by Theorem 9.1, f is continuous on D and regular analytic in the interior D of D. Remark. Proposition 9.12 shows that harmonic analysis of integrable zonal functions on G is very different from the classical Fourier analysis on R. Indeed the spherical transform f off in L1(G)n A is an analytic function on D and can vanish only on a discrete set unlessf(g)=O almost everywhere. Hence in particular there exist no functions of compact supports in the spherical transform of L1(G)n A. This fact was first noticed by L. Ehrenpreis and F. I. Mautner [l]. Proposition 9.13. (The analogue of the Riemann-Lebesgue theorem) Iff belongs to E ( G ) n A, then

lim f ( a + 2)=0 uniformly on D

Ill-+-

.

372

UNITARY REPRESENTATION OF

SL (2,R)

Proof. Since 9 ( G )nA is dense in L1(G)nA, there exists a function p in 9 ( G ) nA such that

Ilf--Colll<~/2 * Then we have (9.21)

~ f ( s ) - q ( sI)<&/2 for every s E D. By Theorem 9.1, Q satisfies Iq(s)lS C(1+ Is12)-1 for every s E D. Hence there exists a constant M >0 such that (9.22) l~(u+U)l
2)

/=$(xky. s)dk = $(x, s)$(y, s ) for every x and y in G and s E C.

3) Put $.(g) = $(g, s). Then

9. * f = f ( s ) $ , for every f E g ( ~ n A ). Proof. 1) I f f E .fj=La(U,dc) satisfies VkO" f=f for all k E K,then we have f = cfo, c E C (Corollary to theorem 2.1). Sincef E 9 ( G )nA satisfies f(k-'g) =fa, we have Vkos*Vfo"fO= Vz0.*fo for every k E K. Hence there exists a scalar E C such that Vr0.'f0=l(f y o. Since V,.:J= Vfo** Vgo,*for every f and g in g ( G ) , we see that 1 is a homomorphism of the algebra 9 ( G ) nA into C. On the other hand,

4f)=(VfO,ih f o ) =

SQf

(V:s'foyfo)dg

=(fYA(s) = C W )

Since (f * g)" =gv *f 2) By l), we have

SQ

=

-

",c, is an algebra homomorphism. $S(X)$*(Ylf(x-llg(Y-')dXdY

f(s)&)= (f*dA(4

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

373

for every f and g in s ( G ) n A. From Lemma 2 which follows below, we obtain 2). 3) By 2), we have

q.e.d. Remark. All the results in Proposition 9.14 are valid for every f in L1(G')n A in stead of s ( G ) n A without changing the proof.

LEMMA2. Put ( g , f )=/fff(x)g(x')dx

for each g E C(c) and f E L(G). 1) If a function g in C(G)n A satisfies (g,f ) =O ,ar every f in (G') nA, then g = 0. 2) If a function g in C(G)n A satisfies (g,f ) =O for every f in g ( G ) n A, then g =0. Proof. 1) For each f in L(G), let f" be the two-sided mean off over Kx K dehed by (9.18). Since g is two-sided K-invariant, we have (g, f ) = (g,fO)=O

for every f E L(G).

Replacingf by L,-g, we get (9.23) g*f =0 for every f E L(G) . Now we prove that (9.23) implies g=O. Let x be a fixed element in G and

374

UNITARY REPRESENTATION OF

SL (2,R)

(V,JnCNbe a fundamental system of compact neighborhood of 1 satisfy8 0 be a continuous function on G satisfying ing V,,2 Vn+l. Let f,,

supp (fn) c V,, and E

s.

f n ( x ) d x = 1. Since g is continuous at x for each

> 0 there exists an integer n E N such that Ig(xy-l)-g(y)l

<E

for every y E V,,

.

Since g*fn =O by (9.23), we have Ig(x)l= Ig(x)-(g *fn)(x)l

s

1

I(g(X)-g(xy-')l fn(y)dy 6

J'n

-

Since E is an arbitrary positive number, the last inequality show that g=o. 2 ) As in l), we have ( g ,f)= O for every f in .g(G). in .g(G) such Since for each f in L(G) there exists a sequence (f,,),,.~ that limIlf,*.f-f&,=O (cf. Ch. I11 $1, Lemma 3), we have ( g , f ) = lim(;Z *f)=0 because f n * f E s ( G ) . Hence g(x) =O for almost every n-m

~

E byG 1).

Proposition 9.15. For each s E C, the homomorphism C8 : f ~ f ( sof) g ( G ) n A into C is not zero. Namely for each s E C, there exists a function f in s ( G ) n A such that f ( s )# 0. Proof. Suppose that there exists a complex number s such that f ( s )=0 for every f in s ( G ) n A. This implies ($.,f)= O for every f e d ( G ) n A in the notation in Lemma 2. Hence we have $#=O. This is a contradiction, because $*(1)= 1. Proposition 9.16. Let OJ =Xoa-Xlz - Y a be the Casimir operator of G(cf. Proposition 6.12). Then we have (9.24)

o $ ~ = s ( ~ - - s ) ~for , every S E C .

Proof. We regard an element X in the Lie algebra g of G as a left invariant vector field. Then we have

375

HARMONIC ANALYSIS OF ZONAL FUN.CTIONS

Since Q =~ V O J ( Ois ) equal to s(s- 1) when Re s = 1/2 (Theorem 6.4), we have ~ # ~ = s ( s - l ) # ,for Re s=1/2. Since both sides of (9.24) are entire functions of s, (9.24) is valid for all s E C.

Proposition 9.17. Let cI be the homomorphism of 9 ( G ) nA into C defined by C 8 c f ) =f(s). If ca= Ct, then either t =s or t = 1 --s. Proof. Fix a complex number s. Then there exists a function h in d ( G ) n A such that &)+O (Proposition 9.15). Applying the Casimir operator o to both sides of the equation

9' * h = A(s)#8. we get I;(s)ogs=W(#a

*A)=#, * o ( h ) = ~ ( h ) ~ ( s ) g , 9

because ~ ( hE ).g(G) n A. Hence we have

(9.25)

O#I=i;(s)-'O(h)A(s)ga=Ta(O(h))ca(h)-l~a.

Therefore if C a =Cr we get

s(l-~)=Ca(~(h))Cs(h)-'=
by Proposition 9.16 and (9.25). Hence either r=s or t= 1 -s.

Definition 5. Let L= {++iJlJrO] and Co(L)be the set of C-valued continuous functions f on L which vanish at the infinity: limf (++ U)=0. .I++-

Co(L)is a Banach space with the uniform norm Ilfll..=suplf(x)l. Proposition 9.18. Let r={fly€s ( G ) n A}. Then r is dense in Co(L). Proof. Since g ( G ) n A is a subalgebra of L'(G) in which the product is defined by the convolution, it follows by Proposition 9.14 that r is a subalgebra of CO(L). r satisfies the following three conditions: 1) r separates the points in L (Proposition 9.17), 2) r is self-adjoint (i.e. if (0 E r then 3 E r),and 3) for each s E L, there exists (0 in r such that (0(s)#O (Proposition 9.17). As for 2), put f*(g)=f(g-'). Then we have ( f *>.(s)=E) for every s E L by Proposition 9.9, 4). Let E= L u { 1/2+ i 00 ] be the Alexandroff compactification of the locally compact topological space L. Then the subalgebra C1 +rof C(E) is self-adjoint, separates the points of , ! and contains 1. Hence by the StoneWeierstrass theorem, C1 +r is dense in C(z). Therefore for each f in Co(L) and E , there exist p E r and a E C such that I]f-(p+u)ll-<E. Since p and f vanish at infinity, we get la1 < E . Therefore Ilf-plI5 I]f -p--all,-t la1 t 2 e . We have proved that r is dense in Co(L).

376

UNITARY RBPRESENTATION OF

SL (2,R)

Theorem 9.2. (Uniqueness of inverse spherical transform). Let dpo(2) +i ~ ) l2n 0}.

= (2s)-lR th nl d2 be the Plancherel measure on L = {s=3 If a function h in L1(L,po) satisfies

,.

then h(s)=0 for almost every s in L. Proof. For each f in g ( G ) nA, we have

Since { f If E g ( G ) n A} =r is dense in Co(L), we have ,-

L

$(s)h(s)dpo(s)=0 for every $ E Co(L).

Hence, by a form of the Riesz representation theorem (cf. Appendix D.5), the measure h(s)d,uo(s)=0 and h(s)=0 for almost every s in L. q.e.d. In the last part of this section, we treat rapidly decreasing zonal functions Iff is a C"-zonal function, then by Proposition 9.7,f[x] =f(ut) (x=cht) is a C"-function on [I, + co) and C"(G) n A is isomorphic to C"([1, a)). Similarly, L1(G)n A is identified with L1([O, + co), 2nshtdt) by the m a p ping f n f ( t ) = f ( u t ) (Proposition 5.3) or with L1([l, -t- a), 2 n d ~ )by the mapping f n f I x ]. By Proposition 7.15, the operation of the Casimir operator o on the zonal functionf is given by

+

(of)(?) = -ctht dfli3t-i3wfldta= -(sht)-l at ( s h t 9 .

Definition 6. Let sp be the set of functionsf in C(G)nA for which t"sht(omf)(t) is a bounded function of t for every pair of integers m 1 0 and nLO. The topology of 9' is given by the family of seminorms pm,,,(f)= sup Itnsht(omf)(t)l for m,n E N. tdO,+=O)

and pm(f)=

SUP I(o"f)(t)l

trCO+")

for m c N .

Then 9 is a locally convex topological vector space. The map fnofis a

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

377

continuous linear map of 9 'into 9'. If cp is a C'-function on G and both 40 and Acp are bounded, then we have

for each f in 9' (The proof is the same as that of Ch. II(6.3)). Moreover if is a zonal function, then we have

(p

(9.26)

/+-(@f)(t)P(t)h(t)

= J 0l f ( t ) ( w m W )

Where dm(t)=2xshtdt. Now we introduce a function space 9' which will be proved to be the image of 9' under the spherical transform. Delinition 7. Let 9' be the class of complex valued functions F(s) defined on the strip D = {sE ClOjRe s j I} which satisfy the following four conditions; 1) F(s) is analytic in the interior b of D. 2) F(s) satisfies the functional equation F(s)=F(1-s). 3) F(s) is a C"-function on D. 4) For all m,n E N,the function

dm -"s"(l -s)nF(s)] drm

is bounded on D. The topology of

9 is defined by the family of seminorms

I

--[S"(I--S)"F(S)] for m, n E N . dSm dm Let (Fn)nr~ be a Cauchy sequence in 9.Then (Fa)converges to a function F uniformly on D. It is easily seen that F belongs to 9 and thus 9' is complete. Therefore 9 is a Frkchet space. We shall prove in Theorem 9.3 below that spherical transform fl ; f n f i s a topological isomorphism of 9 'onto 9'. qm,n(F)=SUP

ttD

Proposition 9.19. Let @(g, s) = (Vgo*'f,fo) and m be an integer2 0. Then we have the following properties:

for all trO and s=u+iA E C. 2) There exists a polynomial P(s) of s and for each compact set C in [0, + a), there exists a constant Mmsuch that

378

UNITARY REPWHNTATION OF

SL (2,R)

for all s E D. Proof. 1) By formulas (5.8) and (9.5), we have #(ac,s) =&Sl'e-8t(at-1-@)d8=(2x)-l

.

lcht- sht cos 81-8d8

0

Hence { -s loglcht -sht cos 81) d8 fr

= ( 2 ~ ) - 1 ( -1)-1 (loglcht-sht cos0))"lcht- sht cos81-sd0 0

Since

e-'= cht - sht 5 lchr - sht cos8/5 cht + sht =el for every t 2 0, we have

= t m$ (at, u> where s = u + 2. 2) Since Icht-sht COSBI-' = (chat+ shatCOSae-shat COS~)-*/', 7

we see that am#(at7 s)/atm is equal to a sum of terms of the form

where PO@)is a polynomial of s and Q(x, y, z ) is a polynomial of x, y , z. If C is a compact subset of [0, + m), then Q(sht, cht, cos8) is a bounded function of (t, 8) on C x [0,2~]. Hence there exists a constant Mm>Osuch that

=MmIP(s)l#(at, a)

d~mlP(s)l

if s= u +i,?E D (Proposition 9.10). Proposition 9.20. The spherical transform Y : f n f l is a continuous linear mapping of 9 into 9.Moreover we have

379

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

(9.27)

(w*f)"(s>=sn(l -s)*f(s)

for every f E 9 and n E N.

Proof. Iffbelongs to 9, thenfbelongs to L1(G)n A. Hence the spherical transform f ( s )=F(s) satisfies the first two properties in Definition 7 (Proposition 9.12). By Proposition 9.19, l), we have

[(am/asm)$(ac, s)] 5 t"

for each t g o and s E D

.

On the other hand, rnf(r) belongs to L1([0, + 03 ), 2 ~ s hdr) t for each f E 9. Therefore for any m E N,the integral s)dm(t)

/;f(t)(a"iasm)p(o,,

converges uniformly for s E D. Consequently the function f ( s ) is a C"function of s in D and satisfies

(d"/ds"lf(s)=S'>(t)(a-las-)$(u,, 0

(9.28)

S)dm(t).

Moreover (dm/dsn)f(s) is bounded on D. By (9.26) and Proposition 9.16, we have

1-

(~"f)^(4 = bnf)(t)$(ut, s) dm@) 0

4)dm(t)

= /d(t)(o.g(al, =sn( 1-s)nf(s)

.

Replacingfin (9.28) by d f and using (9.27), we get (9.29) and (dm/ds*)(sn(l-s>.f(s)) is bounded on D. Hence f belongs to 9. The above formula (9.29) proves the continuity of y.In fact if

Isht(l+~~+~)(o"f)(t)l5&/2n for all t 2 0 , then we have

I(d"/ds")(s"(l-s)"f(s))l

$E

(1 +tn+2)-'tmdt=Ce. q.e.d.

for all s E D.

Proposition 9.21. Iff belongs to 9,then for each integer m 2 0, there exists a constant C , such that Ix logn xf"x]l$ C,,, for all x E [ 1+ a)

.

380

UNlTARY REPRESENTATION OF

f”x] belongs to L1([l,

SL (2,R)

+ m), dx).

Pro05 Iff belongs to 9and m E N,then

s:

ZJ(t) =

Umshuf (u)du

is a bounded function of t. In fact, we have IZmf(t)I 51‘ l(1 +us)umshuf(u)l(l +u2)-1du

r-

The function Jmf(t)=

1:

umshUf’(u)du

is also bounded. For by integration by parts, we get Jmf(t)= tmshtf(t)-

and cht-sht (Itl++

a). Since

s:

(mum-lshu+u”chu)f (u)du

-sht(wf)(t)=(shtf‘(t))’, we have (u”shuf’(u))’ =murn-lshuf’(u)+ umshu(wf )(u) . Since wf belong to 9for eachf in 9, tmshtf‘(t)=mJm-l’(t)+Irnm’(t) is a bounded function oft. Then we see that tmf’(t) is bounded on R. Therefore

is bounded. Since log ( x + JxB--l) -log x, Jthat x l o g 9 * f”x]

-x(x-,

+ a), we see

+

is bounded on [l, m), for each m E N. Since (x log’ x)-1 is integrable on [2, + m), f”x] is integrable on [2, + a).Sincef”x1 is a C--function on [l, + m ) (Lemma l ) , f ” x ] is integrable on [l, 21. Hencef”~]is integrable on [l, a).

+

Theorem 9.3. (The inversion formula for 9). For every f in 9,we have fl*flf=f,

is the spherical transform f-f where the inverse spherical transform defined by

defined by (9.3) and

Y* is

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

9-* ;F Hfk)=

(9.30)

381

F(s)dg, s)dpo(s)*

J L

Proof. I f f belongs to 9,then t%ht f(t) is bounded on R. Thus log*x f [ x ] is bounded on [l, + a ) and f [ x ] is integrable on [l, +a). Hence the Radon transform -m

is defined. The above infinite integral converges uniformly with respect to x in a compact set. Similarly, by Proposition 9.21,f"xl is integrable on [l, a). Therefore the infinite integral

+

/:_f'[x+t?'ld7jl convergesuniformly with respect to x contained in each compact set. Hence F,[x] is a differentiable function of x and we have

Then we have

in exactly same way as in the proof of Proposition 9.8. Put f=2sh(t/2). Then we have 1+ p = c h t , df=ch (t/2) dt, hence (9.31)

F,'[cht]chTdt t

f(l)=G[ -m

Since there exists a constant M>O such that Ixf[x]lsM for all {l, a),we have

+

Hence F,(t) is integrable on R. By Proposition 9.20 Therefore

X E

f belongs to 9.

is a bounded function of A for all m E N and f ( + + i A ) is an integrable function of A. By Proposition 9.6, {(++in) is the Fourier transform

382

UNITARY REPRESENTATION OF

SL (2,R)

of F,(t) multiplied by 4%. By Ch. 111, Theorem 1.8, the Fourier inversion formula holds for F,(t) and we have

1'

+-

=

*

o

f(+ +i

~COSA ) tdA

.

Differentiating both sides of above equality with respect to x=cht, we get

By (9.31), (9.32) and Fubini's theorem, we get

='I 2*

+-

f(++iA)AthlsAdA .

0

The last equality follows from Lemma 3 in $8. By Proposition 9.14, 2), we have

(9.34)

(9.

Remark. The proof of Theorem 9.3 does not depend on the results of 58 except for Lemma 3 of that section which gives the Fourier transform

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

383

of (shr/2)-' on R and is independent of the other parts of $8. Hence Theorem 9.3 affords an another proof of Proposition 9.4.

Theorem 9.4. (Ehrenpreis-Mautner) The spherical transform 9 is a ' onto 9. The inverse mapping 9 - - l topological isomorphism of 9 of 9coincides with the inverse spherical transform 9-*given in (9.30). Proof. By Proposition 9.20, 9is a continuous mapping of 9 'into 9.And Theorem 9.3 proves that Y*Y f=f for every f E 9'. Let F be an arbitrary element in 9, and put f = Y * F . Since

+

+

+

sn( 1 -s)nF(s)$(g, s) = (1/4 Ra)"F(+ iR)$(g, 3 2)

is a bounded function of R for every n E N, and g , F(++iR)#(g, 3+iR) is integrable with respect to the spherical Plancherel measure p o . Hence f= r * F is defined. Now we shall prove the following two facts. (A) Y * F =f belongs to 9'. (B) 9-*is a continuous mapping from 9 into 9. If (A) is established, then Theorem 9.3 proves that 9-* Y f = f = T * F . we have Since 9-fand F belongs to 9, (C) Y f = F i.e. 99-*F=F by the uniqueness of the inverse spherical transform 9-*(Theorem 9.2). The assertions (B) and (C) together with Proposition 9.20 and Theorem 9.3 constitute the content of Theorem 9.4. Therefore it is sufficient to prove the assertions (A) and (B). Let F be an element of 9and put f = Y * F . Since Idm$(at,s)/atml$MmlP(s)l for all 220, m E N and s E D by Propositions 9.10 and 9.19, the infinite integral

1

lII+i"

Ira

F(s)(a "$/a t "1 (at, s)dpo(s)

converges uniformly on any compact subset of 0 4t < + a. Hence f is a C"-function and satisfies

dm

F f(0=

113

Therefore we have

By Propositions 7.16, 9.5 and 9.16, we have

384

UNITARY REPRESENTATION OF

SL (2,R)

Put t

y=sha2

and #(ar,s)=#(y, s).

Then we have #(Y, 4 = ( 1 +y)-*F(s,s, 1; Y/(l +Y))

-

Put z=y/(l +Y)

*

Then by Kunmmer's formula F(u, b, c; z)=(l-z)-aF(u, b, c ; z/(z-1)) #(y, s)=F(s, l-s, 1: -y). Now we put p ( y , s)=y-ar(1-2s)r(l

-s)-V(s,

(cf. ErdClyi [l] p. 64),

s,2s; -J+)

and p y y , s)=y*-T(2s- l)I-(s)-'F(l-s,

1-s, 2-2s; -y-l).

Then we have

.

(9.35) p y y , s)=#"'(y, 1-s) By Gauss' formula r ( ~ ) - ' I ; ( a , b , ~ : z ) = r ( b - ~ ) [ r ( b ) r ( ~ - ~ ) ] - l ( - ~l)--~"+Fa( ,a1,-~+u;z-') +F(U-b) [r(u)r(c-b)]-'(-z)-bF(b, 1-c+ b, 1--a+ b :z-l) (cf. ErdClyi [I] p. 63), we have (9.36) d Y , 8) = 5 w Y , s) +P Y Y , s) * By Euler's formula F(u, b,

C:

s:

z)=Z"(~)[I'(b)I'(c-b)]-~~*-l(l-r)e-a-l(l-r~)dr

(cf. ErdClyi [l], p. 59), we get

s:

=(2~)-'y-~tanns r*-l(l-r)'-l(l

+~y-l)-~dr

using r ( s ) r(l-s)=~/sinns. Since the integrand is an entire function of s, qP(y, s) is a meromorphic function of s in the half-plane Re s>O with simple poles at s= 1/2, 3/2, 5/2,... for y>O. Hence (y, s)= $(l) (y, 1-s) is a meromorphic function in the half-plane Re s< 1 with simple poles at s= 1/2, -1/2, -3/2,.- for y>O. Moreover

385

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

Res $(a)(y,s) = -Res $(l'(y, s) a-l/a

a-111

.

Therefore q P ( y , s)+qP(y, s) is regular analytic in the interior D for y >0. Put W ) ( t ,s)=2-'i(+-s) tan x(+-s)#"(at,

b

of

s)

for j = 1,2 and

@"'(Y,

4=y-'+(y, 8 )

Then by (9.37), we have

Hence +(y, s) is regular analytic in the half-plane u =Re s >0 for y >0. If OO, O s r j l . If 1 / 8 S ~ = R e s ,then l+-sldls~(l+(2~s~)-1)S5~s~. Hence there exists a constant B, e.g. B= 5/4z such that I+(y, $1 d Blsl if 1/8d Re s By Cauchy's formula, we have

.

where contour C is a circle Jz--sl=~-l/8. If 1 / 4 1 0 S 1 we have 1/85 u- 1/8 7/8. Hence there exists a constant C,,,>0 such that

(9.38)

Put

386

UNITARY REPRESENTATION OF

1

SL (2,R)

iia+t-

f ( j ) ( y )=

F(s)@(j)(y, s)ds

(j= 1,2).

iia-to,

Then using (9.35) and the relation F(s)=F(l-s), we have (9.40) f‘a’(Y) =f‘”(v) * Hence by (9.36) and (9.443, we get

f(v)=2 - l ( f C W+f‘”(Y))

=f(’’(y)

9

namely

J

iia+i-

(9.41)

f(t)=

F(s)@“’(at,s)&.

iia-t-

For each integer n >0, we have

Since 0y,=[s(l-s)]~9J and [s(l-s)]”F(s) belongs to (9.41) for the function s”(l-s)”F(s) and get

J

9,we can apply

iia+&-

(9.43)

(o”f)(t)=

s”(l-s)”F(s)cD‘”(ae, s)&

1

iia-i-

For each y>O, F(s)O(l)(y,s) is analytic in 1/2sRe s< 1 and continuous on D s = ( s ~ C ( 1 / 2 S R e s S 1 }By(9.37)and . thefact that F e Q , wehave lim I F(s)W )(y ,s)l =0 IIm J I *-

and the convergence is uniform on D1. Hence we can shift the line of integration in (9.41) and get (9.44)

f ( t )= S1+*-F(s)@“’(u,, l-is)dr

1

1+i-

=

F(s)y-J+(at,s)ds.

l-i-

Since we have proved thatfis a C”-function, the proof of the assertion (A) at the beginning of the proof is reduced to the proof of the following assertion (A). (A) pm,n(f)<+w for m , EN. Since t

(1 + y ) l J Z =ch2

-

t

sh - and log(1+y ) - t 2

(t+

we have tmsht(wnf)(t)--y logmy(wnf)(t) (tn++

W)

+ a).

387

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

By equations (9.41) and (9.43), it suffices to prove (A') for n=O, because the general case is proved by replacing F(s) by sn(l-s)"F(s). The assertion (A') for n = O and (B) are consequences of the following Lemma 3. q.e.d.

LEMMA 3. For each integer m10,we can find a constant Bms0 > O such that if

for all s E D, then IylogmYf(Y)lS BBrn.0 for all y > 0. For the purpose of proving this Lemma, we prove the more general Proposition T,,,,,,. For an m,n E N , and each function F i n 9

is bounded in y>O. Moreover we can find a constant B,,,>O the conditions IF(S)lSe(l+lSl)-',

...,

$E(~+ISI)-~ for all

such that

S ED

imply lylogLySlifmF(s)y-g,,+(y, an

5eBrn.n.

1-t-

Our Lemma 3 is the conjunction of Propositions Tm.ofor all m e N. Let W , be the conjunction of Propositions T,,,, for all n E N. Then we shall prove Proposition W , by the induction on m. Proof of Wo. For each integer n10,we have

for all s in D1={sll/4sRes g 1) and all y>O by (9.38). If F E 9 satisfies IF(s)l S e ( I + IS^)-^ for all s E D,then we have

388

UNITARY REPRESENTATION OF SL42,R)

where go,,= xC,. Proof of W,. Let m >0 and assume W , for all p <m. If F belongs to 9,then the inequality (9.45) holds for all y>O. Since d m F / h mbelongs . for every m, the repeated use of integration by parts shows that to Q

where A, and Ak are some constants. Since d m F / h mand F belong to

9,the induction hypothesis and (9.46) prove Proposition W,. q.e.d. COROLLARY to Theorem 9.4. The space 9is complete and hence a Frtchet space. Proof. 9 'is topologically isomorphic to a FrCchet space 9.

§lo. Irreducible unitary representations of SL(2,R) In sections 2,3 and 4, we have constructed irreducible unitary representations of the group SU(1,l). We have chosen SU(1,l) there instead of SL(2, R) because the behaviour of a maximal compact subgroup K in irreducible representations is easier to describ for SU(1,l). Since the group Go = SL(2, R) is isomorphic to G = SU(1,l) by the isomorphism (10.1)

y:Go

- G,

g' =

(g) = CgC-',

j ,(;

C =-

-ij)

each unitary representation V of G gives a unitary representation V' of Go. V' is defined by V' (g) = V(g'). V' is irreducible if and only if V is so. We put (10.2)

REPRESENTATION OF SL(2, a)

389

However this realization of representations for SL(2,R) is unnatural because the irreducible representations of SU(1,l) are constructed on spaces of functions on the unit disc D or the unit circle U. The group SL(2, R)/{f 1 ) is the automorphism group of the upper half-plane C+ while the group SU(l,l)/{ f1 } is the automorphism group of the unit disc. Hence it is natural to construct the representations of SL(2, R) on the space of functions on the upper half-plane C+ or its boundary, the real line R plus 00. In this section, we construct such representations for the group SL (2, R).

I. DISCRETE SERIES Let c be the Cayley transformation from the upper half-plane C+ = {z = x i y ) y > 0) onto the unit disc D = {CEC~ < 1):

+

z-i C =c(z) =z + i'

(10.3)

The inverse transformation c-l is given by (10.4)

The Cayley transformation c is a complex analytic diffeomorphism of C+ onto D. Let n E -12 and n > 112, and let 8, be the representation space of the 2

discrete series representation U" of SU (1, 1) (cf. $3). Namely sists of all holomorphic functions # on D with a finite norm

where dm (0= dudv for bY

C = u + iv. The representation

8. con-

Un is defined

(10.6)

Let H, be the Hilbert space of all holomorphic functionsfon C+with a finite norm (10.7)

where z = x

Ilfll2= (2n - l$Jc+

If ( z )I 2YZn-2dxdr

+ iy.

Proposition 10.1. 1) Let W,be the mapping from 8, into H , which maps # to f = W,# given by

UNITARY REPRESENTATIONOF SU2,R)

390

+

f ( z ) = ~ ( z i1-h I

(10.8)

z-i (m),

where A = JZ2-(*-I). Then W. is an isometry from @, onto H.. 2) Let D" be the irreducible unitary representation of SL(2, R) defined bY

0;

(10.9)

5:

Then the operator

w, u; 0

0

w;1.

0; is given by

(10.10)

for the element g-' in (10.2).

+

Proof. 1) Since the Cayley transformation C = c (z) = u iv is holomorphic, f = WnI is holomorphic on C+. Moreover we have

by the Cauchy-Riemann equations. A direct calculation gives the equality (10.11)

1

4Y

- IC12=

lz+i12'

Hence we have

X IZ

+ i14 dxdy = (2n -

If ( z ) I zyh-2 dxdy = ~

~ f ~ ~ z .

Hence W. is an isometry of @, into H.. Since (10.12)

z+i=-

2i 1 -C'

for z E C+ and

C E U,

the inverse mapping W,,-I of W. is given by (10.13)

4 (0 = (W,,-l f) (0 = A-I (2i)2n(1 - O-'"f(i l1 -+CC

).

For each function f in H,, the function 4 given by (10.13) belongs to @,, and satisfies W,, 4 =f.Hence W, is an isometry of onto H.. 2) The relations between the matrix entries of g'-I and g-' given in Proposition 1.6,l) imply that

a,,

REPRESENTATIONS OF s y 2 ,

391

R)

(10.14) Since g-' (10.15)

- C),

z = c-l ((g'-') az cz

we have

+b +d

The equalities (10.6), (10.13), (10.14) and (10.15) give

Corollary to Proposition 10.1. Let n E T1Z , n

1 and > -, 2

@-,

be the

Hilbert space on which the discrete series representation U-"is realized (cf. Theorem 3.2) and put W-, = u W,a,

where uf = f i s the complex conjugation. Then W-. is an isometry of &, onto UH,= H-,. H - , is the space of all antiholomorphic functions on C+ with the norm (10.7) finite. Thus also 0;" = W-,U>" W-,-'

(10.16)

is an irreducible unitary representation of SL (2, I?). The representation operator is given by (10.17)

11. Let 1 < u < 1 and

COMPLEMENTARY SERIES

$o

be the Hilbert space of all complex valued

measurable functions on the unit circle U with a finite norm

UNITARY REPRESENTATION OF Sa2,R)

392

where d( = (24-l d8 and & = (279-l d y for C = e'e and q = e ' y . @, is the representation space of the complementary series representation V" (cf. $4). The operator V;, for the element g' in (10.2) is given by (10.19)

Let H, be the Hilbert space of all complex valued measurable functions

f on the real line R with the norm (10.20)

Ilfll'

I X - ~ l - " " - ' " f ( ~ ) f Q d x d< y + 00.

=

PropositiOn 10.2. 1) Let S, be the mapping which maps a function to the function f = So/defined by

4 in @,

(10.21)

where A, = 2('-1)'2n-1.Then S, is an isometry of $, onto H,. 2) Put

c; = s, v;,

(10.22)

0

0

S;].

Then C" is an irreducible unitary representation of SL (2, R) on the Hilbert space H,. The representation operator C ; is given by (10.23)

for an element g - I

=

t f;).

Proof. 1) Put C = c ( x ) and q = c ( y ) (x, y Iy+ il = Iy - i l . Then it is easily verified that (10.24) (10.25)

1 - Re(Cq) =

dC = (2~)-'d8 =

E R)

21x - Y I 2

Ix

+ i I 2 Iy + il' dx

n(1

+ x')

-

and

dx nix

and note that

+ iI2'

By the equalities (10.21), (10.24) and (10.25) we have

393

onto H,. Hence S, is an isometry of 2) Since the inverse transformation S;’ : f-

4

is given by

(10.26)

we can prove (10.23) in the same way as (10.10). In fact we have

( C V )(XI

= [(SU

= Icx

v:, C’)fl(4

+ dl

-2,f

(n). m+b

111. PRINCIPAL CONTINUOUS SERIES The principal continuous series representation VJJ( j E { 0, 1/2}, s E C, Re s = 1/2) is realized on the Hilbert space @ = L2 (U,d o . The representation operator V i a is defined by

The corresponding representation PJ.’ of SL (2, R) is realized on the Hilbeit space $ = L2(R,dx). Proposition 10.3 Let j E { 0, 1/2} and Re s = 1/2. 1) Then the transformation TI,#: 4 f defined by ++

is isometry of $ = L2(U)onto H = L2 (R);and 2) if we put

394

UNITARYREPRESENTATIONOF SL(2,R)

P i J = T/.I VJ9ST-1 (10.29) r' J,r then PJUI is an irreducible unitary representation of SL (2, R). The representation operator p';* for g-' in (10.2) is given by 9

(10.30) (Pi'f) (x) = [sign (cx

+ d)I2JIcx+ dl+f(-).ax + b

Proof: 1) By (10.25) and (10.28), we have

=J- I1(x)I2dx

=

11f1I2

-0.

if Re s = 1/2. Hence Tj,, is an isometry of 8 = L2 (U)onto H = L2 (R). 2) The inverse transformation Ti,.-' is given by

By the equalities (10.14), (10.15), (10.26), (10.28) and (10.31) we have

('j'f)

= [(T,.8

'i;

T,,J-l).f]

Appendix

A. Banach-Steinhaus theorem 1. A subset B of a topological space X i s said to be nowhere dense if the closure has no inner points. A subset M of X is called meager or of the first category if M is the union of a countable family of nowhere dense subsets. If A is not of the first category, A is called of the second category. 2. A toplogical space X is called a Baire space if every non empty open subset of X is of the second category. Theorem (The Baire category theorem). A complete metric space and a locally compact Hausdorff space are Baire spaces. cf. Bourbaki [l] Ch. 9 $5. In particular a Banach space is a Baire space and of the second category. 3. Theorem (The uniform boundedness theorem). Let X be a Banach space and Y be a normed space. Let (Ta)asA be a family of continuous (=bounded) linear mappings of X into Y. If the set {TaxlaE A } is bounded for each x E X , then we have the strong limit lim T,x=O 0*0

uniformly in a E A . In particular {llTaIlla E A } is bounded. cf. Yoshida [l], p. 68, 69. Let X be a Banach space and X * be the dual space of X. X * is the Banach space of all bounded-linear forms on X with- the norm llfll= suplf(x)l. iisnsl

A Banach space Xis regarded as a subspace of the dual space X** of X * . Then Theorem A.3 implies the following theorem. Theorem (Banach-Steinhaus theorem). If a sequence ( x , ) , . ~ in a Banach space is weakly convergent, then the sequence (IIx,ll),.~ is bounded.

4.

B. Hilbert-Schmidt theorem 1. A linear mapping T of a Banach space X into a Banach space Y is called compact (or completely continuous) if every bounded subset B of 395

3 96

APPENDIX

Xis mapped onto a totally bounded subset T(B)of Y (that is, the closure T(B) is compact). Namely T is compact if the image (Tx,),,Nof each bounded sequence (x,),,*N contains a strongly convergent subsequence. 2. If H is a Hilbert space, a linear operator T on H is compact if and only if every weakly convergent sequence (X,,),,~N is mapped to a strongly convergent sequence ( T x ~ ) , , * N . The “if” part is easily proved. In fact if (x,),O such that there exists a subsequence (Xnk)ktN satisfying 11 Tx,, - TxllZ B for all k E N. Since (X,),,
k-

Hence we have y = T x and limllTx,,-Txll=limllTxnk-yII=O. The h-ar

k-or

last relation contradicts with IITx,, - TxllZ B . 3. Let T be a linear operator on a Hilbert space H and 2 be a complex number. Put

H ( l ) = {xEHJTx=;Lx}. If H(;O + {O} , then R is called an eigenvulue of T.If T is self-adjoint, then all the eigenvalues of T are real numbers and H(R) is orthogonal to H(p) if R # p . 4. LEMMA. If T is a bounded self-adjoint operator on H, then we have

Proof. The right hand side of (1) is denoted by m. Since I(Tx, x)ls IITxllllxll4IITII, we have millTII. For any two vectors x and y in H and for any a E R, we have rfr ( T ( a x f y ) ,(Rx&y))$mlllx+yllS

and ;({(Tx,y)+(Ty,x)}$m{Pllxl12+Ilylla}.Put y = T x . Then we have llTxlla5 2-lm {lllxl12 R-lllTxlla}. Put 2 = l l T x l ~ / ~for ~ xx#O. ~ ~ Then we have IITxl12jmllTxllllxll and IITxlld m\lxll for every x E H. Hence we get \]TI\ 5m . 5. THEOREM (Hilbert-Schmidt). Let T be a compact self-adjoint operator on a Hilbert space H and E be the set of eigenvalues of T.Then we have the following.

+

397

APPENDIX

1) E has no point of accumulation except R =0 possibly. 2) dim H(R) is finite if R # 0. 3) If H + {O} , then T has at least one eigenvalue. 4) H=@H(R). ACE

Proof. l), 2). Suppose that there exist sequences (Rn)nrN and (pn)nr~ such that Rn E R, pnE H, Tpn=Rnpn, (pn, Pm)=&,m, lim R n = R # O . Since IIRpnn+m

~ p , # = 2 ~ 3 > 0if n z m , no subsequence of (Rpn)nt~is strongly convergent. By the assumption lim(Rn- 2) =0 and the identity Inpn =R p n (2, -R)P,,, we see that no subsequence of (Rnpn),,.~is strongly convergent. This is a contradiction, because T is compact and TPn=Rnpn. We have proved 1). Putting Rn =2 for all n in the above argument, we see that 2) has been proved. 3) By Lemma in 4, There exists a sequence Cfn)%.~ in H such that

+

Ilfnll=l,

and liml(Tfnn,fn)l=IITII. n+m

Replacing (fn) by a suitable subsequence, we may assume that lim(Tfn,fn) = R where R=IITII or -1lTll. Assume that R=IITll#O. Then we have O$ llTfn-~fnll’=llTfnllf--ZR(Tfnn,fn)+Af42R(R-((Tfn,fn))~O, (n+ a). Since T is compact, there exists a strongly convergent subsequence (Tfn,)*.~of (Tfn). Since lim(Tfn-Rfn)=O and R#O, the sequence cfn,) converges strongly to fo. Then we have llfoll= 1 and Tfo-Rfo=lim(lly;l, k-m -nfn,)=O. Hence R=IITII is an eigenvalue of T. When R = -IITII, -1lTll is an eigenvalue of T. 4) Suppose that H#@H(R). Then we have a contradiction by applying arB

3) to the restrictions of T to the orthogonal complement of @H(R). ICE

C. Measure and integration 1. Let X be a set. A family 123 of subsets of X i s called a 0-algebra or 0-additivefamily if it satisfies the following three properties. 1) XEb, 2) AE123impliesAc=X-AEb,

3) A,Eb(nEN)implies 6 A,,€% n-0 A set A in b is called b-measurable or simply measurable. 2. A function m on a 0-algebra b with values in [0, +a] is called a measure defined on 123 if m satisfies m(#)=O and m( An)= gm(An) n-0 n-0 for every disjoint countable family an),,.^ of sets in 113. The triple (X, b,m) of a set X, a 0-algebra b of subsets of X and a measure m on 123 is

398

APPENDIX

called a measure space. We always assume that Xis a countable union of sets of finite measures. A property P is said to hold almost everywhere if the set of points where it fails to hold is a set of measure zero. 3. Let (X, 8,m) be a measure space. Put R u { k a]. Then an Evalued function f on X is called measurable iff -l((a, a])belongs to 8 every a E R . A complex valued function is called measurable if its real and imaginary parts are measurable. 4. For each A E 8,the defining function x A of A is defined as ~ . ( x ) =1 or 0 according to x E A or x 4 A. A real valued function p is called simple if it is measurable and assumes only a finite number of values. I f p is simple and takes the values cl, ..., cR,then

x=

+

n

(1)

(O =

C~X.~,

where As = { x E X I( o ( x )

=a}.

i-1

5. I f E is a measurable set and p a nonnegative simple function given by (l), then we define the integral of p over E by 11

n

II

pdm =

,,

cim(Ai nE )

.

Let f be a nonnegative h d u e d measurable function on the measure space (X, b,m) and E be a measurable set. Then the integral off on E is defined by

1

f d m = sup

B

oSPS/

1

pdm,

E

where p ranges over all simple functions satisfying Ogp$f. f is called integrable on E if

s.

fdm<

+

CO.

Let f be a measurable function on

X. Thenf is said to be integrable if both f ‘(x)=Max [f (x),0 ) and f -(x) = Max{ -f(x), 0 ) are integrable. I f f is integrable, its integral is defined by fdm= I

E

f+dm/ E

f-dm. / E

f is integrable if and only if If1 = f ’ + f - is integrable. A complex valued function f is called integrable if both Re f ( x ) and Imf(x) are integrable. 6. The following theorem on termwise integration is referred to as Lebesgue’s dominated convergence theorem or simply Lebesgue’s theorem. r~ a sequence of integrable functions on E convergTHEOREM. Let ( f n ) nbe ing to a function f ( x ) almost everywhere. Suppose that there exists an integrable function g on E satisfying 1fn(x)lS g ( x ) for all n and for almost every x. Then f is integrable on E, and

399

APPENDIX

f dm=lim

COROLLARY. (Lebesgue's bounded convergence theorem). Let (fn)nrN be a sequence of uniformly bounded measurable functions on E. Suppose that m(E)< + 00 and Cfn(x))n.~converges to f ( x ) for almost every x E E. Then f is integrable on E and

7. If (X, 8,m) is a measure space and p z 1, we denote by Lp(X, m) the space of all C-valued measurable functions for which

1

f lpdm<

+

00,

considering two functions in L p to be equal if they are equal almost everywhere. LP(X, m) is a Banach space with the norm

Ilf

IlP =

(1 X

If

IPdm)liP

8. Let (X, 8,m) and (Y, Q, n) be two measure spaces. We denote by 23 x Q the smallest a-algebra of subsets of Xx Y which contains all the sets of the form A x B(A E 8,B E 6).It is proved that there exists a uniquely determined measure m x n on 8 x Q such that

(m x n) ( A x B ) =m(A)n(B)

.

m x n is called the product measure of m and n. The integral over Xx Y of an (m x n)-integrable functionf ( x , y ) is denoted by

-

/ x s y f ( x , v) dm(x)dn(v)

THEOREM (Fubini). A (8x &)-measurable function f ( x , y ) is integrable over X x Y if and only if either of the interated integrals

is finite. In such a case, we have

Iff is integrable on Xx Y, then for almost every x E X (resp. Y E Y), the function YH f(x, y) (resp. x ~ f ( xy)) , is integrable on Y(resp. on X).

400

APPENDIX

Reference for Cl-C8. Royden [l], Halmos [l]. 9. Let (X, b,m) be a measure space and H be a Banach space. An

H-valued function 9 on X is called simple if there exists a finite number of elements al, ..., an in H and a disjoint family of measurable sets A, ..., A, such that m(Ar)< + m(1 Si$n) and ad for each X E A( q(X),= 0 for each X E X - ; IA ( .

i

(-1

The integral of q over X is defined by n

n

An H-valued function f on X is said to be m-integrable (in the sense of Bochner), if there exists a sequence (q,)nrNof H-valued simple functions which converges strongly to f(x) for almost all X C X in such a way that r

Iff is m-integrable over X,the integral off over X is defined by x q,dm.

/,fdm=Um/n-*

The strong limit in the right hand side exists and is independent of the approximating sequence (yn)nrN. cf. Yoshida [l], p. 132.

D. Radon measure 1. Let X be a locally compact Hausdorff space. We always assume that X is 0-compact, i.e., Xis a countable union of compact sets. Let R be the family of all compact subsets of X. The smallest 0-algebra containing R is denoted by b. An element of b is called a Borel set in X. Since Xis ucompact, every closed set and every open set are Borel sets. A measure m on b is called a Borel measure if m(C)< + co for every compact set C. 2. Let L =L ( X ) be the space of all C-valued continuous functions on X with compact supports. Every function f in L is integrable with respect to a Borel measure m. The integral

TW=

fdm

J X

is a linear form on L which is positive in the sense that TCf)LO.

fro

implies

APPENDIX

401

3. A Borel measure m is called regular if it satisfies m(E)=sup{m(C)ICE W, Cc E } =inf{m(U)IUc D,Ec U } where D is the family of all open subsets of X . THEOREM (Riesz representation theorem). If T is a positive linear form on L,there then exists a unique regular Borel measure m on X such that T a = sXfdm

for all f in L. cf. Berberian [l], p. 227. A regular Borel measure is also called a Radon measure. The integral with respect to a Radon measure is called a Radon integral. The Riesz representation theorem shows that Radon measures on X correspond bijectively to positive linear forms on L. 4. Let CO(X) be the space of all C-valued continuous functions on X which vanish at infinity. Namely, Co(X)is the set of all C-valued continuous functions on X such that for every E > O there exists a compact set C and If(x)l d E for each x E X- C. Another form of the Riesz representation theorem is given as follows. THEOREM. To each bounded positive linear form T o n CO(X), there exists a unique Radon measure m such that T(f)=

fdm

for all f E CO(X).

S X

cf. E. Hewitt and K. Stromberg, [l], p. 364. Tbeorem. If m is a Radon measure on X , then the space L=L(X) of C-valued continuous functions with compact supports is dense in Lp(X, m) for 16p< + 00. Proof. Since X i s a-compact, there exists an open covering (Un)n,, of X such that 0, is compact and contained in Un+lfor all neN. Then for each elementf 20 in L' and for any E >0, there exists an integer n such that 5.

Let xtl be the d e h i n g function of Un and put fn(x) = xn(x)f(X). Then we have (1)

Ilf-fnllp

<8 *

Since fn is an integrable function 20, there exists a simple function g such that

4 02

APPENJXX

We can assume that G j c Un by (3). Since X is a a-compact locally compact Hausdorf€space, Xis paracompact and hence a normal space. Therefore there exists a continuous function hj on X such that 0 5 hf 5 1 and hj ( x ) is equal to 0 and 1 on F, and X - G, respectively. Then h, belongs to L and satisfies r

By (l), (2) and (4), we get \lf-hllP<3~.Every functionfcan be written asf = c f i + - f i - ) + i G + - h - ) wherefi=Ref,ji=Imfandfif =2-' (Iff1 55). Sinceft* belongs to L p iff belongs to LP, we can drop the assumption q.e.d. f 2 0 in the above proof. 6. Theorem. Let dx=dm(x) be the right Haar measure of a locally compact (0-compact) group G. Then the right regular representation R of G on Lp=Lp(G, dx) is strongly continuous. "he same result holds for left regular representation L. Proof. The operator RP is defined by (&f) (x)=f(xg). Hence we have R,Rh=Rgh and IIR,,fllP= llf[lp for all g and h in G and for all f in LP. Since R, is an isometry of L p , it is sufficient to prove the strong continuity at the identity element e of G. By D.5,for each f in LP and E>O, there exists a function h in L(G)=L such that Ilf-hllp<e/3. Let U be a fixed compact neighborhood of e such that m( U )2 1 and C be the support of h. Then CU is compact and has the measure u=m(CU)& 1. Since the support C of h is compact, h is uniformly continuous on G. Hence there exists a neighborhood N=N-l of e such that N c U and

IIRPh-hll,<~/(3u) for all gcN. Then we have

4 03

APPENDIX

E. Spectral representation 1. Let H be a Hilbert space and b be a cr-algebra of subsets of a set X. A spectral measure E on b in H is defined in Ch. 111, $3, Definition 3. For any two vectors u and v in H, a complex measure p U J A ) = (E(A) u, v) is defined on 2' 3 (cf. Ch. 111, $3). The integral with respect to I.

pu,o is written as

Jx

f(x)d(E(x)u,v). In particular, if b is the a-algebra

b(R) of Bore1 sets in R, put E , = E ( ( - co, 21). Then this integral is a Lebesgue-Stieltjes integral with respect to the function ~ ( 2=) (EAu, v) of bounded variation and is written as [I.j(l)d(Eiu, v).

2. Let E be a spectral measure on b in H andfbe a measurable function on X . Then a linear operator Tcf) is defined on the domain D ( f ) = r

+

{UE HI1 If(x)ladllE(x)ull2< a} with values in H, by X

Id.

cf. Ch. 111, $3. T ( f )is denoted by Tcf>=

f(x)dE(x).

3. The adjoint T* of a linear operator-T with dense domain D(T)is defined in Ch. V. $6, Definition 1. T is called self-adjoint if T = T*. THEOREM (von Neumann). If T is self-adjoint operator in a Hilbert space H, then there exists a unique spectral measure E on b =b(R) in H such. that

.=Iw

RdE,.

(1)

-ca

(1) is called the spectral representation of T. If T is bounded, the integral is taken over a finite interval. 4. THEOREM. Let A be a bounded self-adjoint operator on a Hilbert

space H, and A =

SI.

;tdEi be the spectral representation of A . If M is a

von Neumann algebra (cf. Ch. I, $2) containing A, then EA belongs to., M for all R E R. In particular EAE { A }

".

404

APPENDIX

Proof. If A 2 c (or A 5 -c), then En = 1 (or El = 0) belongs to M. Assume that -c < A. < c. Let E > 0 and pa (x) be a polynomical satisfying 1--e
<pa(x) < 1

--e

--e

for - c $ x d A o ,

+e

+

for loS x 5 Lo & E 5 x 5 c. for

+

<E

For any two vector u, v in H,we have

(P (4u - Elo u,

V)

=

(P. (4 - 1) d(El u. V )

Hence we have

and IIp6

(A)u - E@ll

+ '11

5 &llull

+ (l +

&)IIELo+6

- E&ull

- EAO+6 '11.

Since lim Elo+6= El, (strongly (Ch. 111, proposition 3.9,7)), we see that 1-0

pa (A) converges strongly to EAowhen 8 tends to 0. Since p 6 (A) belongs to the algebra M and M is strongly closed, Elo belongs to M.

F. Weak*-topology Let X be a topological vector space and X* be the dual of X. X* is, by definition, the set of continuous linear forms on X. Every x E X induces a linear formsf, on X* defined by fx (4) = 4 (x) for each 4 E X*. The weak*-topology of X* is the weakest topology on X* which makes the mapping fx continuous for every x E X . The weak*-topology is a locally convex topology on X*.A sequence (#JnN in X* converges to 4 in weak*-topology if and only if lim qJm (x) = 4 ( x ) for each x E X. n-m

G. A theorem of Fatou 1. Let X be a set and '$ ( X ) be the set of all subsets of X. A mapping m* from $' 3 ( X ) to [0, w] is called an outer measwe on X if m* satisfies the following three conditions (l), (2) and (3):

+

405

APPENDIX

+

(1)

0 5 m* ( A ) 5

(2)

A c B implies m* (A)

00,

m* (+)= 0, m* (B),

Let m* be an outer measure on X.Then a subset E of X is called m*measurable if E satisfies the following condition: (4)

m* ( A ) = m* ( A fl E )

+ m* ( A n Ec)

for all A

E 3 (X),

where Ec = X - E. Let 8 be the set of all m*-measurable sets in X. Then 8 is a a-algebra and the restriction m of m* to 8 is a measure on 8.m is called the measure defined by the outer measure m*. 2. Let 9 be the set of all half-open intervals (a, b] = { x E R I a < x 5 b } : 9 ={(a,blla, b E R, a 5 b } . fcontains the empty set 4 = (a, a]. Let g be a monotone non-decreasing function on R and put

I(a,blIP=g(b)-g(a).

Then we can define an outer measure m* = mT by

where I." is the interior of I. , i.e., I." = (a, b) if In = (a, b]. It is easily verified that m* is an outer measure on R. Let 8 be the set of all m*measurable sets. The a-algebra 8 depends on the function g. But 8 contains all intervals, all open sets, all compact sets, and all Borel sets in R. Hence 8 contains the a-algebra goof all Borel sets. Let m be the restriction of m* to Bo.Then ( R , go,m) is a measure space. m is called the Lebesgue-Stieltjes measure defined by g . The integral with respect to m is called the Lebesgue-Stieltjes integral and is denoted by P

P

3. Let f be a complex-valued function on a closed interval I = [a, b]. Let P be a finite subset of I. Then P defines a partition of I into the union of subintervals whose endpoints are elements of P. We identify the partition with P. Let 9be the set of all partitions of I. If a partition P E 9 of I is defined by

406

APPENDIX

. . <x,=b.

U = X ~ < X , < .

Put "

v (p9n=

c If

(XJ

and

-f(xt-dl

1-0

V(f) = SUP ( V V , f ) l

p

E

91.

Then V(f ) is called the total variation off on I. If V (f)is finite, then f is said to be of bounded variation. A function f is of bounded variation if and onlyif bothRe fandImfareso. A real valued function f is of bounded variation if and only if f is the difference of two monotone nondecreasing functions. Hence a complex valued function g of bounded variation is written as g = (g, - gz)

(6)

+ i(g3 - g4)

with four nondecreasing functions gl(1 5 i 5 4). Let g be a function of bounded variation expressed as in (6) on an interval Z = [a. b]. We extend g to a function g* on R which is defined by

s us x 5b

g(4,x

i

g* (x) = g(x),

g (b),b Ix.

Then a complex measure m is defined by m

+ i(m3 - m4)

= (ml- mz)

where ml is the Lebesgue-Stieltjes measure defined by the function g? (1 5 i 5 4). m is called the Lebesgue-Stieltjes measure defined by g. The support of m is contained in I. A function of bounded variation is differentiable almost everywhere.

5 a < 7t12 and ( = de be a point on the unit circle U. The angular domain A , ( O is defined by

4. Let 0

& ( r ) = {ref+/IArg(1

The domain A,

(0is

LEMMA1. Let

- retc#+'))I
illustrated in Fig. 1 of Ch.V. $3.

0 _I a < n/2 and put

B,={z€A,(r))

11 - z I

=cos~}.

Then there exists a constant M such that the following two inequalities hold for all z = rete E &:

407

APPENDIX

Proof. 1) Put 1 - z = pel*. Then we have 11

- zl = p

1 - lzl = 1 - 11 - pef*l = 1 - (1 - 2pcos4

Since z is in the domain A, (l), we have Hence if 0 < p 5 cos a,then we have

-

I1 -zl

1

P

+ +

(1 2 p P2)1/2 5 1 +2cosa - cosa

s

p2)ll2.

I$( 5 a and cos 4 2 cos a > 0. -

1-(1-2pcOs#+py2

- (ZI

+

and

1 - (1 - 2 p cos 4 2cos4-p

+p 2 y

2+P cos a

2+cosa = cos a

Fig. 1

2) Let y be the angle in Fig. 1. Then the law of sines for the triangl Ozl implies that

Hence we have Isinel = I s i n y J - I l-zl

s

( 1 - z J SM(1 -r).

The function

is called the Poisson kernel and has the following properties:

q.e.d.

408

APPENDIX

(I) PI (8) = 0 and P, is a continuous function of z = refofor 0 S r < 1, and 0 5 8 5 2n;

Lr

(II)

2n

--x

P,(8)d8 = 1 for 0 5 r < 1; and

if 0 c 6 < z,then lim sup I P,(8)I = 0.

(III)

it9izd

Because P, (8)4 P, (4 if 6 S 181 5 n. LEMMA 2. Let r (s) and 8 (s) be two continuous functions defined for 0 5 s 5 1 which give a curve in the angular domain A,, (1) (0 5 a < n/2) and r(1) = 1, 8(1) = 0. Put

K, (t) = sin tP: (6' - t )

for r = r (s) and 8 = 8 (s).

Then the functions K, ( 1 ) have the following properties. If 0 < 6 < n, then

6)

lim

sup

r t l dgltl$r

IK, (t)l

= 0.

(ii)

f"

(iii)

J

K,(t) dt is bounded as s

--I

-

1.

Proof. (i) We have sintsin(t - 8) (1 - 2 r cos (0 - t ) t2)2 2rsintsin (t - 8) P,(0 - t ) . 1 - 2rcos (8 - t ) r2

sin t P i ( 8 - t ) = 2 r ( 1

- r2)

+

+

Since the first factor on the right side is bounded when I t I 2 6 and 8 is near 0, (i) follows from the above property (111) of P, (8). (ii) follows from the equality

Lr

2n -*

sin t P ; (8 - t ) dr =

cos t P, (8 - t ) dr = r cos 8.

(iii) By the addition theorem of the sine function, we have

The second integral is equal to

409

and is bounded as r

-

1. Also we have

- _ Isinel a

(--id l+r

1-r

4r Il +. r 1-r2a

because ( 1

+ r)2- 4r = (1 - r)2 2 0 and by Lemma 1,2).

q.e.d.

5. Theorem (Fatou). 1) Let m be a complex Lebesgue-Stieltjes measure on the unit circle U defined by a function g (0) of bounded variation. Put

where P,(0) is the Poisson kernel. f is regarded as a function of z = rde on the unit disc D. Let C = deobe any point on U,where g is differentiable at e,. Then a non-tangential limit limf(r, 0)

= g’

(O,)

(z = rete E A ,

(0).

x-c

exists. 2) I f f (t) is an integrable function on U, in particular i f f is square

integrable, then the Poisson integral

has a non-tangential limit f ( t ) at almost every point e“ on U. Proof. 1 ) For the proof, let us first observe that 1 ) is trivially true for 1

the ordinary Lebesgue measure dm (t) = - dt by the above property 2n (11) of P,(t). So, without loss of generality, we may assume that m (U)= 0 by subtracting a constant multiple of df. Then the function g satisfies g(a)-g(-a)=m(U)=O. Put

410

Then, by integration by parts, we have

Replacing dm ( 2 ) by dm (t - O0), we may assume without loss of generality that O0 = 0. Let r (s) ere(#) (0 5 s < 1) be a non-tangential continuous curve approaching 1 as s 1. Let K, (t) be thefunction

-

K , (t) = sin t

.P:(8 - t)

for r = r (s), 8 = 8 (s)

and put G ( t ) = -- g' (0), sin t

I(s) =

1

"

K, ( t ) dt.

-x

Then the (5)

Since G is continuous at 0 with G (0) = 0, the integral

J

It1 <6

G (0K, ( t ) dt

is small for small 6 > 0 by (iii) by Lemma 2. On the other hand triad

G (t) K , ( t ) dt is small by (i) of Lemma 2. Lastly lim I ( s ) = 1 by (ii) Sf

1

of Lemma 2. Hence we have

liml(r (s), 8 (4)= g' at 1

(0)

by (5). 2) Let f be a function in L1(U)and put

Then g is of bounded variation and differetiable almost everywhere. Moreover the derivative g' satisfies g' (t) =f(t) almost everywhere.

Hence, if we apply 1) to the Lebesgue-Stieltjes measure m = m, defined q.e.d. by g, then we have 2). The above proof of Fatou's theorem is borrowed from Hoffman [l].

Notes

Chapter I. 3 3 . For the general theory of Haar measure, see Bourbaki [3], Ch. VII, Halmos [l], Hewitt and Ross[l], I, Loomis[l], and Weil[l]. Theorem 3.1 has been proved by many authors. Here we follow the proof of S. Ito [2]. If U is a weakly continuous irreducible Banach representation of a compact group, then U is finite dimensional (cf. Shiga [l]).

0 5. Theorem 5.1 cannot be extended to arbitrary continuous functionsj: Since a continuous function f is square integrable, the Fourier series off converges almost everywhere by a theorem of Carleson [l]. On the other hand, it has been proved by Kahane and Katznelson [l] that for every set B of measure zero on T,there exists a continuous function f such that the set of divergence of the Fourier series off coincides with B (cf. Katznelson [l], p. 58). Katznelson’s book is recommended to those readers who wish to consult a more detailed study of the ordinary Fourier series and integrals. Many applications of Fourier series and integrals to various branches of mathematics and physics are explained in the book by Dym and McKean[l]. For the proof of Carleson’s theorem, see Fefferman [l] and Jgrsboe and Mejibro [l]. Chapter 11. 35. For the general theory of Lie groups, see Chevalley [l], Helgason [13], Bourbaki [4], Varadajan [2], etc.

§ 8. The representation theory of compact semisimple Lie groups was established by H. Weyl [l]. A concise summary of his results is to be found in J-P. Sene [l] and Brocker and Tom Dieck [l]. An important addition to the theory concerns the construction of irreducible representations on the spaces of sections of certain line bundles on the flag manifolds or on the higher dimensional cohomology spaces (Borel-Weil-Bott theorem). See 41 1

412

NOTES

Kostant [2], Bott [l], I. Knapp [5]. Theorem 8.1 was proved by Taylor[l]. Theorem 8.2 was proved by gelobenko[l]. See also Sugiura [4].

Chapter 111. §3. In the 194O's, harmonic analysis on locally compact abelian groups was studied by many authors. Positive definite functions and Bochner's theorem play the central role in the theory (cf. Rudin [l], Ch. I). Using positive definite functions and the Krein-Milman theorem, Gelfand and Raikov [ 11 proved the existence of suflFiciently many irreducible unitary representations for every locally compact group (cf. also Godement [3]). The uniqueness of the measure p in Bochner's theorem (Theorem 3.1) may be proved by Levy's inversion formula which expresses p explicitly by the given positive definite function p. cf. Levy [l]. 5j4. The original theorem of Paley and Wiener [l] gave a characterization of Fourier transforms of functions in Lp with compact supports. Schwartz [13 generalized their result to tempered distributions.

Chapter IV

.

§ 2. Irreducible unitary representations of the Euclidean motion group were classified by It0 [l] and Mackey ([3], I). Our proof of Theorem 2.1 is based on the theory of representations of semi-direct products due to Mackey [3]. Vilenkin [l], [2] studied unitary representations of M(2) and their relations to Bessel functions. Theorems 3.2 and 4.2 were proved by him. Other types of formulae for Bessel functions were proved by Orihara [l], [2], using the representations of motion groups of Euclidean and Lobatchevsky spaces. 55. Theorem 5.2 was proved by Kumahara and Okamoto [l].

Chapter V. § 1. The Iwasawa decomposition for general semisimple Lie groups was first given in Iwasawa [l]. See also Helgason [13].

§ 2. The principal continuous series representationsfor SL(2, R) are easily generalized to every connected non compact semisimple Lie group G with finite center. Let G = KAN be an Iwasawa decomposition and M = C,(A) be the centralizer of A in K. Then the subgroup Po = MAN and its conjugates are called the minimal parabolic subgroups of G . Let Q be an

413

NOTES

irreducible unitary representation of A4 and u be a complex linear form on the Lie algebra a of A. Then the representation a@e’@l of P o is defined by (1)

(u@e’@l) (man)= a(m)e’(*O~for meM, UEAand neN.

The representation of G induced by a@e”@l is denoted by U(Po, u,u):

U(Po,Q, u ) = ind~o(o@eu@l). (2) U(Po,a,v ) is a unitary representation if u is purely imaginary valued on a. Let W(G,A) = N,(A)/C,(A) be the Weyl group. Then each element w of W(G, A) acts on u and u by: (3)

(wa)(m) = a(w;’mwo) and ( w )(H) = u(Ad w i ’ H ) ,

where w o E N,(A) is a representative of the coset w. The representation U(Po,Q, u) (u = purely imaginary) is irreducible if wu # D and wv # u for all w # 1 in W. In general the unitary representation U(Po,a,u ) is a direct sum of finite number of irreducible components (cf. Bruhat [I] and Knapp ~51).

In the above construction we can replace the minimal parabolic subgroup P o with an arbitrary parabolic subgroup P which is a conjugate of a closed subgroup of G containing Po. For the classification of the parabolic subgroups, see Matsumoto [l] and Warner ([l] I). A parabolic subgroup P can be decomposed as follows: (4)

P = M(P)A(P)N(P)

where M ( P ) is a reductive group, A ( P ) is a vector group and N ( P ) is nilpotent. Every element x E P is uniquely decomposed as follows: (5)

x = man, m E M(P), u

E A ( P ) and

n

E N(P).

The deomposition (4) is called the Langlunds decomposition. cf. Warner ([I] I). For any irreducible unitary representation n of M(P) and a complex linear form u on the Lie algebra a(P) of A(P), there is defined an induced representation (6)

U(P, a,u) = ind:(a@e”@l).

§ 3. The representations in the discrete series are characterized by the fact that their matrix elements are square integrable on G (cf. Proposition 7.18). Harish-Chandra [ll] proved that a connected semisimple Lie group has a square integrable (i.e. discrete series) representation if and only if it has compact Cartan subgroup H. The discrete series is parametrized by the regular unitary characters of H modulo the Weyl group W(G, H). The values of characters of discrete series on H were given by Harish-Chandra

414

NOTES

[l 13. The values on arbitrary Cartan subgroups have been determined by Hirai in IV of [7l.For the realization of the discrete series representations on the concrete function space, see Harish-Chandra [5], Narasimhan and Okamoto [l], Hotta [l], Parthasarthy [l], Atiyah and Schmid [l], Flensted-Jensen [5], Knapp [5]. Using Zuckerman's tensoring functor, a certain irreducible unitary representations of G are constructed corresponding to some of the singular unitary characters of the compact Cartan subgroup H.These are limits of discrete series. (cf. Zuckerman [l] and Knapp [5]). When G / K is a Hermitian symmetric space some of the limits of the discrete series representations are realized on the Hardy space on GfK. See Knapp and Okamoto [l] and Inoue [I]. §4. If the linear form u is not purely imaginary on a, the representation

U(P,0,u) is not unitary for the usual L2inner product. But for some choices of v , there exists a positive semi-definite Hermitian form invariant under the representation U(P,6,u). The irreducible unitary representations arising from these representations form the complementary series of G. The existence of the complementary series representation is deeply connected with the analytic properties of intertwining operators between U(P, c,u) and V(OP, wo, w ) (w E W(G,A), 8 = Cartan involution). See Gelfand, Graev and Vilenkin [l], Bruhat [l], Schiffmann [3], Knapp and Stein 111, PI, 141 and K ~ ~ PPI.P §5. The GPrding subspace Ho actually coincides with the space H , of C"-vectors for a wide class of Lie groups including semisimple groups with finite center. See Dixmier and Malliavin [l].

9 6. Unitary representations of SL(2, R) were studied in the fundamental work of Bargmann [I]. The results in $9 2, 3, 4 and 6 are due to him. Our notation for irreducible unitary representations differs from that of Bargmann. The correspondence is given as follows: V0&" = C:,.:, 1 1 1 VT FV= C~+,,zZ, U" = Dn,z+, U-"= Dn/z-, V a= Ca(I-o)o. Takahashi [l], [6] gave another proof of Theorem 6.5 by classifying positive definite spherical functions. Koornwinder [5] proved it by a non-infinitesimal method.

9 7. The characters of infinite dimensional irreducible unitary representations were first defined by Gelfand and Naimark [l] for SL(2, C). Later Harish-Chandra ([3] 111) defined characters for an arbitrary semisimple Lie group G as distributions on G. Harish-Chandra [4] calculated the

NOTES

415

characters of principal continuous series representations. Hirai [a] generalized this result to cover a certain type of induced representations including

w ,.I. 0,

Q 8. The results in Q 8 are due to Harish-Chandra [2], [8]. Other proofs of Theorem 8.1 have been given by Pukanszky [l], [2], Schiffmann [l] and Takahashi [l], [6]. The inversion formula for any connected semisimple Lie group with finite center was obtained by Harish-Chandra [15]. His proof depends on a deep study of the Eisenstein integrals and wave packets. A simplified proof was given by Herb [6], [8] and Herb and Wolf [I]. See also Dufeo and Vergne [ l ] . Q 9. Theorem 9.1 and 9.4 were proved by Ehrenpreis and Mautner [l]. The Paley-Wiener type theorem for zonal (i.e. two sided K-invariant) functions on semisimple groups with finite center was proved by Helgason [4] and Gangolli [l]. Similar theorems for arbitrary K-finite functions were studied by many authors in special cases and finally proved by Arthur [4] and Clozel and Delorme [I]. Harish-Chandra [ll] introduced the Schwartzspace W((G) which plays a role for semisimple Lie groups analogous to that of the space 9 ( R ” )do for the Euclidean space R”.

Further Topics The representation theory of semisimple Lie groups bas made remarkable progress in recent years and large number of results have been accumulated. Knapp’s book [5] is recommended as a good survey of the theory. One of the fundamental problems in harmonic analysis is the determination of the unitary dual 8 of a given group G where 8 is the set of all equivalence classes of irreducible unitary representations of G. This problem has been solved for some simple Lie groups (e.g., SL(2, R)(Bargmann [l]), SL(2, C) (Gelfand and Naimark [I]), Spin(n, 1) (Hirai [2]), SL(3, C) (Tsuchikawa [I]), Sp(2, C) and G,(C) (Duflo [2]), Sp(n, 1) (Baldoni-Silva [l]), SU(2, 2) (Knapp and Speh [l]), GL(n, K ) , K = R,C,H (Vogan [13])), but not yet solved for general semisimple Lie groups. The status in 1981 of the classification problem of 8 was described in Knapp and Speh [2]. In recent years Vogan [Ill, [12] and Barbasch and Vogan [4] have made important contributions. See also Clozel [2]. An irreducible unitary representation U of a semisimple Lie group G is called tempered if its character defines a continuous linear form on the Schwartz space B(G).Irreducible tempered representations were classified by Knapp and Zuckerman [2]. They are induced representations U(P,

416

NOTES

Q, u) for which P is cuspidal and Q belongs either to the discrete series or to the limits of discrete series. The results of Langlands [l] and their results give the classification of all irreducible quasisimplerepresentations of linear semisimple Lie groups. Another fundamental problem in harmonic analysis is to decompose a given unitary representation of a group G into the direct integral of irreducible representations. The most important case occures when U is the regular representation on the space L2(G/H). If G is a connected semisimple Lie group with finite center and H is a maximal compact subgroup K,the decomposition of Lz(G/K)into irreducible components were accomplished by Harish-Chandra’s theory of spherical functions (Harish-Chandra [q,[ll]). See also Rosenberg [l] and Helgason [181. If G is semisimple and H is the fixed points subgroup (or its identity component) of an involution of G, the discrete spectrum of L2(G/H) was given by Flensted-Jensen [5] and Oshima and Matsuki [2]. Harmonic analysis on the semisimple symmetric space G/H is an important field of research which remains to be developed. If G is a semisimple group and Tis a discrete subgroup of G, then the decomposition of L y G / f )is deeply connected with the theory of automorphic forms and number theory. See the classical paper of Selberg [l] and Gelfand and Piatetskii-Shapiro [13. For recent developements and the relation to number theory, see the Corvallis Proceedings [C13] and Arthur [41. In recent years many conferences on the representation theory of Lie groups have been held. The proceedings of these conferences are useful sources of knowledge. Therefore we have listed up some of them at the end of the Bibliography as [Cl] to [C29].

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Index A adjoint operator, 281 adjoint representation, 6, 67 algebra, 64 almost everywhere, 398 analytic differential representation, 287 analytic vector, 287 approximate identity, 28 automorphic factor, 236

B %-measurable, 397 Baire space, 395 Banach representation, 219, 272 Banach-Steinhaus theorem, 395 Bargmann's theorem, 305 Bessel's inequality, 4 Bochner's theorem, 129 Borel measure, 400 Borel set, 400 bracket, 62

C Ck-function, 35 C'-vector, 219 Cm-vectorfield, 65 Cartan decomposition, 267 Casimir element, 72, 295, 299, 330 central distribution, 309 central function, 57 character, 25, 51. 178, 311 -Of VJ*',327 - o f Ye,329 - o f U",337 class function, 57 closure of an operator, 296 mmmutant, 10 commutator, 62 compact operator, 17, 395 complementary series, 261 complete orthonormal family, 3 completely continuous, 395

complexification, xv conjugate Fourier transform, 102 convolution of functions, 24, 102 cyclic representation, 12 cyclic vector, 12

D decomposable operator, 120 defining function, 398 degree of a representation, 5, 18 derivation, 64 derivative of a distribution, 39, 155 diagonal element, 122 differentiable action of a Lie group, 74 differentiable function on T,35 differentiable vector, 219 differential of a Cm-mapping,319 differential representation, 67, 219 Dirac measure, 39, 151 direct integral, 120, 121 direct sum of Hibert spaces, 8 direct sum of Unitary representations, 9 d i d set, 149 discrete series, 243 distribution, 38, 151, 178, 309 dominant G-integral form, 94 dual --of a topological group, 19 - o f R, 31 - o f T,32 --of SU(2), 58 -of R",101 - o f SL(2, R), 305 *(c), 306 9 d G L 347

E Ehrenpreis-Mautner theorem, 383 eigenvalue of a linear operator, 396 entire function, 144 equivalent unitary representation, 7 exponential of a matrix, 60 exponential type, 145

449

450

INDEX

F factor group, 5 Fatou's theorem 409 Ff,Ff",341 i k t (principal) series, 218 formal degree, 335 Fourier-Laplace transform, 146, 199 Fourier-Plancherel transform 117, 184 Fourier series, 1 --of a piecewise C'-function, 35 -of a squareintegrable function, 32 -of a distribution, 41 - o n a compact group, 22 - o n a compact Lie group, 96 - o n SU(2), 76 Fourier transform - o f a distribution on R",153 - o f a distribution on T,41 - o f a Radon measure, 126 - o n a compact group, 22 - o n M(2), 169, 188 - o n Rn,102 - o n SL(2, R),359 Fubini's theorem, 399

G GHrding subspace, 275 G-integral form, 94

H Haar integral, 14

--of M(2), 169 - o f SL(2, R),266,267,325 - o f SU(2), 56 harmonic oscillation, 2 harmonic polynomial, 85 highest weight, 69, 94 Hilber-Schmidt norm, 22,69, 177 Hilbert-Schmidt operator, 177 Hilber-Schmidt theorem, 395 holomorphic function, 143 homomorphism, 5 H-S norm Hilbert-Schmidt norm H-S operator Hilbert-Schmidt operator

-I

induced representation, 162, 164

infinitesimalgenerator, 284 integrable function, 398 integral, 398, 400 intertwining operator, 9 invariant subspace, 8 inversion formula of the Fourier-transform, 106 -for R", 106 -for M(2), 175 -for SL(2, R),359 irreducible unitary representation, 8 Ito-Mackey theorem, 165 Iwasawa decomposition, 205

J Jacobi identity, 64 Jacobi polynormial, 334

K K-finite differential representation, 280 K-finite representation, 292 K-finite vector, 279 K-invariant, 362 K-weight, 298 Killing form, 71

L Lebesque's (dominated convergence) theorem, 398 left regular representation, 7 Lie algebra, 64, 65 Lie group, 35 Lie product, 62 Lie subalgebra, 64 limit of discrete series 255 linear Lie group, 65 logarithm of a matrix, 61

M maximal compact subgroup, 206 maximal torus, 54 maximal weight, 69 measurable complex valued function 398 measurable vector valued function, 119, 120 measure, 397 -space, 398 motion group, 156 multiplicity, 30, 273

451

INDEX

N nilpotent Lie algebra, 226 normalized Haar integral, 15 nowhere dense, 395

0 one dimentional torus group, 5 orthonormal family, 3 orthogonality relations, 18 - o f characters, 51

P Parseval equality, 3, 22, 32, 107, 177, 361 Payley-Wiener theorem, 146, 150 Peter-Weyl theorem, 19 piecewise C'-function, 35 Plancherel theorem, 115 -for M(2), 183 positive definite function, 122 positive linear form, 400 product measure, 399 principal continuous series for SL(2, R) 218 principal series for M(2), 162

Q quasi-invariant measure, 165

R radial part of Casimir operator, 349 Radon integral, 14, 401 Radon measure, 401 rapidly decreasing function, 36, 79, 103, 170, 195 regular Bore1 measure, 401 regular elements, 317 representative algebra, 27 representative function, 26 representation space, 5 Riemann-Lebesque theorem, 34, 112 right Haar integral, 14 right invariant differential operator, 275 310 right regular representation, 7, 402 root of a compact Lie group, 94

S o-additive family, 397

a-algebra, 397 Schur's lemma, 11 second category, 395 second (principal) series, 218 self-adjoint operator, 281,403 spherical function, 364 semisimple Lie algebra, 72 simple function, 398 slowly increasing function, 40 slowly increasing measure, 152 solid harmonics, 85 spectral measure, 131 spectral representation, 403 spherical harmonics, 91 spherical transform, 364 Stone's theorem, 138 strict inductive limit, 150 strongly continuous representation, 6 summable, 3 symmetric linear operator, 281 .Y(R"), 103 Y(G), 170

T tangent space, 319 tangent vector, 319 tempered distribution, 152 tempered function on Z, 40 tensor product of Hilbert spaces, 125, 126 three dimensional proper Lorentz group, 21 1 topological group, 4 torus group, 5 total variation of a complex measure, 133 trace of an operator, 172 trace class, 172 Tschebyscheff polynomial, 257

U uniform approximation theorem, 27 uniform boundedness theorem, 395 unimodular locally compact group, 14, 25 uniqueness of Fourier transform -on a compact group, 28 - o n R",111 unitary representation, 5 , 18 universal enveloping algebra, 70

V von Neumann algebra, 11

4 52

INDEX

w W*-algebra, 11 weakly analytic, 201 weak*-topology,404

weight of representation, 69, 94

Z zonal function, 362 zonal spherical function, 364

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