Noncommutative Harmonic Analysis

NON COMMUTATIVE HARMONIC ANALYSIS MICHAEL E. TAYLOR

MATHEMATICAL SG RVEYS

AND MONOGRAPHS

NUMBER 22

Published by the American Mathematical Society

MATHEMATICAL SURVEYS AND MONOGRAPHS

NUMBER 22

NONCOMMUTATIVE HARMONIC ANALYSIS MICHAEL E. TAYLOR

American Mathematical Society Providence, Rhode Island

1980 Mathematics Subject Classification (1985 Revision). Primary 43-XX, 35-XX; Secondary 22-XX.

Librasy of Congress Cataloging-in-Publication I)ata Taylor, Michael Eugene, 1946— Noncommutative harmonic analysis. (Mathematical surveys and monographs, ISSN 0076-5376; no. 22) Bibliography: p. Includes index. 1. Harmonic analysis. I. Title. II. Title: Noncommutative harmonic analysis. ILL Series. 1986 QA403.T29 515'.2433 86-1 0924 ISBN 0-8218-1523-7

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Executive Director, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940. The owner consents to copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that a fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 27 Congress Street, Salem, Massachusetts 01970. When paying this fee please use the code 0076-5376/90 to refer to this publication. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotion purposes, for creating new collective works, or for resale.

Copyright ®1986 by the American Mathematical Society. All rights reserved. Printed in the United States of America The American Mathematical Society retains all rights except those granted to the United States Government. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. This publication was typeset using the American Mathematical Society's TEX macro system.

1098765432

959493929190

Contents Introduction

ix

0. Some Basic Concepts of Lie Group Representation Theory 1. One parameter groups of operators 2. Representations of Lie groups, convolution algebras, and Lie algebras 3. Representations of distributions and universal enveloping algebras 4. Irreducible representations of Lie groups 5. Varieties of Lie groups

1 1

9 17

27 34

1. The Heisenberg Group 1. Construction of the Heisenberg group H" 2. Representations of H" 3. Convolution operators on H" and the Weyl calculus 4. Automorphisms of H"; the symplectic group 5. The Bargmann-Fok representation 6. (Sub)Laplacians on H" and harmonic oscillators 7. Functional calculus for Heisenberg Laplacians and for harmonic oscillator Hamiltonians 8. The wave equation on the Heisenberg group

42 42 46 50 54 58 61

2. The Unitary Group 1. Representation theory for SU(2), SO(3), and some variants 2. Representation theory for U(n) 3. The subelliptic operator on SU(2) +

87 87 92 98

3. Compact Lie Groups 1. Weyl orthogonality relations and the Peter-Weyl theorem 2. Roots, weights, and the Borel-Weil theorem 3. Representations of compact groups on eigenspaces of Laplace operators V

67 81

104 104 110

119

vi

CONTENTS

4. Hannonic Analysis on Spheres 1. The Laplace equation in polar coordinates 2. Classical PDE on spheres 3. Spherical harmonics 4. The subelliptic operators + iaL3 on S2 +

128 128 130 133 140

5. Induced Representations, Systems of Imprimitivity, and Semidirect Products 1. Induced representations and systems of imprimitivity 2. The Stone-von Neumann theorem 3. Semidirect products 4. The Eucidean group and the Poincaré group

143 143 146 147 150

6. Nilpotent Lie Groups 1. Nilpotent Lie algebras and Lie algebras with dilations 2. Step 2 nilpotent Lie groups 3. Representations of general nilpotent Lie groups

152 152 154 158

7. Harmonic Analysis on Cones 163 1. Dilations of cones and the ax + b group 163 2. Spectral representation and functional calculus for the Laplacian on a cone 172

8. SL(2,R) 1. Introduction to SL(2, R) 2. Classification of irreducible unitary representations 3. The principal series 4. The discrete series 5. The complementary series 6. The spectrum of L2(1'\PSL(2, R)), in the compact case 7. Harmonic analysis on the Poincaré upper half plane = A2 + B2 + on SL(2, R) 8. The subelliptic operator

177 177 181 188 193 195 196 199 202

9. SL(2, C) and More General Lorentz Groups 1. Introduction to SL(2, C) 2. Representations of SL(2, C) 3. The Lorentz groups SO(n, 1)

204 204 209 221

10. Groups of Conformal Transformations 1. Laplace operators and conformal changes of metric

2. Conformal transformations on R', S", and balls 11. The Symplectic Group and the Metaplectic Group 1. Symplectic vector spaces and the symplectic group 2. Symplectic inner product spaces and compact subgroups of the symplectic group 3. The metaplectic representation

226 226 227 235 235 239 241

CONTENTS

12. Spinors 1. Clifford algebras and spinors

2. Spinor bundles and the Dirac operator 3. Spinors on four-dimensional Riemannian manifolds 4. Spinors on four-dimensional Lorentz manifolds 13. Semisimple Lie Groups 1. Introduction to semisimple Lie groups 2. Some representations of semisimple Lie groups Appendixes A. The Fourier transform and tempered distributions B. The spectral theorem

C. The Radon transform on Eudidean space D. Analytic vectors, and exponentiation of Lie algebra representations

vii

246 246 256 260 264 268 268 279

287 292 298 300

References

313

Index

327

Introduction This book began as lecture notes for a one semester course at Stony Brook, on noncommutative harmonic analysis. We explore some basic roles of Lie groups in linear analysis, concentrating on generalizations of the Fourier transform and the study of naturally occurring partial differential equations. The Fourier transform is a cornerstone of analysis. It is defined by (1.1)

=

=

f

dx,

where, given I E L', one has / E L°°. if / and all its derivatives are rapidly so is f. if you define

decreasing (one says f E (1.2)

=

=

dx

then one has the Fourier inversion formula (1.3)

rY=lr=I

onS,

and hence I extends uniquely to a unitary operator on L2(R?t); this is the Plancherel theorem. These facts are proved in Appendix A, at the end of this monograph. The Fourier transform is very useful in deriving solutions to the classical partial differential equations with constant coefficients, because of the intertwining property (1.4)

1(8/Ox3) =

which implies (1.5)

IP(D) =

is any polynomial. For example, the fundawhere D1 = and mental solution to the heat equation P(t, x), defined by (1.6)

(8/Ot)P(t, x) = x), P(O,x) = 5(x), ix

INTRODUCTION

x

transformed to an ODE with parameters for the partial Fourier transform P(t, is

(d/dt)P(t,

(1.7)

P(t,

=

=

and computing the inverse Fourier transform of this Gaussian (see Appendix A) gives (1.9)

P(t,x) =

=

It is suggestive, and quite handy, to use the operator notation P(t, x) =

(1.10)

where, for a decent function is defined to be From here we can apply a nice analytical device, known as the subordination

identity, to derive the Poisson integral formula in the upper half space. The subordination identity is (1.11)

j

=

e_v2/4te_tA2t_3/2 di,

if A > 0.

We will give a proof of (1.11) at the end of this introduction. By the spectral theorem (or the Fourier integral representation for A = this identity also holds for a positive selfadjoint operator A. We can apply it to A = substituting (1.9) for to get

j

(1.12)

=

di

[

di

Jo = cny(y2

+

the last integral is easily evaluated using the substitution s = 1/i. If n 2, then maps the space of compactly supported distriwhere

butions e'(R") to the space of tempered distributions by integrating (1.12) with respect to y. We obtain

=

(1.13)

and is obtained

+

We can analytically continue these formulas to complex y such that Re y> 0, and then pass to the limit of purely imaginary y, say y = it, to obtain a formula for the solution of the wave equation on R x

(1.14)

(02/812

—

= 0,

u(0,x) = 1(x),

=

INTRODUCTION

xi

which is (1.15)

u(t, x) = cos

+

sin

In fact we have

=

(1.16)

lim(1x12 — dO

(t —

Taking real and imaginary parts, we can read off the finite propagation speed,

and the fact that the strict Huygens principle holds when n 3 is odd. For example, if n =3, some elementary distribution theory applied to (1.16) yields (1.17)

sint(—iX)'/28(x) = (4irt)18(IxI — t).

In the last paragraphs we have sketched one line of application of Fourier analysis, which is harmonic analysis on Eucidean space, to constant coefficient PDE. One reason such analysis works so neatly is that such operators commute with translations, i.e., with the regular representation of RTh on L2(R") defined by (1.18)

R(y)u(x) = u(x + y),

x,y E if'.

The Fourier transform provides a spectral representation of R(y), i.e., intertwines

it with (1.19)

7

=

and thus any operator commuting with R(y) is intertwined with a multiplication operator by 7. This restatement of (1.5) is obvious since (1.4) is just a differentiated form of (1.19). From this point of view we can look upon the functions of differential operators such as (1.12) or (1.16) as being obtained by synthesis of very simple operators (scalars) on the irreducible representation spaces into which the representation (1.18) decomposes. Many partial differential equations considered classically, particularly boundary problems for domains with simple shapes, exhibit noncommutative groups of symmetries, and noncommutative harmonic analysis arises as a tool in the investigation of these equations. The connection between solving equations on domains bounded by spheres and harmonic analysis on orthogonal groups is one basic case. Sometimes symmetries are manifested not simply as groups of motions on the underlying spatial domain, but in more subtle fashions. For exwe have the ample, for the harmonic oscillator Hamiltonian + 1x12, on rotation group SO(n) acting as a group of symmetries in an obvious manner, but actually SU(n) acts as a group of symmetries, by restricting the "metaplectic representation," introduced in Chapter 1 and discussed further in Chapter 11. As another such example, we mention the action of the "ax + b group" on L2 functions on a cone, which will be explored in Chapter 7. Harmonic analysis, commutative and noncommutative, plays an important role in contemporary investigations of linear PDE. For example, pseudodifferential operators are certain variable coefficient versions of Fourier multipliers

xii

INTRODUCTION

(convolution operators). They can be used to impose an approximate translation symmetry on elliptic operators, such as Laplace operators on general Riemannian manifolds, and are particularly effective in treating various analytical aspects of elliptic equations and elliptic boundary problems. Fourier integral operators allow one to impose a translational symmetry on an operator with simple characteristics (and real principal symbol). However, when a Fourier integral operator is applied to an operator with multiple characteristics, in cases where the characteristic set can be put in some sort of normal form via a symplectic transformation, one is often led to a problem in noncommutative harmonic analysis. These notes will not be concerned much with such aspects of harmonic analysis. Presentations in the commutative case can be found in [121, 234, 239], and an introduction to "noncommutative microlocal analysis" in [2351. Although we do not consider the general theory here, one thread to be followed through these notes is the use of noncommutative harmonic analysis to treat certain important model cases of operators with double characteristics. The basic method of noncommutative harmonic analysis, generalizing the use of the Fourier transform, is to synthesize operators on a space on which a Lie group has a unitary representation from operators on irreducible representation spaces. Thus one is led to determine what the irreducible unitary representations of a given Lie group are, and how to decompose a given representation into irreducibles. This general study is far from complete, though a great deal of progress has been made on important classes of Lie groups. These notes begin with an introductory Chapter 0, dealing with some basic concepts of Lie group representation theory. We lay down some of the foundations for the study of strongly continuous representations of a Lie group G, and representations induced on various convolution algebras, including the algebra of compactly supported distributions on G, and the universal enveloping algebra of G, which includes its Lie algebra. Members of the Lie algebra, or more general elements of the universal enveloping algebra, are typically represented by unbounded operators, a fact which requires some technical care. Spaces of smooth vectors for a representation, introduced by Gkding originally to complete some arguments of Bargmann, and also analytic vectors, provide useful tools to handle these technical problems. Chapter 0 also includes some descriptions of various types of Lie groups one runs into, such as nilpotent, solvable, semisimple, etc. In Chapter 1, we begin our study of analysis on specific Lie groups with a look

at the Heisenberg group. This nilpotent Lie group has a representation theory that is at once simple and rich in structure. Its irreducible unitary representations were classified by Stone and von Neumann many years ago, when the group was studied because of its occurrence in basic quantum theory. Much terminology associated with the group, in addition to its name (actually, many physicists

call it the Weyl group), celebrates this. For example, certain of the group's irreducible unitary representations are called Schrödinger representations, and other unitarily equivalent forms are called Bargmann-Fok representations. We

INTRODUCTION

xii

obtain an analogue of the Fourier inversion formula easily in this case, and postpone proving that all irreducible unitary representations are given until Chapter

5. A study of the Heisenberg group naturally introduces other important Lie groups, in particular the symplectic group, acting as a group of automorphisms of the Heisenberg group, and the unitary group, arising as a maximal compact subgroup of the symplectic group. It might seem unorthodox to treat the Heisenberg group before the unitary group or other compact Lie groups, since most of its irreducible unitary representations are infinite-dimensional. However, it is this group for which it is easiest to develop harmonic analysis in closest analogy to the development on Eucidean space R". This may be traced to the fact that the Heisenberg group really has the simplest representation theory, compared to compact groups like SU(n) or SO(n), for which combinatorial considerations arise even to list all their irreducible representations. That the classification is particularly simple for SU(2) and SO(3) still does not make harmonic analysis on these groups as easy as on the Heisenberg group, partly for the basic reason that integrals are easier to work with than infinite series, and, not unrelated, partly because of the richer set of automorphisms of the Heisenberg group, particularly groups of dilations.

Chapter 2 studies the unitary groups U(n), and Chapter 3 pursues some general results about compact Lie groups. The complete description of the irreducible representations of U(n) and STJ(n), for general n, spills over into Chapter 3, where it is produced as a consequence of the Theorem of the Highest Weight. In fact, we give a complete proof of this theorem only for SU(n). Chapter 3

contains a number of general results about compact Lie groups, such as the Peter-Weyl theorem and the Borel-Weil theorem. We have also emphasized the study of compact group actions on eigenspaces of Laplace operators in Chapter 3. The interplay between the orthogonal group and harmonic analysis on spheres is discussed in Chapter 14. Here we have tried to include short, simple derivations for a number of classical identities from the theory of spherical harmonics. In Chapter 5 we discuss a major general tool for constructing representations

of Lie groups, the method of induced representations. We discuss Mackey's results on systems of imprimitivity and applications to some important classes of Lie groups, such as the groups of isometries of Eudidean space and some other

groups that are semi-direct products. This chapter also gives a proof of the Stone-von Neumann theorem classifying the irreducible unitary representations of the Heisenberg group. Chapter 6 pursues the study of nilpotent Lie groups. The theory of the Heisenberg group plays a crucial role here; the representation theory in the general case is reduced by an inductive argument to that of the Heisenberg group. Chapter 7 gives a brief study of harmonic analysis on cones. The basic analytical tool for this study is the Hankel transform (1.20)

=

f

INTRODUCTION

xiv

dr) onto L2

The map H1, is unitary from L2

and the Hankel

inversion formula says

H1,(H1,f)(r) = 1(r).

(1.21)

We produce the Hankel transform as an operator intertwining two irreducible unitary representations of the "ax + b group," a two-dimensional solvable Lie group whose irreducible unitary representations were classified in Chapter 5. We remark that special cases of the Hankel transform arise in taking Fourier transforms of radial functions on R". In fact, if /(x) = 1(r) on R", a change of

variable yields

=

(1.22)

f

f

Sn-i

=

/1 —

=

)(fl_3)/2 ds

(r)

in view of the integral formula (1.24)

J1,(z) =

+

f(i

—

dt

for the Bessel function. The unitarity of H,, and the Hankel inversion formula, for ii = — 1, follow from the Fourier inversion formula and Plancherel theorem. These results are not accidentally special cases of the result (1.21), but conceptually so, since Eucidean space R" is the cone over S"1, and both sets of formulas arise from the spectral representation of the Laplace operator. In Chapter 8 we study the group SL(2, R). This group covers the Lorentz group SOe(2, 1), and hence is important for relativistic physics. We derive the classification of the irreducible unitary representations of SL(2, R) achieved in the famous paper of V. Bargmann [12], and study a number of topics in harmonic analysis on this group, and also on the two dimensional hyperbolic space. Chap1), and describes ter 9 studies SL(2, C), which covers the Lorentz group its repesentations, given by Gelfand et al. [71]. More general Lorentz groups SOe (n, 1) are also introduced. Our study of their representations is somewhat less complete, though we do describe the principal series, which provides "almost all" the irreducible unitary representations for n odd. Chapter 10 considers the actions of 1) as groups of conformal transformations on balls and spheres. We show that the existence of a certain (nonunitary) representation of SOe(fl, 1) implies Poisson's on the space of harmonic functions on the unit ball in integral formula for the solution to the Dirichiet problem in the ball.

In Chapter 11 we return to the symplectic group Sp(n, R), introduced in Chapter 1, and make a detailed study of the metaplectic representation. Our

INTRODUCTION

xv

study uses the Cartan decomposition of the symplectic group, a special case of

a decomposition that plays an important role in the general study of semisimple

Lie groups. The Cartan decomposition is useful in studying the metaplectic representation, because the action of the double cover of U(n) C Sp(n, R) takes a particularly simple form in the Bargmann-Fok representation. Chapter 12 is devoted to spinors. The spin groups are double covers of SO(n), and more generally of q). We define Dirac operators on manifolds with

spin structures, but do not really pursue harmonic analysis for these objects, though there is a great deal that has been done. Some of this is hinted at in the final chapter of these notes, and material given here on spinors, it is hoped, will make this intelligible. Chapter 13 is a very brief introduction to the general theory of noncompact semisimple Lie groups, illustrated to some degree by the material of Chapters 8—11. We describe some general results on principal series and discrete series, with no proofb but references to some basic sources. Much more extensive introductions to this vast topic can be found in the last two chapters of Wallach [253], and in the two volume treatise of Warner [256]. This subject is under

intensive development at present, some of which is described in the more recent reports [137, 140]. Also see the new book [136]. The course from which these notes grew was given to students who had been through one semester with the book of Varadarajan [2461. The present mono-

graph is addressed to people with a basic understanding of Fourier analysis (commutative harmonic analysis) and functional analysis, such as covered in one of the basic texts, e.g., Reed and Simon [201], Riesz-Nagy [202], or Yosida [267].

A concise statement of the ideal preparation would be "Chapters 6, 9, and 11 of Yosida," together with the first few chapters of a standard Lie group text, such as [246] or [100]. We have included four appendixes, one on the Fourier transform and tempered distributions, one on the spectral theorem, emphasizing the use of tempered distributions in the proof of the spectral theorem, one on the Radon transform on Euclidean space, proving some results needed in Chapter 6, and one on analytic vectors, discussing some technical analytical points often encountered in passing from a representation of a Lie algebra to a representation of a Lie group. These notes focus on ways specific Lie groups arise in analysis. The emphasis

is on multiplicity, rather than unity. We do treat some general approaches to certain classes of Lie groups, especially the classes of compact and of nilpotent Lie groups, and to a lesser extent the class of semisimple Lie groups, but our emphasis is on the particular. One unifying theme is the relationship between harmonic analysis and linear PDE, the use of results from each of these fields to advance the study of the other. In addition to certain operators with double characteristics, mentioned earlier, we also study the "classical" PDE, the Laplace, heat, and wave equations, in various contexts. For example, we show how exact solutions to the wave equation store

xvi

INTRODUCTION

up a great deal of information about harmonic analysis on spheres and hyperbolic space. For the past twenty years or so it has been an active line of inquiry to see how the spectral analysis of the Laplacian on a general manifold (especially in the compact case) stores up geometric information, and how such analysis can be achieved by parainetricea for the heat equation and for the wave equation. To give another illustration of this point of view, we mention a PDE approach to the proof of the subordination identity (Lii). We begin with the observation

that a direct proof of (1.12), for n = 1, will imply the identity (Lii), on the operator level, with A replaced by (—&/dx2)'/2, which in turn implies (1.11) for all positive real A, by the spectral theorem for obtain a proof of the Poisson integral formula for ii = (1.25)

Thus it suffices to 1.

But

= (21r)_'f = (2ir)'

J

+

0

= (2irY'[(y — ix)1 + (y + ixY'] = ir'y/(y2 + x2). This proves the subordination identity (1.11). A more classical method of proof is to take the Mellin transform with respect to y of both sides of (1.11), deducing that (I.ii) is equivalent to the duplication formula

= 22z_lr(z)r(z + for the gamma function. The subordination identity will make another appear-

ance, in Chapter 1, in a study of various PDEs on the Heisenberg group. The reader will find that, once the material of the introductory Chapter 0 is understood subsequent chapters can be read in almost any order, and by and large depend on each other only ioosely. For example, if one wanted to read about the theory of nilpotent Lie groups, one would need only 1 and 2 of Chapter 1 and and 2 of Chapter 5, before tackling Chapter 6. In particular, later sections of chapters with numerous sections, dealing with special problems, can be bypassed without affecting one's understanding of subsequent chapters.

CHAPTER 0

Some Basic Concepts of Lie Group Representation Theory The purpose of this introductory chapter is to provide some background for the analysis to be presented in subsequent chapters. We give precise notions of strongly continuous representations of Lie groups, and show how they give rise to various sorts of representations of convolution algebras, Lie algebras, and universal enveloping algebras. We point out some special features of irreducible representations, particularly irreducible unitary representations. We will be mainly concerned with unitary representations in this monograph, but we have included general discussions of Banach space representations in this introductory chapter, since on the general level considered here it does not matter

usually; we do not hesitate to retreat to unitary representations when a more general situation would entail the slightest additional complication. We also recall some of the order that has been imposed on the panoply of Lie groups. We assume the reader has had an exposure to Lie groups, so many of the subjects of this introductory chapter should be familiar. We have concentrated

on some of the technical elementary aspects, which tend not to be covered in introductory Lie group texts, having to do with the fact that infinite-dimensional representations generally have unbounded generators. §1 gets things started, with a discussion of one parameter groups of operators and their generators.

1. One parameter groups of operators. In this section we look at representations of the Lie group R of real numbers. If B is a Banach space, a one parameter group of operators on B is a set of bounded operators (1.1)

tER,

V(t):B—*B,

satisfying the group homomorphism property (1.2)

V(s + t) = V(s)V(t),

for all a, t E R,

and (1.3)

V(O)=I. 1

BASIC CONCEPTS

2

We also require a continuity property. The appropriate requirement is strong continuity, which means (1.4)

t,

t

V(t,)u —' V(t)u in B, for each u E B.

A one parameter group of operators will by definition satisfy (1.1)—(1.4). An important set of examples is the set of translation groups L"(R) —' L"(R),

(1.5)

1 p < 00,

defined by

= f(x — t).

(1.6)

The properties (1.1)—(1.3) are clear in this case. Note that = 1 for each t. Also, we see that = 2 if t t', by applying the difference to a — function f with support in an interval of length It — t'I. It takes a little effort to verify the strong continuity (1.4), and we now show how to do this.

Note that C8°(R) is dense in each Banach space LP(R), for 1 p < 00. = f(x—t,) t, —+ t, then all the functions

Furthermore, if f E

have support in a fixed compact set, and converge uniformly to f(x—t), so clearly (R). The proof that we have the convergence in (1.4) in norm for each f (1.5)—(1.6) is strongly continuous is completed by the following simple lemma. LEMMA 1. 1. Let 7'3: B1 —' B2 be a uniformly bounded set of linear operators

be a dense linear subspace of B1, and suppose

between Banach spaces. Let

T3tz—T0u

(1.7)

in the B2-norm, for each u e

Then (1.7) holds for all u E B1.

PROOF. Given u E B1,e > 0, pick v lJTjfl

such that lu — vu <e. Suppose

Then JT,u — Toull = 117ju — T7v

+ Tjv — T0v +

Toy

—

Toull

— T3v11 + llT3v — Tovil + llTov — ToujI

0 is

We note that a locally uniform bound on the norm (1.8)

IIV(t)iI

< M for ti 1

for some M E [1,00) holds for any strongly continuous group, as a consequence of the uniform boundedness theorem. As a consequence of (1.8), we have, for all E R, (1.9)

IIV(t)ll
K where IIV(t)II this is impossible unless (u,w) = 0. This proves the proposition. Proposition 2.2 provides a very nice tool for obtaining results on essential selfadjointness of iA, when V(t) is unitary, and also of all powers (iA)k. See Appendix B for more on this. As slick as arguments using Proposition 2.2 can be, it is also instructive to see directly that, for an appropriate approximate identity E C8°(G), if u E P(ir(X)), then ir(f1)u —+ u in the topology of P(ir(X)), i.e., ir(X)ir(f1)u converges to ir(X)u in B. A direct attack on this involves comparing ir(X)ir(f1) with ir(f3)ir(X). This gives us an excuse to analyze analogues of (2.15) and (2.33), with the order of the operator factors reversed. For ir(f)ir(gj), 91 E G, we have, in place of (2.15), (2.37)

ir(f)ir(gi)u

= j f(g)ir(ggi)udg.

Now the Haar measure dg was constructed to be left invariant, not necessarily right invariant. But a right translate of dg is clearly still left invariant, so, as

BASIC CONCEPTS

14

stated before, it must be a scalar multiple of dg. Hence, for any integrable function h on C, h(ggi) dg =

(2.38)

where the factor

L h(g) dg

called the modular function, defines a homomorphism

(2.39)

of C into the multiplicative group of positive real numbers. Hence (2.37) yields

ir(f)ir(gi)u =

(2.40)

IG

Now, for X E g, 'yx(t) = exptX,

=

(2.41)

+

is the left invariant vector field on C matching up with X at TeG. generates the flow of right translation by Note that, since where

div

(2.42)

=

Consequently, from (2.40) we derive, for u E V(ir(X)),

(2.43) where

+

ir(f),r(X)u =

=

=

is the formal adjoint of

+

as a first order

differential operator. We remark that the Lie bracket on g is defined so that 1'

TIX,Y]

—

v v Eg. 4'k,l

rTX

The difference in sign in (2.34) is explained as follows. Consider the diffeomor= g1. This map produces a map ic on vector phism ic: C C given by fields, and, for X E g = TeC, we have (2.45)

=

preserves Lie brackets, this shows that (2.44) and (2.34) are equivalent. One could imagine reversing conventions, i.e., reversing the signs in (2.34) and (2.44), but this would mess up the signs in (2.36). This gives one reason for the convention (2.44). Suppose a sequence 1, E C0°°(C) is picked as follows. Fix some coordinate Since

chart U about e E C, identified with the origin; you could use exponential coordinates. Put Lebesgue measure on U, matching up with the coordinate expression for Haar measure at e = 0. Let fi(x) E C000(U), fi(x)dx = 1, and set f,(x) =f'fi(jx), n = dimC. Then = 1, and = —, 1 as j —' oo. We now establish the following result, which provides a second proof of Proposition 2.1.

BASIC CONCEPTS

15

PROPOSITION 2.3. If u E V(ir(X)),X E 9, then

ir(f,)u

(2.46)

u

in D(ir(X))

j -4 00. PROOF.

What we must show is that

(2.47)

ir(X)ir(f,)u —' ir(X)u in B.

Since we know ir(f,)ir(X)u —' ir(X)u, consider the sequence of commutators (2.48)

K,u = ir(X)ir(f,)u — ir(f,)ir(X)u,

defined a priori for u E P(ir(X)). We want to show that (2.49)

K1u

for u E D(ir(X)).

0

By (2.33) and (2.43), we have (2.50)

K1u =

—

—

Clearly each K, extends uniquely to a bounded operator on B. In fact, we claim

(K, } is a unformly bounded family of operators on B. This follows because is a vector field on C whose coefficients vanish at e. From the given — form of in local coordinates, it easily follows that (2.51)

n/2, then u is bounded and continuous. I/k = 1 > 0, then D°u is bounded and continuous for all al

Ico

1.

PROOF. By Proposition 2.5 of Chapter 0, u is a Ck vector for if

(2.12)

Dau

Thus

for al

(2.13)

for all hal

k, i.e.,

(1 +

€

p

+ 1, Ico > n/2,

k.

if and only

THE HEISENBERG GROUP

48

But for k > n/2, using Cauchy's inequality, we have

f

=

f(i +

+ 1/2

1/2

(2.14)

.

0 there is an interval [—N,N1 = I such that fR\I 1u3(x)I2dx < e for all j. Hence u31, —+ u in the L2(R) norm. It follows that (6.7)

(I + H)—' is a compact operator on L2(R).

Thus L2(R) has an orthonormal basis consisting of eigenfunctions for H. Each eigenspace has finite dimension (dimension one for (6.5), as we will see in a moment), and the eigenvalues of H must tend to +00. In a moment we will see exactly what they are. Generalizing (6.6), we find (6.8)

V(Hlc) = {u E L2(R):

E L2(R) for j + 1 2 k}.

In particular, (6.9)

flv(Hlc) = S(R),

so all the eigenfunctions of H belong to S (R). As is emphasized in quantum mechanics texts, one easy way to analyze the spectrum of (6.5) is to use the operators (6.10)

= d/dx —

A = —d/dx — x,

x.

A calculation gives

AA=H+1.

(6.11)

The first identity shows 1 is the smallest possible eigenvalue of H. These identities imply (6.12)

AH=AAA+A,

and hence (6.13)

(A, H] = 2A.

Similarly, (6.14)

[A*,H] = —2K.

The identities (6.13)—(6.14) are equivalent to (6.15)

HA = A(H - 2),

HA = A(H + 2).

Hence, if the eigenspace decomposition of H is (6.16)

L2(R) = H

THE HEISENBERG GROUP

63

then (6.17)

and

A: V —i

(6.18)

In particular, if p E spec H, then either p —

2 E spec H or A annihilates From (6.10) we see A annihilates only the linear span of

ho(x) = R._1/4e_X2/2

(6.19)

The factor ir"4

has been thrown in to make h0 a unit vector in L2(R). Thus we see that Vi is the linear span of h0 and that

specH={1,3,5,7,...}.

(6.20)

By (6.11), A: is an isomorphism for p 3 an odd integer, so each V2k+1 is one-dimensional and is the linear span of (6.21)

hk(X) = Ck(d/dX — x)ce_E2/2 = ckHk(x)ez/2,

where Hk(X) is the Hermite polynomial

Hk(x) = Ik/21

(6 22)

= i=o

The constants ck are chosen so II hk = 1; they will be specified shortly. Since the eigenspaces V,.1 are mutually orthogonal, we know the hk(x) are mutually orthogonal, i.e.,

(6.23)

j

if j

dx = 0

k.

We can evaluate the ck in (6.21) by noting that, with hk C V2k+1, (6.24)

IIA*hkII2

= (AA*hk, hk) = 2(k + 1)IIhkII2.

Consequently, if IIhkD = 1, in order for

hk+1 =

(6.25)

to have unit norm, we need (6.26)

'Yk+l = (2k + 2)_h/2.

Thus the constants ck in (6.21) are given by (6.27)

Ck =

Another approach to (6.27) is to use the recursion formula (6.28)

Hk+1(x) — 2zHk(x) + 2kHk_l(x) = 0

(convention: H_i(x) = 0),

THE HEISENBERG GROUP

64

which can be deduced from the generating function identity

= e2xt_t2.

(6.29) See

Lebedev [154] for details on this. Note that (6.29) follows immediately from

(6.22).

Having shown that the spectrum of —d2/dx2 + x2 consists of the positive odd integers, all simple eigenvaluea, we deduce that the spectrum of + 1x12 on consists of all the positive integers of the form n + 2j, = 0,1,2,...: (6.30)

specH={n,n+2,n4-4,n+6,...}.

Note that, for H =

+

1x12

on

R", an orthonormal basis of the n + 23

eigenspace of H is given by

aI=j}.

(6.31)

The combinatorially rather complicated form of the Hermite functions given

by (6.21)—(6.22) induces one to seek a simpler approach. In fact, use of the Bargmann-Fok representation provides a cleaner route to understanding the spectrum of H. Note, from formula (5.4), 13i(Co)

= >J(çjO/ôc + O/ôcjçj)

(6.32)

=2

c,O/Ocj + n =

W,

which is equivalent to saying H and W are intertwined by the unitary operator K defined by (5.6)—(5.14).

KHK1 = W.

(6.33)

Now it is very easy to verify that

a 0,

= is an orthonormal basis of the Hilbert space by (5.1), and, by (6.32), (6.34)

=

+

of entire functions on

=

defined

(21a1 +

so the spectrum of W is precisely {n + 2j: j = 0, 1,2,. . .}, and an orthonormal basis for the 2j + n eigenspace of W is given by (Wa: IaI = j}. The formula (6.32) also implies that the unitary group generated by the skew adjoint operator iW is given by (6.35)

=

f E )(.

This is a neat formula. The formula for eit H , which we will consider in the next section, is more complicated.

THE HEISENBERG GROUP

65

We can use this result on the spectrum of the harmonic oscillator Haxniltonian H to determine invertibility of the operators (La), where

La = Lo + iaT.

(6.36)

Note, from (2.14) and (6.3), Aa = —A(H ± a).

= —AH

(6.37)

From the analysis of the spectrum of H, we have (La) is invertible, for all X E (0, oo), if and only sf

PROPOSITION 6.1. (6.38)

avoids the set {n+2j:j=0,1,2,...}.

As will be seen later, this condition is equivalent to hypoeffipticity of the operator La. We now consider more general second order differential operators on H', namely operators of the form Po =

(6.39)

a,kY,Yk j,k=1

where

= L,

(6.40)

= M,,

1 I n,

and (a,k) is a symmetric, positive definite matrix of real numbers. By (2.14) we have (6.41)

1 j fl,

= ±iA"2z1, so if Po is given by (6.39),

= —AQ(±X,D),

(6.42)

where

Q(z,

= j,k=1

with

The operator Q(±X, D) appearing in (6.42) is a positive selfadjoint second order differential operator, and we want to find its spectrum. This is gotten from studying the interplay between the quadratic form Q(x, on and the symplectic form (6.43)

(x', c')) = z.

We define the Hamilton map of Q(z, (6.44)

on

—

to be the linear map F on

c(u,Fv) = Q(u,v),

u,v E

given by

THE HEISENBERG GROUP

66

where Q(u, v) is the symmetric bilinear form on polarizing the quadratic form Q(u), i.e., Q(u) = Q(u, u). We are assuming Q is positive definite. So we see that F is skew symmetric and invertible, so its eigenvalues must all be pure imaginary, nonzero, and occur in complex conjugate pairs, i.e., be of the form

1 j n,

p, > 0. It turns out that we can pick a symplectic basis of (6.45)

diagonalizing Q, as

follows.

LEMMA 6.2. If Q is positive definite, there is a symplectic basis of {e1, f,: 1 j n}, i.e., a basis satisfying

a(e,,ek) = 0 = o(f,,fk),

(6.46)

a(e,,fk) =8jk,

such that, if u

(6.47)

a3e3

+

= then

+

Q(u, u)

(6.48)

= with

given

by (6.45).

PROOF. F' is characterized by a(u,v) = Q(u,F'v), so F' is skew symmetric with respect to the inner product Q. Thus there is a basis {E1, F,: 1 orthonormal with respect to Q, such that j n} of

F'F,=A,E1, Set e3 = of since

A,>0.

f, = )ç"2F,. Then {e,, f,: 1 j

0), and, on the boundary of this region, i.e., the imaginary axis, P3(t,0) continues analytically as long as —s2/A avoids the points jir, = 1,2,3 Thus = is analytic as long as 4ltljir, = 1,2,3 Note also that P,8(t,0) and P_88(t,0) agree for a2/41t1 < ir, so vanishes for ti > 82/47r. This is a special case

j

j

of the finite propagation speed which we will derive. We want in general to deform the contour so that its image g('y') will hug

a segment of the positive real axis. Let us note that, for x E R, z cot z is monotonically decreasing from 1 to —oo for x E [0, ir), with derivative zero at

x=

0, going to —00 as x —+ ir. Thus, assuming A > 0, for x E [0, ir), g(x) increases from g(0) = B to a maximum at z = zO and then decreases to —00 as z ir. The point z0 = xo(A, B) E [0, ir) is defined by g'(x0) = 0, i.e.,

(8.12)

Asin2z0 = B(xo —sinxocosxo),

since (8.13)

g'(ç) = [A sin2

—

B(ç

—

sin ccos ç)]/ sin2 ç.

Thus deforming the path so it crosses the real axis at x0 and proceeds for a while along the curve orthogonal to the real axis through x0 along which g(ç) is real-valued, effects a deformation of the contour g(-y) to a curve which hugs a segment of the real axis [E, E + E). See Figure 8.3. Here (8.14)

E = E(A,B) =

max{OA,B(x):

0 x

tdtere

E(A,B)

is defined by (8.14).

Note the simple estimate

E(A,B)B+coA

(8.15)

for some positive c0. We proceed to make a more gjobal deformation of the path

to a path on which the function g(ç) is real. Start at x0 and go along the path y' orthogonal to the real axis at x0, on which g(ç) is real. Continue until you hit a point where g'(c) = 0. Then make a turn of w/2 counterclockwise (if 9" $ 0 there) and continue, still keeping g(ç) real. In order to analyze what sort of path we get, it is useful to have the following result.

LEMMA 8.2.

is

real and g'(ç) = 0, then ç is real.

PROOF. Note that

g(ç) = (c/sinç)(Bcosç + Asinc)

(8.16) and

= (Bcosc+Asinc — Bc/sinc)/sinc.

(8.17)

Thus, (8.18)

if g'(c) =0, = B(ç/sinç)2 = B'(Bcoeç + Asinç)2.

THE HEISENBERG GROUP

85

In particular, if g(ç) is real, then B cos ç + A sin ç is either real or pure imaginary. It follows that, if

ç=u+iv,

(8.19)

u,vreal,

then either is real, or

Bcosu + Asinu = 0 or

(8.20)

—

Bsinu + Acosu =

0.

Now, write

cosç = cosucoshv — isinusinhv, sinç = sinucoshv + icosusinhv.

(8.21)

if g'(ç) =

0,

we must have, by (8.17),

ç=

(8.22)

cos ç

sin ç + (A/B) sin2 ç,

and substituting in (8.21) and equating real and imaginary parts, respectively, gives

(8.23)

u = (sin u cos u + (A/B) sin2 u) cosh2 v + (sin u cos u — (A/B) cos2 u) sinh2 v, v = (cos2u — sin2u+ 2(A/B)sinucosu)sinhvcoshv.

Now, the first possibility in (8.20) yields (8.24)

(8.25)

u = —(A/B)[1 + (A/B)2]sin2usinh2v, v = —[1 + (A/B)2] sin2 usinhvcoshv,

and dividing these equations gives (8.26)

u = (A/B)v tanh v.

Note that (8.24) implies (B/A)u 0 and (8.26) implies (B/A)u 0. Thus we see that the first possibility in (8.20) does not allow for a nonreal ç satisfying the hypotheses of the lemma. On the other hand, the second possibility in (8.20) yields (8.27) (8.28)

u = (A/B)[1 + (B/A)2]sin2ucosh2v, v = [1 + (B/A)2] sin2 usinhvcoshv.

Also cos u = (B/A) sin u implies 1=

[1

+ (B/A)2] sin2 u,

which, together with (8.28), yields (8.29)

v/ sinh v = cosh v.

But the left side of (8.29) is 1 and the right side is 1, with equality only at v = 0. Again we get no nonreal ç satisfying the hypotheses of the lemma, so the proof is complete.

THE HEISENBERG GROUP

86

It remains to study the real numbers x such that = 0. We have already discussed the unique such point in the interval [0, ir), denoted xo(A, B). From the way çcotç decreases from +oo to —oo on any interval (jir, (j + 1)ir), j 1, one can see that on each such interval, vanishes either nowhere, or on a pair of points ;(A, B) = (Y,[H,X]) =ia(H)(Y,X),

and, by (2.7), this identity is equivalent to (2.8). Note that

a(H0) = (HO,HO) >0.

(2.10)

We are now in a position to prove PROPOSITION 2.2.

=

dim

1

for each root a.

PROOF. Suppose dim Øa 2. Pick linearly independent X, Z E Øa, and pick VE Granted Øo, and are not orthogonal, which we check in a moment, we can suppose normalization made so that (X, Y) = 1, (Z, Y) = 0. This implies [X, Y] = [Z, Y] = 0. Now an inductive argument (exercise) shows

=

(2.11)

+

By (2.10), it follows that all the elements are nonzero. This E would imply dim g = oc, a contradiction. It remains to show and 9a are not orthogonal with respect to (, ). Indeed,

x +iy E g0(x,y E g)

x

—

iy

E

But (x+iy,x —iy) = (x,x)+ (y,y) >0.

This completes the argument. For each root a, pick nonzero vectors = 1. Thus

e

We can normalize so (en,

= iHa.

(2.12)

Note that, by (2.1), we can write (2.13)

e±a=xa±iya,

XQ,y0Eg.

The commutation relations (2.12) are equivalent to (2.14)

[Xa,ya] =

We turn now to the representation theory of C. Let ir be a unitary representation of C on a complex vector space V, dimV < oo. Then ir gives a skew adjoint representation of the Lie algebra g, which in turn extends to a complex linear representation (also denoted ir) of the complexified Lie algebra Cg. As in

COMPACT LIE GROUPS

112

Chapter 2, we also find it convenient to analytically continue ir on C. Let Oc denote the simply connected Lie group with Lie algebra Cg. Cc has a natural complex manifold structure. Let C1 be the Lie subgroup of Cc with Lie algebra g. We claim C1 covers C. To prove this we can suppose that C, being compact, is faithfully represented on a finite-dimensional space W, p: C —, End(W). Then p defines a representation of g, hence of Cg, and hence a representation of Cc, which restricts to the desired covering C1 —+ C. The kernel of this covering homomorphism is a discrete subgroup r of the center of C1. Analytic continuation implies it belongs to the center of Cc, so Cc = Gc/r is a group which is the complexification of C. A complex linear representation ir of the complex Lie algebra Cg of Cc (whose restriction to g exponentiates to C) exponentiates to a holomorphic representation (also denoted ir) of Cc; i.e., with respect to a basis of V, the matrix entries of are holomorphic functions on Cc. denote a maxiTo pursue our analysis of a representation ir of C on V, let mal torus of C. Thus the action of on V can be simultaneously diagonalized. Thus, given A E (2.15)

if we set VA = {v

E V: ir(h)v = iA(h)v, for all h E

we have

V=®VA.

(2.16)

If A E and VA 0, we call A a weight, and any nonzero v E VA a weight vector. For eQ the root vectors in Cg defined above, set E0, = ir(eQ).

(2.17)

We call E0, a raising operator if a > 0 and a lowering operator if a < 0. We want to study the action of the operators E0 on the VA's. To do this, we use the commutation relations (2.18)

[h,eQ] = ia(h)eQ,

hE I),

to get (2.19)

lr(h)EQ = EQ7r(h) + ia(h)EQ.

This implies

PROPOSITION 2.3. For each root a, we have (2.20)

EQ: VA —' VA+Q.

In particular, if A is a weight, and a is a root, then either EQ annihiLates VA, or A + a is a weight. PROOF. if

E VA, we have

(2 21)

which proves the proposition.

= E0(ir(h)e) + ia(h)E0e = i(A(h) + a(h))EQe,

COMPACT LIE GROUPS

113

The ordering we have put on Ij' induces an ordering on the weights. For a given finite-dimensional representation ir, with respect to this ordering there will be a highest weight Am and also a lowest weight From Proposition 2.3 we see

that 221/

for all raising operators

EQ = 0 on

=

0

on VA, for all lowering operators

In general, call a weight A nonraisable if VA is annihilated by all raising operators and call it nonlowerable if VA is annihilated by all lowering operators. Shortly we will show that, if ir is irreducible, then the only nonraisable weight is maximal. At present, we record the following progress.

PROPOSITION 2.4. If ir is a unitary representation of the compact Lie group G on V, dim V < oo, then there exists a highest weight vector and in particular V annihilated by all raising operators. there exists a nonzero weight vector

This result gives a tool for showing that certain representations of G are irreducible, namely,

COROLLARY 2.5. Let ir be a unitary representation of G on V, dim V 0, and let N_ be the subgroup of Cc whose Lie algebra .W_ is generated by the lowering operators a 0), then v231, . . . ,V2,j,n_i on the next Vn...2 columns, etc., and

COMPACT LIE GROUPS

119

FIGURE 2.1

write the action of

as

Viii

Vn_1,1,1

(2.57) vi,i,n—i

Many columns will be shorter than the first column, and the pattern of the spots occupied by the Vjkj will look like Figure 2.1, called a Young tableau. This particular example is for SU(6), with i/5 = 2, i14 = 0, i/3 = 3, v2 = 2, and zi1 = 3. v2e3 The representation space WA associated to the highest weight = is contained in (2.55), 80 all vectors in WA c ®N C" = (ØN(Cny)l are antisymmetric under the interchange of any two vectors in any one column of (2.57). An additional symmetry requirement is suggested by our analysis of the representation with highest weight in (CII)(k), which is on the space SkCn. We are

motivated to look at the following two projections in End(ØN C"). Let Q be the projection obtained by antisymmetrization with respect to elements in any common column, and let P be the projection obtained by symmetrization with respect to the elements in any common row. P and Q do not commute, and it is a remarkable, though elementary, fact that C = QP for some C2 = it> 0, so is a projection. It is easy to see that the highest weight vector (2.56) is in the range of this projection. Since the SU(n) action on ®N commutes with the action of the permutation group SN, it follows that the range of is invariant under SU(n). It can be shown that the action the projection of SU(n) on the range of is irreducible, so the irreducible representation i'1e3 is identified. For the details on of SU(n) with highest weight = this last argument and further use of Young tableaux, we refer to [261, 262], or [268].

3. RepresentatIons of compact on elgenspaces of Laplace operators. Suppose a compact Lie group C acts transitively on a compact

Riemannian manifold M as a group of isometries. Then if we set (3.1)

1(g)f(x) =

qFGx

COMPACT LIE GROUPS

120

we have a unitary representation of C on L2(M). Note C°°(M). Now if is the Laplace operator, we have 1(g)/s =

(3.2)

that

1(g): C°°(M) —'

g E C.

Thus if the eigenspaces of A are denoted (3.3)

VA =

{u E C°°(M):

=

—A2u},

we have (3.4)

for each A such that —A2 E specs. We introduce the notion of zonal

K0 be the connected component of the identity in K.

dimensional subspace of C°°(M)

invariant

3(V) = {v E V: l(k)v =

(3.6) and

P0 E M, and let

K= {geG:

(3.5)

Let

function. Fix

Let

V be a finite-

under 1(g). We define v

for all k E K0},

call this the space of zonal functions in V.

PROPOSITION 3.1. If V PROOF. Define

0, there is a nonzero element of 3(V).

E V' by

1eV.

(3.7)

0 since V contains a function not identically zero and C is transitive on M. Thus ker c V is a hyperplane; it is invariant under K0. Thus L = (ker c)-'- c V is also invariant under K0. We have dimc L = 1. It follows from (3.1) that L = C ® LR, dimR LR = 1. Since K0 is connected, and acts on it must act trivially, so the proof is complete. Then

COROLLARY 3.2. If dim3(V) =

1,

then 1(g) is irreducible on V.

PROOF. If 1(g) acts on each factor of V1 V2 = V, we get two linearly independent zonal functions v3 C by Proposition 3.1. We can use this to show that, under certain circumstances, C acts irreducibly on all the eigenspaces VA of DEFINITION. We say M is a rank one symmetric space if K0 acts transitively

on the set of unit tangent vectors to M at po.

PROPOSITION 3.2. If M is a compact rank one symmetric space, then C of on M.

acts irreducibly on each eigenspace VA

PROOF. It suffices to show that dim3(VA) =

(3.8) Suppose —p

0

1.

v and w are two linearly independent elements of 3 (VA). Pick a sequence

such that v(x) and w(x)

are

nonzero for dist(po, x) = r3. (Note that if

COMPACT LIE GROUPS

121

M is a rank 1 symmetric space, then elements 1(x) of 3(VA) depend only on dist(po, x), for x near po.) Since elements of VA are real analytic, this can be arranged. Let Il, = {x E M: dist(po,x) 0, t > 0.

—

sinht(2cosht —

We suppose n — 1 2. In fact, (2.9) already furnishes the formula for (2.5), the solution to (2.1). Note the formal equivalence therefore between the Laplace equation (2.1) on x and the Laplace equation (1.4) on the unit ball in with solution (1.9).

Our next goal is to analyze the wave equation (2.3). By (2.7), it will be desirable to analyze and which we will obtain from (2.8), (2.9) by analytic continuation. In fact, both sides of (2.8), (2.9) are holomorphic in {t E C: > 0}, so, for any e >0, we can write

=

(2.10) (2.11)

=

—

e)

—

sinh(it — e)[2cosh(it

—

—

HARMONIC ANALYSIS ON SPHERES

131

Now we can pass to the limit e j 0, to obtain

=

(2.12) (2.13)

—

—

=

—

2isinhesint

—

Now we have

sint,, =

cost,.' =

(2.14)

so (2.12), (2.13) provide formulas for these kernels. For example, on S2 (n = 3), we have

=

(2.15)

which implies that, for Iti

—2A3(2 cost —

2cos0)"2

ir,

= j — 2A3(2cos0 — 2cost)'/2,

(2.16)

tO,

01

< ti,

101> ItI,

with an analogous expression for general t, determined by the identity Zr1 sin(t + 27r)z/ =

sin

on

The last line on the right in (2.16) reflects the well-known finite propagation speed for solutions to the hyperbolic equation (2.3). On the other hand, on odd dimensional spheres (where n is even), the exponent on the right side of (2.12) is an integer, so the distribution kernel for must vanish if Iti 0. In other words, the kernel is supported on the shell 0 = Iti. This is the strict Huygens principle, well known for the wave equation on Euclidean space (for odd space dimensions). In case n — 1 is odd, one obtains, from (2.12), (2.13),

(2.17) z.r'sintz.'f(x) =

1

ô

1

(n—4)/2

(sin

and (2.18)

cos tz.'f(x) =

1

1

ô

sins (—-—

sJ(z,

where (2.19) We

7(x,s) = mean value of Jon E3(x) = {y E S"': 0(x,y) =

IsI}.

can examine general functions of the operator ii by the functional calculus

=

(2ir)"2

(2.20)

= (2ir)_1"2

f

J(t)e1"' dt

1(t) costz'dt,

the last identity holding provided f is an even function. We can rewrite this, using the fact that, for n even, cos tii has period 2ir in t, while, for n odd, cos tv

HARMONIC ANALYSIS ON SPHERES

132

has period 4ir in t. This periodicity follows from (2.13), and is equivalent to the assertion that the spectrum of ii consists of integers for n even and half integers for n odd. For more on this, see the first paragraph of §3. In the case of odddimensional spheres S"' this fact, together with (2.18), shows that the kernel of f(v) (for f even) is given by (n—2)/2

(2.21)

f(ii) =

(21r)_h/2

As an example, we calculate the heat kernel on odd-dimensional spheres. Take = Then 1(8) = (2lrt)_h/2e_82/4t, and

f(s + 2irk) = (4irt)'12 (2.22)

k

lv

=

19(8,t)

where i9(s,t) is the theta function. Thus the kernel of

on

(n—i odd)

is given by 1

(2.23)

A similar analysis on

ô

1

= 'for n —

1

(n—2)/2

t9(O,t).

even gives an integral, with the theta

function appearing in the integrand. We omit the details. We end this section with a remark on how to get the fundamental solution to the wave equation on hyperbolic space the simply connected Riemannian manifold of constant curvature —1. This can be achieved by analytic continuation of the metric tensor. In fact, let prj be the origin (north pole) of S" 'in geodesic polar coordinates, and consider the one parameter family of metrics giving the

spaces of constant curvature K. The unit sphere corresponds to K = 1; K E (0,1) corresponds to dilated spheres, and the fundamental kernel for v'sin tz' with (2.24)

=

+ K(n — 2)2/4)1/2

can be obtained explicitly from that on the unit sphere by a change of scale. The explicit representation so obtained analytically continues to all real values of K, and at K = —1 gives the following formula for the wave kernel (2.25)

,c' sinti, =

—

e)

—

where r denotes the distance between the source point and the observation point. Here (2.26)

ii

=

— (n — 2)2/4)1/2.

Thus for odd-dimensional hyperbolic space we also have the strict Huygens principle, i.e., sin iii and costi' are supported on the shell r = ti. Compare with the derivation of Lax and Phillips 1151], which involves an "inspired guess" when

HARMONIC ANALYSIS ON SPHERES

133

1 = 3 (which becomes plausible once one suspects the strict Huygens principle should hold), and a variant of the method of descent to handle n — 1 = 2. That the solution to

n—

82R/8t2

— K(n

—

—

2)2/4)R = 0,

(8/ôt)R(0, x) =

R(O, x) = 0,

(z)

on the space of curvature K depends analytically on K, in an appropriate sense (which can be deduced from the Cauchy-Kowalewski theorem), implies that the formula for z' 1 sin (x) depends analytically on K. It follows that (2.25) is valid.

In connection with the appearance of the operator (2.26), it is useful to note that the Laplace operator on of constant curvature —1, has the following bound on its spectrum: C [(n — 1)2/4, oo).

(2.27)

will be identified with the upper half space in

We give a proof of this.

0, with metric

x;2

(2.28)

A derivation of this metric can be found in Chapter 10, §2. Then the Laplace operator, given on a Riemarmian manifold by = is of the form

=

(2.29) The volume element on (2.30)

we have

E

((—is — (n — 1)2/4)u,u) =

— ((n

dx1

—

+

.

. . .

On the other hand, integration by parts implies — ((n

—

dx1 . .

.

(2.31)

= I J

so, for u E

..

the expression (2.30) is 0. This proves (2.27).

3. Spherical harmonics. We want to determine the spectral projections Ek of ii onto its eigenspaces V,, which of course are the eigenspaces of the Laplace operator Suppose lal, which of course can be derived from the residue theorem, we get (3.46)

=

—

[rxi

2rx1 +

— 1

+v'l—2rxi + r2]'

on S2, where

(3.47)

x2

+ ix3 =

+

denotes the angular coordinate about the x1-axis in R3. Comparing with the case j = 0, one can deduce so

(3.48)

zk,(x) =

—

on

with

=

(3.49)

(_i)i(i —

These functions are called associated Legendre functions. Note that the kernel Ek(x, y) of the projection Ek clearly satisfies

Ek(x,y) =

(3.50)

we have normalized the basis functions by setting 4, = Zkj/IIZk, II. If we substitute the result of (3.48) on the right side of (3.50) and the result of where

(3.12), (3.24) on the left, we get the "addition theorem" for Legendre functions, obtained by Legendre in 1782. We note the following alternate method of producing an orthonormal basis of Vk, in L2(S2). If angular coordinates ,j,O are chosen on S2, with (1,0,0) = P0

the north pole, 9 the geodesic distance from p0, given by (3.47), then the regular representation of SO(3) on L2(S2) defines the operators (3.51)

= 8/Oi,b,

Lj, =

+ icot9ô/ôe,b).

HARMONIC ANALYSIS ON SPHERES

140

As we saw in Chapter 2, all the weight vectors, giving an orthonormal basis of Vk, can be obtained by applying repeatedly the lowering operator L_ to the highest weight vector (3.52)

Wk

= sinC

= (x2 +

Alternatively, one could apply both lowering and raising operators repeatedly to the 0-weight vector zk (x) = Pk(cos 0). In either case, we rederive the formulas (3.48)—(3.49) for the associated Legendre functions, up to normalizing factors. Normalization can be achieved by using the formulas for operator norms of L±, acting on eigenspaces of L1, given by (1.21)—(1.22) of Chapter 2. We leave the details to the reader.

We end this section with one more simple fact about the representation of SO(3) on L2(S2), which is special to this case. PROPOSITION 3.2. The decomposition (3.53)

L2(S2) =

contains each irreducible unitary representation of SO(3), exactly once.

PROOF. As we have seen in Chapter 3, §3, since S2 = SO(3)/SO(2)

is

a rank one symmetric space, the decomposition (3.53) contains precisely once each irreducible representation of SO(3) with a nonzero vector invariant under SO(2), i.e., for which 0 is a weight. In Chapter 2, we classified all the irreducible representations of SU(2) and saw that, among them, precisely those for which 0 is a weight arise from representations of SO(3), so the proof is complete.

4. The subelliptic operators + + iaL3 on 52• If L3 generates the group of rotations about the x3 axis in R3, we can regard L, as an operator on V'(S2), and the Laplace operator on 52 is (4.1)

We want to study the operator (4.2)

P0 =

+

and more generally (4.3)

Recall that [L1, L2] = L3. The operators Pg,, are not elliptic. They have characteristics on the equator {x ES2: X3 0). In fact, charP0 c T(S2) \0 consists of elements (x,e) such that X3 = 0 and such that annihilates any tangent vector V1 E T1(S2) which

is orthogonal to the equator x3 = metric on 52

0,

with respect to the natural Riemannian

HARMONIC ANALYSIS ON SPHERES

We want to show an equivalence between the study of the operator on SO(3) defined by

141

and the study of

+ iaX3, = X? + where X, E SO(3) is the left invariant vector field on SO(3), of unit length, generating the one parameter subgroup of rotations about the z3 axis in R3. This operator (on SU(2) rather than SO(3)) was studied briefly in Chapter 2. Of course, the general theory of pseudodifferential operators can be applied to (4.4)

both of these operators, but we aim to keep the analytical techniques elementary. The correspondence is a special case of the map T from left in'—+ variant operators on C°° (SO(3)) to operators on C°° (S2) which commute with the Laplace operator, which we now define. We can the space of left invariant operators on C°° (SO(3)) with V'(SO(3)), by right convolution. Given f E V'(SO(3)), write

iri(f)=f

(4.5)

f(g)ir,(g)dg,

SO(3)

where {irj} runs over all the representations of SO(3), each contained once in L2(S2), and set

Tf=>irj(f)Pj,

(4.6)

where P3 is the orthogonal projection on L2(S2) whose range is the space where SO(3) acts like irs. Thus,

Tf=JSO(3) f(g)A(g)dg

(4.7)

where

is the regular representation of SO(3) on

=

(4.8)

.

x),

g E SO(3), x e S2.

In other words, Tf = .\(f). The action of Tf on

e

C°°(S2) is therefore given

by

(4.9)

If we let (4.10)

(Tf)ço(x)

f(x)ço(g' x) dg.

= JSO(3) tend to a delta function, we get

Tf(x,y)

J

.

x) dg

SO(3)

=

JK(x,y) 1(g) dg

where (4.11)

K(x,y) = {gESO(3):

gy=z}

is a great circle in SO(3). In (4.10), we are C°°(S2) which commute with with (4.12)

{u e V'(S2 x S2):

the set of operators on

=

HARMONIC ANALYSIS ON SPHERES

142

Note that if f E C°°(SO(3)), then Tf E x S2). We see that (4.10) is a sort of Radon transform. We want to obtain an inversion formula, in order to recover functions of La from functions of the operator Pa. Our first goal will be to impose a Hubert space structure on some subspace of D'(S2 x S2) containing the smooth functions, so that T maps L2(SO(3)) isometrically into this Hubert space. Then T* will map the range of T isometrically onto L2(SO(3)) and will provide the inverse. Note that the orthogonality relations of Chapter 3 imply (4.13)

Ill 11L2(SO(3)) =

+

the dimension of the representation space of ir1 is d, = 2j + 1. Recall that is equal to —3(2 + 1) on V,,. Thus the Hilbert the Laplace operator A on (S2 x S2) should satisfy the identity space norm I 11111 on since

IIITfIII2 = (4.14)

=

+

=

II(—2A +

where we denote by A the operator A1 + A2, a Laplace operator on 52 x Thus we pick our pre-Hilbert space norm on C00(S2 x S2) to be (4.15)

111u1112

= ((—2A + 1)'12u,u)L2(s2Xs2).

We can now construct the adjoint of T. Given u E C°°(S2 x S2), / E C°°(SO(3)), we should have (4.16)

(Tu,f)L2(so(3)) = ((—2A + 1)'12u,Tf)L2(s2Xs2).

If we let f tend to the delta function T*u(g) 4 17

=

D'(SO(3)), we have, by (4.10),

JfS3xS2 (2A + 1)hI'2u(z, y) fK(z,y)

dg' dx dy

I (—2A + 1)'/2u(g. y, y) dy. = JS2 This is our inversion formula.

Returning to our operators La and Pa, we have

PROPOSITION 4.1. Let f(La)öe = kj(g) E V'(SO(3)) and let f(Pa) have kernel k2(x,y) E V'(S2 x S2). Then (4.18)

k2(x,y) =fK(x,y) ki(g)dg,

K(x,y) = {g E SO(3): g• y =

and (4.19)

ki(g) = j(_2A + 1)'/2k2(g.y,y)dy.

CHAPTER 5

Induced Representations, Systems of Imprimitivity, and Semidirect Products In the preceding chapters, representations of groups have been constructed by various straightforward devices. Here we introduce a more sophisticated technique, both for constructing representations and for demonstrating when one has a complete set. In particular, we will be able to show that the set of irreducible unitary representations of H" discussed in Chapter 1 is complete, as well as tackle other groups, such as Euclidean motion groups, which are semidirect products of simpler groups.

1. Induced representations and systems of imprimitivity. Let C be a Lie group, H a closed subgroup, and let

M=G/H.

(1.1)

Suppose for the moment that M has a C-invariant measure Let L be a unitary representation of H on a Hilbert space V. We construct a Hilbert space consisting of measurable functions / on C with values in V such that (1.2)

f(gh) = L(h')f(g),

h E H,

and (1.3)

< oo,

JM

where [g] denotes the image of g in M, under C product (1.4)

C/H = M. We use the inner

=

Then we denote by UL the representation of C on EL defined by

(1.5)

U"(g)f(x) = f(g'x),

g,x E C, / E EL.

One verifies this is unitary. It is called the representation of C induced by the representation L of H. Sometimes it is convenient to use a more complete 143

INDUCED REPRESENTATIONS

144

notation, such as (1.6)

There are important cases one wants to consider where G does not leave invariant a smooth measure on M. A weaker condition to impose is that the smooth measure satisfy the following: if B), then = = pg(x) E for each g E C. (1.7) This always holds in the Lie group case. In this more general case, one can define ML as above and set (1.8)

U"(g)f(z) = p9(z)"2f(g'z).

A function f satisfying (1.2) corresponds naturally to a section of the vector bundle EL over M, constructed from the principal H-bundle C —' M by the homomorphism L: H Aut(V). Thus the fibres of EL are copies of V. The natural C action on EL then defines a representation of C equivalent to UL, via the natural analogue of (1.5) or (1.8). We remark that strong continuity of this representation is easily checked for f a smooth compactly supported section of EL. Strong continuity on ML follows, by Lemma 1.1 of Chapter 0. If M has no smooth measure invariant under C, we can also define using sections of the vector bundle EL® IA1I2 over M, where IA"2 I is the bundle of half-densities (see Guillemin and Sternberg [87]). We will not go into details on this. We next define a system of imprimitivity. Let C be a Lie group, U a unitary representation of C on a Hilbert space M, M a C-space (possibly not transitive), and P a projection valued measure on the Borel sets of M, P(B) being orthogonal projections on M. We say (C, U, M, P) defines a 8y8tem of imprimitivity if P(M) = I and

U(g)'P(B)U(g) = P(g.

all g E C, B C M Borel set. If C acts transitively on M, so M = C/H, we say we have a transitive system of imprimitivity. An important example is provided by M = C/H, U = an induced representation as above, on L2 sections of the vector bundle EL over M, and pL defined by PL(B)f(x) = xa(x)f(x), f section of EL over M. (1.10) In fact, in many important cases, this is essentially the only example, as is shown by the following important theorem of Mackey. (1.9)

B),

IMPRIMITIVITY THEOREM. Let (C, U, M, P) define a transitive system of imprimitivity. Then there exists a representation L of H and a unitary map (1.11)

K:M—+ML

such that (1.12)

KU(g)K' =

for all g E C, B C M Borel.

UL(g),

KP(B)K' =

INDUCED REPRESENTATIONS

145

We give a brief indication of a proof. For more details, see the book [161] of Mackey, or [162]. First, applying the strong version of the spectral theorem, we can suppose = L2(M, W) for some auxiliary Hubert space W, and

P(B)f(x) = x8(x)f(x).

(1.13)

Now define the regular representation R of C on L2(M,

R(g)f(x) =

(1.14)

W) by

x).

Consider now the operator valued function on

Q(g) = U(g)R(g)'.

(1.15) Since

R(g)1P(B)R(g) = P(g• B),

(1.16)

we see that (1.17)

Q(g)P(B) = P(B)Q(g) for all g E G, Borel B C M.

It follows that the unitary operator Q(g) on L2(M, W) is realizable as a multiplication operator, i.e., as the operation of multiplication by a function on M taking values in the set of unitary operators on W: (1.18)

Q(g)f(x) = Q(g,x)f(z);

Q(g,x) unitary on W.

Now for U(g) to be a representation, we require Q(g, x) to satisfy the identity

Q(gig2,x) =Q(gi,x)Q(g2,gi .x).

(1.19)

Let us extend Q from C x M to C x G, writing

= Q(g,x),

(1.20)

x = g'H E M.

Now define

B(g) =

(1.21)

with go fixed. Then

= B(g')'B(gg').

(1.22)

Note that (1.23)

g'h) =

g') for all h e H implies

B(gg'h)B(gg')' = B(g'h)B(g')', for all g, g' E C, h E H,

so if we set (1.24)

L(h) = B(gh)B(g)'

we see that L(h) is independent of g e C. This defines a representation L of H on W. From here one verifies that, if U = UM, p = pM and the process above is used, then one obtains L = M. Next, let (U, P) and (U', P') both lead to the

INDUCED REPRESENTATIONS

146

same representation L of H. Then one can set up P' to act on L2 (M, p, W) exactly as P, and define the operator valued functions Q', Q', B', in analogy with (1.15), (1.20), (1.23). So we have (1.25)

B(gh)B(g)' = L(h) =

which implies

K(g) = B'(g)B(g)

(1.26)

is invariant under right multiplication by h E H, so we have defined a unitary operator valued function K(x) = K(g), z = gH E M, and the desired unitary operator K on L2(M, W) is given by

Kf(z) = K(z)f(x).

(1.27)

2. The Stone-von Neumann theorem. In this section we will prove that conthe set of irreducible unitary representations of the Heisenberg group is given structed in Chapter 1 is complete. Recall that the group law on by (2.1)

where t E R, q E R", p E (2.2)

Denote by C the subgroup

C = {(t,0,0): t E R}.

is scalar Any irreducible unitary representation U of C is the center of and on C. If it is trivial on C, it must represent the commutative group given in Chapter 1: so be one of the one-dimensional representations q,p)

(2.3)

=

So we can restrict attention to U nontrivial on C, say (2.4)

U((t,0,0)) =

We claim that if U is irreducible, it must be equivalent to IrA, constructed in Chapter 1, or equivalently, any two irreducible unitary representations of satisfying (2.4), with the same nonzero A, must be unitarily equivalent. Using the dilations 5±A of H's, we can suppose A = 1. Thus, suppose U is any irreducible unitary representation of (2.5)

satisfying

U((t,0,0)) = e*tI.

Let (2.6)

V(q) = U(0,q,0),

W(p) = U(0,0,p),

so V and W are each unitary representations of the commutative group by (2.1), (2.7)

U(t, q, p) =

and,

INDUCED REPRESENTATIONS

147

Conversely, if V and W are unitary representations of R" on a Hubert space )1, (2.7) defines a representation of H" if and only if W(p)V(q) =

(2.8)

Now apply Stone's theorem to the unitary group V(q), to get a projection valued measure P on R" such that V(q)

(2.9)

=

/

dP(y).

The commutation relation (2.8) is equivalent to (2.10)

W(p)P(B)W(pY' = P(p+ B),

for B c R" a Bore! set. Now R" acts transitively on R", by translation, so (R", W, R", P) is a transitive system of imprimitivity. Note that the subgroup of R" fixing some given point in M = R" is the trivial group 0. The imprimitivity theorem implies there is a unitary map K: E —+ L2(R") such that (2.11)

KWK' =

KP(B)K' =

= U',

where

(2.12)

P'(B)f(z) = xs(x)f(z)

and of course U' is the regular representation of R" on (2.13)

U'(p)f(z) = f(x+p).

This gives the uniqueness of U up to unitary equivalence, and concludes the proof of the Stone-von Neumann theorem.

3. Semidirect products. Let K be a Lie group and A be abelian. In fact, suppose A = R'. Suppose K acts on A by automorphisms (3.1)

K —' Aut(A).

We then define the semidirect product (3.2)

G=Axç,K

to be set-theoretically A x K, with the group law (3.3) We

(a, k). (a', k') = (a +

kk').

want to analyze the irreducible unitary representations of C in terms of those

ofK andA. If U is a unitary representation of C, then, since (3.4) we

(a,k) = (a,e) . (0,k),

can set

(3.5)

V(a) = U(a,e),

W(k) = U(0,k),

INDUCED REPRESENTATIONS

148

and we have U(a, k) = V(a)W(k).

(3.6)

V is a unitary representation of A and W is a unitary representation of K. Suppose conversely that V and W are unitary representations of A and Also,

K, respectively, on a Hilbert space Then (3.6) defines an operator valued function on C = A xi,, K, and it is a representation of C provided for all a E A, k E K.

= V(a),

(3.7)

We will obtain a system of imprimitivity, as follows. Use Stone's theorem to write the commutative group V as (3.8)

V(a)

=

f

dP(a),

a E A.

We see that (3.7) implies (3.9)

W(k)—1P(B)W(k) = P(co(k)' . B)

for any Borel set B C A'. We now aim for conditions yielding a transitive

system of imprimitivity, so we want to force the projection valued measure P to be supported on a single orbit of K in A'. For starters, suppose the representation U of C is irreducible. Now let S be any K-invariant Bore! set in A'. Then (3.9) shows P(S) commutes with W(k)

for all k E K. But certainly P(S) commutes with V(a) for all a E A, so P(S) must commute with U(g) for all g E C = A xv, K. If U is irreducible, this implies either P(S) = 0 or P(S) = I. Now suppose the following condition is satisfied: (3.10)

There is a countable family B1 of K-invariant Borel sets in A'

such that if 0, 0' are disjoint orbits in A', then there are disjoint B11, B12 such that 0 C B,, and 0' C B,2. If this condition is satisfied, we say the K-orbits in A' are countably separated.

It is clear that this condition guarantees that, if (3.11)

J = {j: P(B,) =

then

(3.12)

0 = fl B, is an orbit, IEJ

and (3.13)

P(O) = I.

Thus (K, W, 0, P) defines a transitive system of imprimitivity, under the hypotheses above, so by the imprimitivity theorem there is a representation L of the subgroup Ka of K fixing some a E 0, such that (3.14)

W is equivalent to

=

INDUCED REPRESENTATIONS

149

Note that

0 = K/KC, = G/AKQ,

(3.15)

and the identity (3.6) says U is equivalent to

(3.16)

= is defined by

where the representation XQL of

XQL(a,k) = eaaL(k),

(3.17)

kE

We summarize the result in

THEOREM 3.1. Let G = A x K be a semidirect product with A = R1. Assume the K-orbits in A' are countably separated. Let U be an irreducible unitary representatson of G. Then there is a K-orbit 0 C A', and if K fixing an element a E 0, there is a (necessarily irreducible) unitary representation L of Ka such that U is unitarily equivalent to the induced representation UXa L =

L•

We state, without proof, a result which complements Theorem 3.1.

See

Mackey [162].

constructed above is irreTHEOREM 3.2. The induced representation and are equivducible if L is an irreducible representation of and UXSL2 cannot be alent if and only sf L and V are equivalent. Also equivalent unless xa and are in the same K-orbit in A'. If they are in the is equivalent to UXSL2 for some L2. same orbit, then every When applied to specific examples, the assertions of Theorem 3.2 will be fairly straightforward, and so we will mainly depend on Theorem 3.1 to guarantee that the set of irreducible representations we produce is complete. The first example we consider is called the "ax + b group," the group of affine

transformations of the line. This group is

C2 = R

(3.18)

R is additive, (3.19)

is multiplicative, and

cp(a)b=ab,

in R' = R:

Note that there are three orbits of (3.20)

= {0},

= (O,oo),

0_ =

(—oo,0).

fixing 0 is all of so AK0 = = C2. The For 00, the group K0 C representations obtained are hence the one-dimensional representations of C2. The orbit 0÷ contains 1, i.e., the representation Xi of R given by Xi (a) = e'0. fixing 1 is trivial, so the only irreducible unitary In this case, the subgroup of representation is L = 1. Thus 0+ gives the representation (3.21)

=

INDUCED REPRESENTATIONS

150

By the same argument, O_ gives the representation

=

(3.22)

Thus C2 has exactly these equivalence classes of irreducible unitary representations: the one-dimensional representations and the two infinite-dimensional representations and U_, which can be seen to be contragredient to each other. We note that the Heisenberg group is a semidirect product. To see this, we find it convenient to use the group (isomorphic to Rn), with group law

(t,q,p)Ø(t',q',p')=(t+t'+q•p',q+q',p+p').

(3.23)

Then we see that

=

(3.24)

with

R",

given by

—,

= (t+p•q,q).

(3.25)

Since we have already analyzed the representations of in §2, we will not do so again here. Another important class of semidirect products is furnished by the Euclidean groups, which will be the subject of the next section.

4. The Euclidean group and the Poincaré group. The Eucidean group E(n) is the group of isometries of

E(n) =

(4.1)

where

trices on

Thus it is a semidirect product

0(n)

0(n)

is given by the standard action of orthogonal maThe component of the identity in E(n) will be denoted Eo(n),

so

Eo(n) = R"

(4.2)

S0(n).

According to the results of §3, we can classify the irreducible unitary representations of Eo(n) by looking at S0(n)-orbits in These orbits consist of ={O} and IzI=r},rE(O,00). Itisclearthatthese orbits are countably separated. The orbit Co gives rise to the finite-dimensional representations of Eo(n), arising from Eo(n) SO(n). Given r E (0, oo), pick = to lie on the xe-axis. Then the subgroup Ka of S0(n) fixing aE a is naturally identified with S0(n — 1). (If n = 2, S0(1) consists of only the identity.) For each irreducible unitary representation of S0(n — 1), we can form (4.3)

U,.,A =

al =

and by the results of §3, these exhaust the irreducible unitary representations of

Eo(n).

INDUCED REPRESENTATIONS

151

Note that, for n = 2, SO(n — 1) = {e}, so there is just a one parameter family of infinite-dimensional irreducible unitary representations of Eo(2):

=

(4.4)

For n = 3, SO(n —

1)

= SO(2) =

IaI = r. S1.

In this case, the representations of S' are

given by lrk(ezO) =

(4.5)

so the infinite-dimensional irreducible unitary representations of E0 (3) are given by (4.6)

al = r,k E Z.

=

n = 4, SO(n — 1) = SO(3), whose representations were derived rather explicitly in Chapter 2, §1. For n = 5, SO(n — 1) = SO(4), which is covered by SU(2) xSU(2), so its representations are also explicitly described by the results of Chapter 2. We can similarly treat the inhomogeneous Lorentz group, or Poincaré group, For

(4.7)

£(n + 1) =

O(n, 1),

of isometries of with the Lorentz metric (x, x) = x? + . . + this case the orbits fall into four classes: .

—

In

(4.8)

0b={o};

incasesA>O, A=O, AczO.

The orbit 0b gives rise to the irreducible representations of £ (n + 1) arising from those of 0(n, 1) via £(n + 1) —' 0(n, 1). Given A E R, the subgroup of 0(n, 1) stabilizing a given point in the orbit is isomorphic to 0(n) if A < 0, to O(n — 1,1) if A > 0, and to a group we denote MN which is a product of M = 0(n — 1) and an abelian subgroup N of 0(n, 1), if A = 0. Thus the representations of the inhomogeneous Lorentz group are analyzed as before, in terms of irreducible representations of the groups 0(n, 1), 0(n), O(n — 1, 1), and MN. This result was obtained by Wigner in his famous paper [264], which pointed out the physical significance of those representations of £(n +1) arising from representations of the compact group 0(n), whose representations were well understood at that time. In Chapters 8 and 9 we shall discuss the representation theory of SO(2, 1) and SO(3, 1). The group MN is isomorphic to the Euclidean group E(n — 1).

CHAPTER 6

Nilpotent Lie Groups The Heisenberg group H" is the simplest example (other than R") of a nilpotent Lie group. An inductive process enables one to reduce the representation theory of a general nilpotent Lie group to that of H". This is particularly simple for "step two" nilpotent groups, which we study separately in §2 before considering the general case in §3. Inductive methods are also effective on solvable Lie groups. This works easiest on "exponential" solvable groups, for which exp: g —+ C is a surjective diffeomorphism. More general solvable groups need not be "type I," but the type I cases have been identified and their representations classified. In the previous chapter we considered two solvable Lie groups, the Euclidean group E(2) and

the "ax + b group," which will appear again in the next chapter, but we will not go into a general study of solvable Lie groups. See [9, 10, 177, 200] for information on this topic.

1. Nilpotent Lie algebras and Lie algebras with dilations. A Lie algebra g is said to be nilpotent if, for each X E g, the linear transformation ad X is nilpotent, i.e., (adX)k = 0 for some k. on g, defined by adX(Y) = [X, Engel's theorem says that g satisfies this condition if and only if, with (1.1) we

0(0)

=L

0(1) = [0,0],

g(i) =

have, for some K,

) 0(K) = 0. In fact, something stronger is true. If g is nilpotent, there exist ideals g, such that dimg, = j, j = 0,1,... ,dimg, and [g, g,] C This fact in turn is a (1.2)

0(0) J 0(1)

special case of

PROPOSITION 1.1. If p is a representation of a Lie algebra g on a finitedimensional vector space V such that p(X) is nilpotent for each X E g, then there are linear subspaces

with

(1.3)

such that dimV, = j and p(g)V, C 152

NILPOTENT LIE GROUPS

153

This proposition can be proved using the following simple lemma. V

LEMMA 1.2. Under the hypotheses of Proposition 1.1, there is a nonzero E V such that p(X)v = 0 for all X E 9.

To deduce Proposition 1.1 from Lemma 1.2, let V1 be the linear span of such a v. Then g acts on V/V1. Pick a nonzero E V/V1 annihilated by g; say i32 is the image of v2 E V. Let V2 be the linear span of v2 and v; g acts on V/V2. Continue in this fashion. We give a proof of Lemma 1.2, using induction. Let N = dim g, and suppose

the assertion is true for all Lie algebras of dimension less than N. Let a be a proper Lie subalgebra of p(g) of maximal dimension. If we consider the natural representation r of a on p(g)/a, we see by the induction hypothesis that there

a, such that r(a)X1 E a. This implies (Xi) + a is a

is an X1 E p(g), X1

subalgebra of p(g), hence is all of p(g). Thus a is an ideal in p(g) of codimension 1. Let W = {v E V: Yv = 0 for all Y E a}. Then, by induction, dim W 1.

If v E W, Y E a, then with X1 as above YX1v = [Y,Xj]v + X1Yv = 0, so X1(W) C W. Since = 0 for some k, there exists a nonzero v E W such that X1v = 0. Hence p(g)v = 0, and the lemma is proved. A connected Lie group G whose Lie algebra g is nilpotent, is also called nhlpotent. We note without proof the standard fact that, if C is mlpotent and simply connected, then exp: g —p C is a diffeomorphism of g onto C. Thus C manifolds. Furthermore, Lebesgue measure on g gives and g are identified as a bi-invariant Haar measure on C. For these facts, see the standard references, e.g., [100, 254]. Important examples of nilpotent Lie algebras are given by Lie algebras with dilations. We now introduce this concept. Suppose 9 is a Lie algebra on which a one parameter group of automorphisms a(t) acts. Then X E 9, 5(X) = (d/dt)a(t)X1t0, defines a derivation on g. In other words, the identity (1.4)

Y]) = [a(t)X, a(t)Y]

(1.5)

implies

o([X,Y]) = [X,6(Y)] + [S(X),Y].

(1.6)

Suppose all the eigenvalues of the linear transformation S on g are real. For bE B = specS, set

= {X E g: X is a generalized b-eigenvector of 5}.

(1.7)

Then we have

linear (not Lie algebra) direct sum.

g=

(1.8)

bEB

Note

(1.9)

that X E 9b if and only if a(t)X =

x polynomial in t,

NILPOTENT LIE GROUPS

154

the polynomial taking values in g. If X E 9a, Y E 9b,

we have

Y] = [a(t)X, a(t)Y] = e(0+b)t x polynomial in t.

(1.10)

It follows that (1.11)

[go,gb] C 90+b•

Of course, if 6 is scalar on 9b for all b E B, this is also an immediate consequence of (1.6). An immediate consequence of (1.11) is

(or if B C R).

g is nilpotent if B C

(1.12)

Generally, we will say a one parameter group a(t) of automorphisms of a Lie where 6 algebra g (necessarily nilpotent) is a group of dilation8 if spec 6 c is the derivation defined by (1.4). We say it is a group of strict dilations if in addition a(t) acts as a scalar on each Ob, b e spec 6. The Lie algebra

of the Heisenberg group is the basic example of a nilpotent

Lie algebra. Recall from Chapter 1 that I" has a basis T, L,, M, (1 i n), with (1.13)

[M,, I,,] =

— [L,,

M1] = T,

other commutators zero.

We have

= (T),

(1.14)

the linear span of T, and (1.15)

A dilation group on

which we introduced in Chapter 1, is given by

a(t)(sT + q L + p M) = e2tsT + etq L + etp. M,

(1.16)

where q. L =

M is similarly defined. Thus and strict dilations. In the notation of (1.7), we have (1.17)

specö={1,2},

gi=(L,,M3:

has a group of

g2=(T),

and in the notation of (1.1) we have (1.18)

9(0) =

9(1) = (T),

0(2) =

0.

The simplest sort of generalization of IJ" is the class of Step 2 nilpotent Lie algebras, which we will study in the next section.

2. Step 2 nllpotent Lie groups. We introduce the following concept. DEFINITION. A Lie algebra g is called nilpotent of Step 2 if we have (2.1)

using the notation of (1.1).

9(2) = 0,

NILPOTENT LIE GROUPS

155

Such algebras always possess dilations. Indeed, we have

PROPOSITION 2.1. If g is nilpotent of Step 2, there is a group of strict dilations whose assoczated derivation satisfies spec 5 = (1,2).

PROOF. With 9(1) =

[g,

g], let V be any linear subspace of g which is

complementary to 9(1). Define a(t) on g by

XEV, YEO(l).

cx(t)(X+Y)=etX+e2tY,

it is easy to check a(t) is a group of automorphisms. Clearly

Since [V, V] C

spect5={1,2},andgi=V, 92=9(1). The following are examples of (2n + 2)-dimensional Step 2 nilpotent Lie alge. and define a Lie bracket so

bras. Pick generators Xj,. that

[X,,Xk] = [Y,,Yk] = [X,,Tk) = [Y,,Tk]

(2.2)

=0

and (2.3)

[X,, Yk] = S,k(aJTl + b3T2),

where a3, b3 E R are given. The reader will notice quite a bit of structure such Lie algebras share with (j'2. That there is a close relation generally is illustrated by

PROPOSITION 2.2. Let g be nilpotent of Step 2; g = 01+02. If V is a linear subspace of 92 of codimension 1, then, for some n, k,

g/V

(2.4)

Rk

(Lie algebra direct sum).

PROOF. With g' = g/V, we see that g' is a nilpotent Lie algebra of Step g2/V, one-dimensional. Picking a nonzero T E we 91, see that, for X, Y E [X, Y] = o(X, Y)T, where o-(X, Y) is an antisymmetric bilinear form on the vector space Let E = {X E Y) = 0 for all V E O'i }. Let F be any linear subspace of complementary to E. Then o(X, Y) is nondegenerate on F. It follows that dim F = 2n is even, and choosing a symplectic basis of F shows that 2, with O'i

=E

(F + (T)) = E

If2

(Lie algebra direct sum).

Here E is a commutative Lie algebra, so E = Rc with k = dim E. The proof is complete. We remark that an arbitrary Step 2 Lie algebra is constructed as follows. Let V and W be finite-dimensional vector spaces, and (2.5)

B:V®V-+W

an antisymmetric bilinear map. Then we define a Lie algebra structure on g = (2.6)

[vj+wl,v2+w2]=B(vl,v2),

V,EV, w3EW.

156

NILPOTENT LIE GROUPS

We have = W. The reader might consider conditions on a pair = V, B1, B2 of bilinear maps which give rise to isomorphic Lie algebras. We now consider the irreducible unitary representations of the simply connected Lie group with Lie algebra 9, assumed to be nilpotent of Step 2. Let be any such representation of C, giving rise to a skew adjoint representation of g, also denoted ir. If we write 9 = +92 as above, since g2 is contained in the center of g, by Schur's lemma, ir must act by scalars on

(1/i)ir: 92 —' R.

(2.7)

Now either = 0 or V,. = {X E 92: ir(X) = 0} has codimension 1. In the first case, with C2 the subgroup of C generated by the Lie algebra 92, we have ir factoring through: 'U(H) C

G/G2

Rm, i.e., C/C2 is abelian and simply connected. Thus in this case the Hilbert space H is one-dimensional and the set of irreducible unitary representations of C/C2 Rm is well known. In the second case, ir factors through However, clearly C/C2

C

'F

wU(H)

C generated by the Lie algebra

According to

Proposition 2.2, (2.8)

C/H,

H" + Rk for some n, k,

and again the set of irreducible unitary representations of this group is known, by the Stone-von Neumann theorem. We take a closer look at a nilpotent Lie group of Step 2 which is of particular interest, namely the group Nk,2 with Lie algebra

V=Rk,

(2.9)

the bracket operation on Nk,2 being given by (2.10)

= (0,vAw).

Lie algebras are universal, in the sense that any nilpotent Lie algebra of Step 2 is a quotient of one of them. One says Nk,2 is the (simply connected)

These

nilpotent Lie group, free of Step 2. If ir is a nonscalar irreducible unitary representation of Nk,2, annihilating a codimension one subspace H of A2 V, then ir comes from a representation ire' of = Vef\2 V/H = V +WH, where we have set WH = A2V/H; dimWH = 1. Put a standard eucidean metric on V, A2 V, and hence Nk,2. This determines

NILPOTENT LIE GROUPS

157

a linear isomorphism WH an element T WH of norm 1 corresponds to 1 R. The Lie algebra structure on V + WH is characterized by

[v,wJ=w(v,w)T,

(2.11)

v,wEV.

The bilinear form w is a general element of (A2 V)1 A2 VI, i.e., a general antisymmetric bilinear form on V, normalized (with norm 1 in A2 V'). In the case dim V = 21 is even, the generic such w is nondegenerate; when w is nondegenerate, we have OH = V + WH isomorphic to and we get a family of irreducible representations lrA,H, A E R\O, characterized to within unitary equivalence by lrA,H(T) = iA. We will work out rather explicitly the Plancherel formula and inversion formula on N21,2. We will deduce the Plancherel formula for N21,2 from the Plancherel formula for the Heisenberg group H1 proved in Chapter 1, (2.12)

=

Ill

I7rA(f11?lsIAI1 dA.

C1

Now if we identify N21,2 and )121,2 as C°° manifolds, with Haar measure equal to

Lebesgue measure, and if Lebesgue measure is imposed on gjj, let the factor (for H such that OH Ij') such that =

(2.13)

denote

iio(H)f

Now we introduce the partial Radon transform

w, H)

(2.14)

=

L f(z, y + w) dy

where x V, y H, w E Wi',, and V + WH is identified with a linear subspace of )121,2 by identifying T E WH with a vector in A2 V orthogonal to H, of norm 1. The formula (2.13) gives

f

(2.15)

lRf(x,w,H)l2dxdw = cQ(H)f

V+WH

—00

Now the Plancherel formula for the Radon transform (see Appendix C) implies

(2.16)

where 9

=

Ill is

Cj

fL

w,

dxdwdvol(H),

the Grasamannian manifold of hyperplanes in A2 V, and

_1=212_i_i.

a=dimA2V —1

(2.17)

= represented as Al by

Since formula

is

(2.18)

Ill

=

Cl

we deduce from (2.15) the Plancherel

(I) ll?is

dAd vol(H).

NILPOTENT LIE GROUPS

158

From here, the inversion formula is a simple consequence:

f(ç) = cj

(2.19)

fJ

dA dvol(H).

leave it to the reader to work out analogous formulas for N21+i,2. More general results on representations of nilpotent Lie groups are given in the next section, but the discussion here has had the advantage of being very simple. For connections between the study of Step 2 nilpotent Lie groups and hypoelliptic operators, see [174, 1081, and Rothschild and Stein [204] for further We

generalizations.

3. Repreaentations of general nilpotent Lie groups. In this section we show how to produce a complete set of irreducible unitary representations of a simply connected nilpotent Lie group C, following the theory of Dixmier [43] and Kirillov [135]. As in §2, this will be achieved by application of the Stonevon Neumann theorem on the representations of the Heisenberg group, but the argument in this case will be somewhat more elaborate.

Let g be the Lie algebra of C, and let

be the center of g. If 7t is an

irreducible unitary representation of C on a Hilbert space H, it induces a skew adjoint representation ir of g. This must represent 3 by scalars 7r(Z) =

(3.1)

If

= ker A, 3i generates the subgroup H1 of C, then ir comes from a unitary representation p of C/Hi; ir = p o where C —+ C/Hi is the natural projection. Let us suppose, therefore, that dims = 1 and A 0 on Fix a nonzero Z C which therefore generates 3. where

A is a linear functional on

We begin our analysis by producing a subalgebra of g isomorphic to LEMMA

3.1. There exist X, Y E g such that [X, Y] = Z.

(3.2)

j... j j

PROOF. Using Proposition 1.1, with p the adjoint representation, we have

idealsg, of.g, g Oo =0, with dimg, =j and [g,g1]c 9,—i. Pick any nonzero V C 92\gi. Since V 3, there exists X E 9 Hence 9' = 3. with [X, Y] 0. But [X, Y] E = so [X, Y] is a multiple of Z. Rescaling X gives (3.2).

With X and V chosen so (3.2) holds, let

C={W€g: [W,Y]=O}.

(3.3)

We note that (3.4)

£ is a Lie algebra, and V and Z are in the center of £.

NILPOTENT LIE GROUPS

159

It is also useful to note

LEMMA 3.2. We have g = (X) + £. PROOF. Since [9,92] C = j and dimj = 1, it is clear that codimL < 1. Since X does not belong to £, by (3.2), the result follows. Now define to be the linear span of X, Y, and Z. Thus is a Lie subalgebra of g, isomorphic to I", and ir is a skew adjoint representation of which is given by (3.1) on the center of The Stone-von Neumann theorem, in the form established in Chapter 5, implies

LEMMA 3.3. There is a unitary equivalence of H with L2(R, H1), for some Hilbert space H1, such that, for

/ E L2(R,H1),

(3.5)

we have (3.6)

ir(exp(tZ + qY +pX))f(s) = =

Here we identify

with

+ p).

R\O.

We next examine the behavior of ir on L, the subgroup of C generated by .C. First of all, by the defining relations (3.3), we see that each 1 L commutes with

exptY, so (3.7)

ir(l)ir(exptY) = ir(exptY)ir(l),

1 E L.

However, if f E H is represented by (3.5), since ir(exp tY) is given by multipliif ir(l) commutes with all these multiplication operators, it must cation by be a multiplication operator, so we must have (3.8)

ir(l)f(s) = U(l, s)f(s)

where U(l, s) takes values in the unitary operators on H1. A priori, U(l, s) is a measurable function of s. Since £ is a subalgebra of g of codimension 1, it must be an ideal. Hence, for L, we have exp(—toX)lexp(toX) L. Since 1 ir(exp( —toX)l exp(toX)) = ir(exp(—toX))ir(l)ir(exp(toX)),

and since, by (3.6), ir(exp(toX))f(s) = f(s+to), we see that, almost everywhere, (3.9)

U(l,t — to) = U(exp(—toX)lexp(toX),t),

1€ L.

It follows that, possibly modifying U(1, t) on a set of measure zero, we obtain U(l, t) strongly continuous in t, and (3.10)

U(l,t) = U(exp(tX)lexp(—tX),O).

Another simple property of U(l, t), which follows from ir(ll') = that U(l1', s) = U(1, s)U(l', s) and in particular (3.11)

U(ll',O) = U(1,O)U(1',O).

is

NILPOTENT LIE GROUPS

160

Consequently, if

Uo(l) = U(l,0),

(3.12)

I e L,

then (J0 gives a unitary representation of L on H1. The following result gives a simple but incisive relationship between the representation ir of C which we set out to analyze and the representation U0 of L. PROPOSITION 3.4. The representation ir of C is unitarily equivalent to (3.13)

PROOF. This follows almost immediately from the definition of an induced representation. Indeed, acts on the Hubert space H of measurable functions u from C to H1 satisfying u(gl) = g E C, I e L, such that d(gL) 0, independent of k. One consequence of this is the following. If the spectrum of (1/i)E, which must consist of integers, contained both even and odd integers, if we set H = H, with k even

then both

k odd

and H, would be invariant under the operators (2.25) for (t1, t2, close to 0, and hence invariant under ir(g) for all g E SL(2, R). Since it is assumed

SL(2, R)

184

to be irreducible, this means that the spectrum of (1/i)E is contained in either the set of even integers or the set of Qdd integers. Note by (2.16)—(2.17) that, in (2.23), each Vk —+ Vk+2 is a scalar multiple of a unitary operator, as is each R_: Vk —+ Vk_2. Such a multiple could be zero. However, if 0 and R÷ = 0 on Vk, then, comparing (2.18) and (2.19), we

see that also R_ = both forces

V1 and

0

on Vk+2. Thus the sequence (2.23) is severed at Vk, and V1 would be invariant under ir. If ir is irreducible, this

0. By the same on Vk, Vk =Oforalll< k. Inotherwords,

0

1

reasoning,ifR_ =OonVkand Vk

(2.23) is either two-sided infinite, with all and R_ isomorphisms, or one-sided infinite, of the form Vk_2,, or of the form Vk+2,, with R+Ivk = 0 or R_ IVk = 0, in these respective cases, and all other R± isomorphisms.

The remaining possibility, that (2.23) is finite, does not ever occur, except when ir is the trivial representation. Indeed, from (2.18)—(2.19), if R_ IVk = 0 then (k—i)2 = while = 0 then (k+l+1)2 = 1—A. That would imply k + 1 + 1 = ±(k — 1), i.e., either 1 = —2 or 1 = —2k. Since 1 0, I = —2 is not possible. If I = —2k, then the string (2.23) is V_1k1 We will reach a contradiction after making a general comment. Since = —R_, the scalars in (2.16)—(2.17) must be 0, for all k e spec(1/i)E. If spec(i/i)E contains k = 1, or k = —1, this forces A 1. If spec(1/i)E contains k = 0, it forces A 0. If ir is finite-dimensional, the identity (k—i)2 = i—A, which implies (IkI+1)2 = 1 — A, forces A < 0 unless Iki = 0. Thus ir must be representation on which E and R±, hence A1 and B1, act trivially, and thus a trivial representation. We remark that it is easy to see directly from simple general principles that SL(2, R) cannot have any nontrivial finite-dimensional irreducible unitary representation. Such a representation would induce a Lie algebra homomorphism (2.26)

ir: sl(2, R) —' u(n).

Since ker ir is an ideal and sl(2, R) is simple, this means ir is injective on sl(2, R)

if ir is not trivial. Now any bi-invariant Riemannian metric on U(n) induces an Ad-invariant positive definite form on u(n), which by (2.26) would pull back to an Ad-invariant positive definite form on sl(2, R). However, as mentioned in §1, any such form must be a multiple of the Killing form, which is certainly not definite, as it has Lorentz signature. This same sort of argument shows that a general noncompact (connected) semisimple Lie group has no nontrivial finite-dimensional unitary representations. To return to an analysis of the infinite-dimensional irreducible unitary representations of SL(2, R), we have four possibilities for the spectrum of (1/i)E: (2.27) spec(i/i)E = {n, n + 2, n + 4,.. .}, (2.28) spec(i/i)E = {. . , n —4, n — 2, n}, .

(2.29) (2.30)

spec(i/i)E= {...,—4,—2,0,2,4,...}, spec(i/i)E = {. , —5, —3, —1, i, 3,5,. . . .

SL(2, R)

185

All the maps R+: Vk —' Vk+2, R_: Vk —' Vk_2, between nonzero eigenspaces of E, are isomorphisms. We are now ready to prove that

if k E spec(1/i)E. In fact, pick any Ico E spec(l/i)E, pick a nonzero vko E Vk0, consider its iterated images under belonging to for k> k0, and its iterated images under R_, belonging to Vk for k < k0. The representation ir induces a representation of the Lie algebra sl(2, R) on the finite linear span of these vectors. By the analyticity (2.24) of all the vectors in each Vk, this generates a unitary representation of SL(2, R) on the closure of this linear span (see Appendix D). In fact, this representation simply covers ir. Since ir is supposed to be irreducible, this closed linear span must be all of H, so (2.31) is established. We are now ready for a case by case analysis of (2.27)—(2.30), which will produce the classification of the irreducible unitary representations of SL(2, R). In each case, pick a unit vector Vk E Vk for each k E spec(1/i)E, producing an orthonormal basis of H, uniquely determined up to phase, which can be adjusted arbitrarily. We may as well specify (2.31)

dimVk =

1

R+vk = akvk+2 where ak is any complex number whose absolute value is equal to the quantity computed in (2.19), i.e., (2.32)

(2.33)

lakI

=

1)2 + A

—

111/2.

Then the action of R_ on Vk+2

is

(2.34)

R_vk+2 = /3kVk, = (k + 1)2 + A —

where the identity (2.17) forces (2.35)

uniquely determined:

1

=

41ak12, i.e.,

13k =

Picking 13k so this holds also makes (2.16) satisfied, and of course (2.16)-(2.17)

imply —4[R+, R_] = —4k on Vk, so we have the commutation relation given by the last part of (2.3). It is routine to verify that (2.32)—(2.35) implies all the commutation relations (2.3) are satisfied on the finite linear span Vk, and also (2.35) implies = —R_, so whenever (2.32)—(2.35) hold we have a Lie algebra representation of sl(2, R) by skew-symmetric operators on H0 = Vk, a dense subapace of H, consisting of analytic vectors. In the case (2.27), we have R_ = 0 on V,, which, by (2.18), implies (2.36)

(n—1)2=1—A,

i.e., (2.37)

ir(D) = 1—(n— 1)2 =A.

If n were 0, then either 0 or —1 would have to belong to spec(1/i)E. If —1 belongs, then (2.33) forces A> 1, and if 0 belongs it forces A > 0, in both cases contradicting (2.37). Thus, for the case (2.27) to occur, we need (2.38)

n 1.

SL(2, R)

186

Conversely, as long as (2.38) holds, then (2.33) can be implemented for k = n+2j, 0, 1, 2, .. ., i.e., lakI =

(2.39)

+ 1)2 — (n

—

1)211/2,

and a Lie algebra representation of sl(2, R) by skew symmetric operators is defined. As we will see, this can be exponentiated to a representation of SL(2, R), for each integer n satisfying (2.38). The representations so obtained are denoted and are called members of the holomorphic discrete series of representations of SL(2, R).

The case (2.28) is handled similarly. We have R÷ = (n + 1)2 = 1 — i.e.,

0

on

so by (2.19),

ir(D)=1-.-(n+1)2=A.

(2.40)

The same argument as before forbids 0 or 1 from belonging to spec(1/i)E, so we must have

n —1,

(2.41)

and conversely if (2.41) holds then we can implement (2.33) for k = n 0, 1, 2, .. ., choosing so that

=

(2.42)

=

—

2j,

+ 1)2 — (n + 1)211/2 —

1)2

—

—

1)2]1/2

for k < n. This can be exponentiated to a representation of SL(2, R), for each integer n satisfying (2.41), denoted 7ç, and said to be a member of the antiare related via holomorphic discrete series. it is easy to see that and the automorphism a of SL(2, R) given in (1.22):

=

(2.43)

irt and

called "mock discrete series" representations, rather than discrete series representations, for reasons briefly discussed in §4. Consider now the case when spec(1/i)E is given by (2.29). Since 0 belongs to spec(1/i)E, (2.33) implies Actually,

are

ir(D) =

(2.44)

A

> 0.

Conversely, whenever (2.44) holds, we can implement (2.33). In fact, we could take (2.45)

with s = (2.46)

provided

1.

In such a case, the representation is denoted (2.47)

X=1+s2,

5ER,

SL(2, R)

187

and called a member of the first principal 8eries; one typically takes both signs of s in (2.47), but notes that (2.48)

If 0 < A < 1, then we still get a representation of sl(2, R) by skew symmetric operators, but we cannot use (2.45) in order to achieve the desired identity

where 1 — A

+ 1)2 —

=

(2.49)

=

In this case, the representation is denoted

A=1—s2,

(2.50)

sE(—1,1)\{0},

and called a member of the complementary series. Again by convention we take s of both signs, but note

=

(2.51)

Finally, there is the case when spec( 1/i)E is given by (2.30). Since —1 belongs

to spec(1/i)E, (2.33) implies

ir(D) = A>

(2.52)

1.

Conversely, whenever (2.52) holds we can implement (2.33). In fact, we could take

=

(2.53)

with s = ±v'A — (2.54)

1.

In such a case, the representation is denoted A=

1

+ 82,

s E R\{0},

and called a member of the second principal series. Again one takes both signs of s, and (2.55)

lr°_,8.

We remark that you can formally pass to the limit in (2.54) as s —+ 0. In such a case, R+: v_i —# V1 vanishes, and the representation splits into a sum of representations considered in cases (2.27) and (2.28), with n = ±1, i.e., (2.56)

As we have stated, each of the Lie algebra representations of sl(2, R) constructed above exponentiates to a representation of SL(2, R). In fact, we will give explicit realizations of these representations in the next three sections, and this will provide one justification of such a statement. We can also use the general results discussed in Appendix D, to exploit the fact that the vk E Vk are all analytic vectors. This implies each such representation of sl(2, R) considered above exponentiates to a unitary representation * of the universal covering group SL(2, R) SL(2, R). Note that the kernel of the natural projection SL(2, R) consists precisely k E Z. By (2.11), *(g) = for g in the kernel, so

I

SL(2, R)

188

gives rise to a representation ir of SL(2, R), as desired. The dense set of analytic vectors also implies the uniqueness of the exponentiated unitary representation for any such given Lie algebra representation. We now state the main result on the classification of the irreducible unitary representations of SL(2, R), first derived by Bargmann [12). THEOREM 2.2. Any nontrivial irreducible unitary representation of SL(2, R) is unitarily equivalent to one of the following types:

nEZ+,

nEZ+,

8ER,

ir,

8ER\{O},

8E (—1, 1)\{O}.

Each of these representations is irreducible, and there are no unitary equivalences except for (2.48), (2.51), (2.55). In particular, an irreducible unitary representation ir of SL(2, R) is determined uniquely by the spectra of ir(D) and ir(Z).

The only point remaining to be proved is the actual irreducibility of all the representations on this list, and we turn to that task. From (2.10) and (2.31), we deduce that any closed invariant linear subspace of the representation space H of such a representation must be an orthogonal direct sum of some collection of Vk's. Thus, if any such ir were not irreducible, there would exist vk E Vk, vi E V1, l > k, such that (2.57)

= 0 for all g E SL(2, R).

(lr(g)vk,

Differentiating repeatedly and evaluating at g = e, which is permissible since Vk E C00(ir), yields (lr(P)vk,vz) = 0 for each P in the universal enveloping algebra of sl(2, R). In particular, = 0.

(2.58)

But by the construction (2.32)—(2.35) this is impossible. The irreducibility of each representation listed in Theorem 2.2 is proved. We close this section with a compact relisting of the action of each of these representations on the Casimir operator, from (2.37), (2.40), (2.47), (2.50), (2.54): (2.59) (2.60) (2.61)

= 1 — (n — = 1— (n — =1+

1)2, 1)2,

n nE a E R,

(2.62)

ir(D) =

1

—

S

E (—1, 1)\{0}.

3. The principal series. The simple form (3.1)

for the coefficients occurring in the construction (2.32)—(2.35) of the principal

SL(2, R)

189

series representations and derived in (2.45) and (2.53), makes it tempting and easy to realize these representations of the Lie algebra sI(2, R) as a Lie algebra of vector fields (plus zero order terms) on the circle S', if we let E L2 (S') (with square-norm (2ir) — If(0) 12 dO) correspond to vk. Consequently, the representation ir13 = of sl(2, R) on the space of trigonometric polyno® mials on S1 is given by

= 8/80,

(3.2) and

(3.3)

ir13(X+) = (1/2i)e2°t9/8O +

(3.4)

ir13(X_) = _(l/2i)&2108/89 +

and hence, since 2A =

+X_, —2iB =

+ is)e2'°,

—

+ is)e2'°,

X_,

2ir13(A) = sin 200/00 + (1 + is) cos 20,

(3.5)

and

= cos200/00

(3.6)

—

(1

+ is) sin 20.

These last two operators are formally skew symmetric first order differential operators with analytic coefficients, as is ir(Z), so it is clear that the analytic vectors for this Lie algebra are precisely the real-analytic functions on S', which of course includes the set of trigonometric polynomials. Rather than directly

tacide the exponentiation of this Lie algebra representation, which would in principle just amount to integration along orbits of appropriate vector fields, we will take a detour on the way to explicitly realizing the associated representation of SL(2, R). Namely, the forms of (3.5)—(3.6) make it natural to consider these operators as given by real vector fields on R2, acting on functions of the form (3.7)

especially in view of the role of the Mellin transform, discussed below. In fact, acting on (3.7), the operators (3.2), (3.5), (3.6) coincide respectively with (3.8)

R(Z) = 8/80,

2R(A) = sin 208/80 —

(cos 20)rO/Or,

2R(B) = cos 208/80 + (sin 20)rO/Or, or, in Cartesian coordinates on R2, z = rcos0, y = rsin0, (3.9)

R(Z) =

—yO/Oz + xO/Oy,

2R(A) = —z8/8z + yO/Oy,

2R(B) = yO/Ox + z8/Oy. Now this Lie algebra representation of sl(2, R) by vector fields on R2 with linear coefficients exponentiates explicitly to the regular representation of SL(2, R) on L2(R2), defined by (3.10)

R(g)f(x) = f(gt . x),

g

SL(2, R), x

R2.

SL(2, R)

190

We now decompose this representation into the irreducibles ira, ira. Note that R(g) commutes with the group of dilations

D(t)f(x) = etf(etx),

(3.11)

and with the inversion map Jf(x) = f(—x). We show that the spectral decomposition of D(t), plus splitting functions into even and odd parts, will effect an irreducible decomposition of R(g). The spectral decomposition of D(t) can be effected via the Mellin transform: .M/3(s) =

(3.12)

f13(t)tt8dt.

Note that a change of variable t =

gives

)vt(3(s) =

(3.13)

so the Fourier inversion formula and Plancherel theorem give, respectively, /3(t) = (1/2ir)

(3.14)

j

ds,

and

= 21rj

(3.15)

113(t)I2tdt.

If we define (3.16) we

P3 1(x)

f

=

f(tx)t' dt,

see that, for f E C8°(R2), P3f belongs to the space

We

r > 0, and j

gfrx) =

(3.17) N3 = {g E

make N3 a Hilbert space, with norm

g12

loP; note that the homogeneity

condition

g(rr) =

(3.18)

makes g

uniquely determined by its restriction to S1. We see from (3.15)

that L2(R2)

(3.19)

ds.

There is the "Plancherel theorem" (3.20)

Ii!

= (1/2ir)

d8,

and the inversion formula (3.21)

f(rw) = (1/2ir)

f

ds.

SL(2, R)

191

The representation R(g) decomposes as a direct integral of representations Ps on

= f(gt x),

(3.22)

/E

113.

We can realize all these representations on L2 (S1). Indeed, the restriction map I '—p Is' takes 113 isometrically onto L2(S1). Using (3.9), we see that this defined by unitary map intertwines with

I

(3.23)

=

w),

hgt

f E L2(S'),

where A(g): 5' —+ S' for each g E SL(2, R) is defined by (3.24)

A(g)w =

In (3.23), it is clear that and are invariant. Note that the group SO(2), generated by Z, operates by rotation:

=

(3.25)

f(k'w),

k E SO(2).

Thus it is clear that, if we write (3.26)

1r23 =

and are explicit realizations of the principal series representations described in §2. then

Note that the image of the Casimir operator 0 under R is given by (3.27)

R(D) =

—X2 ± 1,

where X generates D(t), so X = rä/8r + 1. These representations and of SL(2, R) on spaces of functions on S', on which SL(2, R) acts transitively, can be seen to be examples of induced representations. For more on this perspective, see Chapter 9, §3. If we have a principal series representation of the form acting on a Hubert space 11, pick fo, a generator of the one-dimensional space on which K = SO(2) acts trivially, and for each / E 11 consider the function on C = SL(2, R) defined by (3.28)

(T3f)(g) =

Then it is clear that (T3f)(gk) = (T3f)(g) for all k E K, so we define a function by r3f e C(G/K) = (3.29)

r3f(x) = (T3f)(g),

x = gK E ftp.

Note that the map r3: 11 —' C(11÷) is injective. Indeed, kerr3 must be invariant under and cannot be all of 11, since Jo kerr3, so the irreducibility of implies kerr8 = 0.

Note that (3.30)

=

SL(2, R)

192

is the Laplace operator on

is the Casimir operator on C and with its Poincaré metric (1.9), i.e., where U

= y2(32/8x2 + 82/8y2).

(3.31)

In view of the results (2.61) on (3.32)

see that

we

= —1(1

+

In other words, the image of N under r3 consists of C°° functions u on

satisfying the equation

=

(3.33)

+ 82)U.

If we denote the space of C°° functions on

r,:

(3.34)

—

4,

satisfying (3.33) by e8, we have injective.

Note that

= r3(f)(g' x).

(3.35)

Let f,,, span the one-dimensional subspace of ': on which K = SO(2) acts as = Then, in geodesic polar coordinates the character and let centered at the origin eK E (3.36)

=

=

+

satisfying (3.36) It is not hard to show that the space of smooth functions on up to some scalar factor, is one-dimensional, so these conditions characterize which depends on 8 and n. If one expresses the Laplace operator in geodesic polar coordinates (it is easier to think of geodesic polar coordinates on the Poincaré di8c, centered at the origin), one sees that (3.37)

(x) =

r),

where is the associated Legendre function. It will follow from the inversion formula to be proved in §7, that if R1 denotes

the regular representation of SL(2, R) on (3.38)

Ri(g)f(x) = f(g' x),

g E SL(2,R), xE

then

fir, ds,

(3.39)

with the corresponding decomposition of Hilbert spaces (3.40)

L2(O+)

we see If we compare (3.39) and (3.19) and recall the equivalence is equivalent to that the even part of R, defined by (3.10), acting on a sum of two copies of the regular representation R1 defined by (3.38).

SL(2, R)

193

An explicit operator intertwining these representations can be constructed by the following device, used by Ehrenpreis [54, 551. See also Strichartz [228, 229] for related ideas. First, we can identify even functions on R2 with functions on C, the forward light cone in R3 with the Minkowski metric. Then the intertwining operator takes a pair of functions on (LF, identified with the upper sheet of a two sheeted hyperboloid in R3, and uses this pair of functions as Cauchy data to solve the wave equation within the forward light cone, then taking the limiting values on the light cone. Inverting this process involves solving a Goursat problem for the wave equation. For details we refer to [54).

4. The discrete series. There is a natural representation of SL(2, R) on L2(R2) which contains each member of the holomorphic discrete series, suggested

by some calculations mentioned at the end of §1 of Chapter 7, on harmonic analysis on cones. If we view R2 as the cone over S', these observations specialize to the following. Let (4.1)

X=rô/8r+1,

V=ir2.

Then X, U, V are skew adjoint operators, satisfying the commutator relations (4.2)

[X, V] = 2V,

[X, U] = —2U,

[U, V] = 4X.

Let (4.3)

Z=

- V)

A=

= + V) =

B=

+ 1x12), 1),

+ (x12).

Then we have (4.4)

[Z,A] = 2B,

[Z,B] = —2A,

[A,B] =

so these skew adjoint operators on L2(R2), span a Lie algebra isomorphic to sl(2, R). Thus they generate a representation of C, the universal covering group of SL(2, R). Note that the kernel of the covering map C —+ SL(2, R) consists of exp(2irkZ), k E Z. The first formula in (4.3) shows that the spectrum of (1/i)Z consists of all positive integers, since, as we saw in Chapter 1, the harmonic oscillator hamiltonian + x2 on R2 has as its spectrum all even positive integers. Thus exp(2irkZ) is represented by the identity operator on L2(R2), so the representation of C actually comes from a representation of SL(2, R). Let us denote this representation by Note that all the operators (4.3) commute with rotations. Thus we can decompose 'y on L2(R2) by (4.5)

L2(R2)

=

k=—oo

where (4.6)

Hk =

E

SL(2, R)

194

As already shown in Chapter 7 on cones, the ax + b group, generated by X already acts irreducibly on each Hk, this representation being and U = intertwined with a "standard model" via a Hankel transform. It follows a fortiori that SL(2, R) acts irreducibly on each Hk, via -y. Since spec( 1/i) Z consists of positive integers, it is clear from the classification of §2 that each irreducible component 'Yk must be a holomorphic discrete series representation. We claim that, on Hk, the smallest element of spec(1/i)Z is kI + 1. In fact, consider our operators on Bargmann-Fok space I, a Hilbert space of entire functions on C2 with orthonormal basis a = (al, a2) 0, on whicH (1/i)Z acts as multiplication by + 1 (see Chapter 1, §6). Then the rotation group action 1(x) '—' f(kx), k SO(2). f EE L2(R2), is intertwined to a subgroup u(az), a SU(2). As we know, each of the natural SU(2) action on E, u(z) irreducible representation of SU(2) is contained exactly once in I, on one of the

spaces E, = span of al = j}. On the space E,, the operator (1/i)Z is the scalar j + 1 and the skew adjoint generator of the SO(2) action on E3 takes the values —j, —j + 1,. .. — 1, j, multiplied by i. This proves that the smallest eigenvalue of (1/i)Z on the ±ij eigenspace of the generator of the SO(2) action is j + 1, as asserted. ,

j

It follows that (4.7)

exactly twice, for each n 2, and it contains irt once. o a, where a is the automorphism of SL(2, R) given Via the identity =

Thus 'y contains

in § 1, we obtain the antiholomorphic discrete series. Another realization of the holomorphic discrete series is given on (4.8)

= {u(z) holomorphic in D: ID Iu(z)12(1 —

< oo}

where D is the unit disc

D= {z€C:IzI < 1}.

(4.9) We

can write

=

(4.10)

(73z

+

.

z)

where e SU(1, 1) SL(2,R), = ( with action on D given by the linear fractional transformation, as described in § 1. Note that the compact subgroup K is given by (4.11)

(4.12)

K = {k9

=

:0 €

SL(2, R)

195

and

=

(4.13)

Thus spec(1/i)Z has n as its smallest element. The Hilbert space is nontrivial precisely for n 2, so we obtain all the holomorphic discrete series representations with n 2. We do not get itt in this fashion. In fact, one usually says that itt is not a discrete series representation, but rather a mock discrete series representation. The reason for this is that, for any f, f' in the representation space of the function

= f') belongs to L2 (SL(2, R)) if n 2, but not if n = 1. We will skip the details on this, but remark that it implies will occur in the Plancherel formula on (4.14)

do not make such an appearance. SL(2, R), discretely, for n 2, while The Plancherel formula is an identity of the form

,2

(A

L2(C)

J

—

I

2

,

,'

J where O denotes the set of equivalence classes of irreducible unitary representations of C, and ,.t is a certain measure on C, called the Plancherel measure. = tr(TT) is the squared Hilbert-Schmidt norm. A general formula of this type holds for a broad class of groups, the unimodular type I groups, a class which contains SL(2, R) and, indeed, all semisimple Lie groups. This general result was derived by Sega! [2121 and Mautner [168]. The explicit Plancherel formula for SL(2, R) is the following: (4.16)

=

Ill

—

f

+

tanhirsds

f 0

— 1) + + n=2 We will not give a proof of this Plancherel formula. We refer to Lang [147] for a treatment of this formula.

The Plancherel formulas (4. 15)—(4. 16) might be called Plancherel formulas for the decomposition of the regular representation of C on L2(G) into irreducibles. In §7 we will establish a Plancherel formula for the "regular" representation of is the Poincaré upper half plane. This formula SL(2, R) on where will be similar to (4.16), but simpler, as no discrete series representations occur.

5. The complementary series. The complementary series ir, s E (0, 1), seems a bit more mysterious than the other series. We will confine ourselves to a brief description of one realization of Namely, let (5.1)

H3

=

ff

Ix

—

< oo},

SL(2, R)

196

and set (5.2)

ir(g)u(x) =

!cx

+ dI8_lu(g_l x)

=

(a b)

where

(53) and

x = (ax + b)/(cx + d).

(5.4)

6. The spectrum of L2(r\ PSL(2, R)), in the compact case. Suppose We r is a discrete subgroup of C = PSL(2, R) such that X = 1'\C is also assume r acts on C/K without fixed points. Then M = X/K = an arbitrary compact surface of constant negative curvature, i.e., an arbitrary compact Riemann surface whose first Betti number exceeds two, and X is the unit circle bundle over M. X is a homogeneous space for C. It has a natural Lorentz metric, inherited from the Killing form on G, which is preserved by the C action, and hence its natural volume element is preserved by the C action. Thus we have a unitary representation of C on L2 (X): is

(6.1)

p(g)f(x) = f(g1 . x),

g E C, x E X.

In this section we want to say something about the decomposition of p into irreducibles.

Let us denote the operators p(Z) and p(D) on X by Z and 0. Since 0 belongs to the center of the universal enveloping algebra of C, these operators on X commute. Note that the operaor 2Z2 —0 is elliptic. Thus it has a discrete spectrum, with eigenspaces of finite dimension, and both Z and 0 leave each of these eigenspaces invariant. Thus L2(X) =

(6.2) where

(6.3)

=

{f E C°°(X): Zf = inf,

Of = Af},

and we know the sum is countable and each is finite-dimensional. Let us denote by E the set of (n, A) such that 0. For each A, define (6.4)

(n, A) E E}.

EA =

Then we have (6.5) and

L2(X) =

0 acts on EA as the scalar A. Thus each EA in (6.5) is invariant under the

action of p, and if we denote by PA the restriction of p to EA, each PA is a finite

SL(2, R)

197

sum of irreducible representations of G. From our computation in §2, we see that the representation which occurs in PA is

ifA=1—(n—1)20, ifA=1—82E(O,1),

(6.6)

4 if A= 1+s2 E[loo) An important question is to determine the multiplicity with which such an irreducible occurs in We describe here a classical result of Gelfand and Graev relating this multiplicity to some phenomena arising from the compact Riemann surface M. First, suppose A > 0 and or occurs in PA, where either A = 1 — belongs to (0,1) or A = 1 + E [1, oo). Then we know that each irreducible component has a one-dimensional subspace where Z = 0, so

multiplicity in p, = dim V0,,.

(6.7)

Now elements of V0,, are invariant under the SO(2) action, so there is an isomorphism (6.8)

VO,,

V0,A

to a linear subspace of C°°(M) = C°°(X/K), which is a restriction of the isomorphism L2(X, K) L2(M), where L2(X, K) denotes the space of elements of L2(X) invariant under K. Note that the Casimir operator 0 on C°°(X, K) is taken to on C°°(M), where is the Laplace operator on M, with respect to its natural metric. Thus

v0,, =

(6.9)

{f E

=

or in other words

is an eigenspace of the Laplace operator on M. Those eigenvalues of in (—oo, — give rise to principal series representations of C on V (X), and those eigenvalues in (— 0) give rise to complementary series representations. In either case, (6.10)

multiplicity in

= multiplicity of the eigenvalue — A/4,

for A > 0.

We remark that there is considerable interest in determining for which compact Riemann surfaces M does have spectrum in (— 0). Of course, the trivial representation of C occurs once in L2 (X), and this corresponds to the 0 eigenvalue of on M. Next suppose A 0. Then the irreducible representation occurring in must be a discrete series representation, or ire, where n = 1 + n = 2m must be even. Suppose for a moment it is the other case is handled similarly. Then we know that each irreducible component in PA has a one-dimensional space spanned by a lowest weight vector, so in this case (6.11)

multiplicity in PA =

SL(2,R)

198

are not constant on the fibers of X M, but they belong to the space C°°(X, n) = {f E C°°(X): f(ke x) = Since SL(2, R) is a double cover of PSL(2, R), this space is naturally isomorphic to the space Now elements of

C°°(M, ic(m))

(6.12)

of smooth sections of the vector bundle sc(m) —' M obtained from the principal SO(2) bundle X —' M via the representation e'0 s—' elmO of SO(2) on C. ic(m) is a holomorphic line bundle on the Riemann surface M, the mth power of the canonical bundle ic.

In fact, the subspace Vs,,, of C°° (X, n) is characterized by the property of being annihilated by the lowering operator R_ = p(A + iB). This gives rise to a first order operator on C°°(M, ic(m)), which is verified to coincide with the operator. Thus (6.13)

the space of holomorphic sections of ic(m). Hence, when n = 2m, (6.14)

multiplicity of

= dim O(M, ,c(m)).

Similarly we see the multiplicity of is equal to the dimension of the space of antiholomorphic elements of C°° (M, Thus and occur with equal multiplicity. The dimension of O(M, ic(m)) can be deduced from the Riemann-Roch theorem. We have (6.15)

dim O(M, ic) =

dim O(M, sc(m)) = (2m

g,

—

1)(g — 1),

m 2,

where (6.16)

g=

is the genus of M. See Gunning, Lectures on Riemann Surfaces [90], page 111. To close this section, we note a group theoretic interpretation of the geodesic flow on the compact Riemann surface M = X/K. In fact, the geodesic flow on the Poincaré disc D = C/K is a one parameter family of transformations on the unit tangent bundle T1D C, which commutes with the left C-action of C. In other words, the infinitesimal generator of is a left invariant vector field on C, so there is a one parameter subgroup of C such that (6.17)

Ftg=g.-y(t),

gEG.

If we identify C with SU(1, 1)/{±I}, acting on D by (1.7), we see that if we identify the unit tangent vector over 0 E D pointing along the real axis with e E C, then (6.18)

y(t)=Fte=

(cosht sinht\ ESiJ(1,1), cosht)

SL(2, R)

199

with infinitesimal generator

Io

A'=

(6.19)

i\ Esu(1,1)

o)

that is taken by the isomorphism (1.12) to (6.20)

—2A

E

=

sl(2, R).

—

Using these observations, we can establish that the geodesic flow is ergodic for any compact surface M of constant negative curvature.

PROPOsITION 6.1. If u E L2(X) is invariant under

on T1M

t E R, then u is

constant.

PROOF. In view of the discrete decomposition of V (X) into irreducibles, it suffices to show that if it is any nontrivial irreducible unitary representation of G = PSL(2,R),

kerir(A)=0.

(6.21)

this is a Given the classification of these representations discussed in straightforward task, whose details we omit. A great deal of work has been done on L2 (1'\G), for quotients either compact or noncompact, but with finite volume, work centered around the Selberg trace formula. We refer to Hejhal [99], Lang [147], Lax-Phillips [151], and Terras [238] for more on this large topic.

7. Harmonic analysis on the Poincaré upper half plane. We wish to produce the spectral decomposition of the Laplace operator on the Poincaré From comments in §3, it follows that a decomposition upper half plane of the regular representation of SL(2, R) on V ((1k) into irreducibles will be achieved.

or equivalently of the operator

The spectral decomposition of (7.1) is

—

I, =

given formally by

Pg =

(7.2)

t5(u

—

a)

=

d8.

If we use the fact that spec v C [0, co), then, for a 0, we can write

=

=

=2aJoI r'sinsvsincsds.

SL(2, R)

200

tv, the fundamental solution Now we use the formula for the operator zi to the wave equation 82u/8t2 — = 0, derived in Chapter 4, §2: +

zT1sintv—

(74)

—

fC2(sgnt)(2cosht—2coshr)1/2 ifr
ItI,

1.0

r denotes the geodesic distance between the source point and the observation point. We obtain where

Pau(x)

(7.5)

u(y)p0(x,y)dvol(y)

=

with

p0(x, y) = pg(r),

(7.6)

r = dist(x, y),

where

p, (r) =

(7.7)

2oC2

(2 cosh a — 2 cosh r) — 1/2 sin

d8.

This is a Legendre function. In fact, there is the integral formula

r) = (2/ir) cot(v+

(7.8)

sinh(v+ )s ds;

f (2 cosh 8—2 cosh

see Lebedev [154], page 173, valid for r E R, —1

S

R+ (Z'ei,i_i)

•

FIGURE 2.3

which implies that

Z(Z'ejm) = 2imZ'eim.

(2.41)

We first apply (2.39) to analyze Z'e,,j_i; applying both sides of (2.39) to eli and using

R_eu = we have

=

(2.42)

+ 2Z'eu,

into which we substitute (2.36). Using

R+R_eu = R+R_ei+i,j =

—2leu, —2(21

+ l)e,+ij,

from (2.10), we have an explicit forimila for R+(Z'ez,z_i):

=

(2.43)

—(2(1

—

—

where aj and are defined in (2.37). We indicate our situation in Figure 2.3. except Now R+ is injective on each Note that R+(Z'ei,z_i) E In view of (2.41), we see that (2.43) uniquely the highest weight spaces determines Z'e1_1,j, modulo an element of H?_ i• We want to identify this element, and show that it actually belongs to Mg..1,j_i, i.e., is a multiple of 1,1—1.

Specifying the component of Z'ej,j_j in 4—1,1—1 is easy; skew symmetry of Z' implies Ih.'*

' = el,l_1, el_1,l_l)

)

I

7?

—Lel,i_1, L

and the right side of (2.44) is defined by (2.36), with 1 replaced by 1 —

1:

8ilZ'e1...1,1_1 = p2el_i,1_i +

On the other hand, suppose so the right side of (2.44) is equal to —ii1_ is orthogonal to ej_ 1,1_i. We have fz—i,i—i E I'71

I

t

=

I

'7?!

Ji—1,i—i

SL(2, C) AND MORE GENERAL LORENTZ GROUPS

...

215

.

.

. •

S

S

S

S

S

FIGURE 2.4

Now Z'f1_1,1_1 is analyzed as in (2.36), and via (2.28) is seen to be orthogonal to ezj_i, so (2.45) vanishes. This proves (2.46)

Z': M1,1_1

In Figure 2.4, we record schematically our progress in understanding Z'. From here, for each m 1 — 2, the recurrence (2.39) uniquely determines the ® li+i,m. We claim this is everything, i.e., component of Z'eim in Mi—i,m ®

Z': llm

(2.47)

6_i,m

Indeed, any extra components of Z'ezm must belong to The analysis we have so far shows for p 1+ 2, and so skew-symmetry of Z' implies for A I — 2. This proves (2.47) and shows that Z' has

I

been uniquely determined on k. Then, 4 and Y_ are uniquely specified on by the identities Y_ = i[R_, Z'],

Z'],

Y÷ =

(2.48)

from (1.28). Note that, by (2.47), (2.49)

Y+: )IZm

'

®

and (2.50)

Y_: Iim

l1-1,m-1

li-t-i,m-i.

Thus, the complexified Lie algebra Csl(2, C) acts on LEMMA 2.2. Each elm E )1im is an analytic vector for the action of Csl(2, C) induced by ir.

PROOF. This follows from estimates of operator norms of R±, Z, 4, Z' on the spaces )4m. We know the norms of R± and Z. To estimate those of 4 and Z', we use the following consequence of (1.45): (2.51)

—2Z'2 —

+

= p' — 2K =

+ 81(1 + 1).

SL(2,C) AND MORE GENERAL LORENTZ GROUPS

216

Applying this to elm and taking the inner product with elm, using and = —Y_, we have (2.52)

211Z'ejm 112 + IIY_eim 112 + IIY÷eim 112 = IL1

=

—Z'

+ 81(1 + 1),

which provides adequate estimates to prove analyticity. Details are left to the

reader.

Since power series expansions for exp(siZ+s2A+s3B+s4Z'+s5A'+ssB')eim are consequently convergent for Is2 12 small, it follows that )7, the closure of in H, is invariant under ir. If ir is irreducible, this implies = H, so we have proved

= Him,

4m =

(2.53)

and in particular

dimHjm=1

L254

dimHj=21+l

0 only for integers (resp., nonintegral half-integers) lo, if is an integer (resp., nonintegral half-integer). Furthermore, if ir is not the

We see that H1

trivial representation, H1 0 for each such 1. For if not, say if a possibility is H1 is a (finite-dimensional) proper invariant subspace excluded, then 1 for ir, which would have to be all of H. But an argument as in Chapter 8 shows that SL(2, C) cannot have any nontrivial finite-dimensional irreducible unitary representation, so this possibility is excluded. In our analysis so far, we have not excluded the possibility that some = 0, i.e., Y+ejj = 0 for some 1 = lo + k. We now take care of this. LEMMA 2.3.

is minimal such thatHj0

0, k E Z+u{0} = {0, 1,2,...),

I = Io + k, then (2.55)

Y+eu

0,

i.e., Y+eu is a nonzero multiple of el+l,l+l.

PROOF. If = 0, then, by (2.36), Z'eu ej+j,l, so, by skew-symmetry, Z'ej÷i,i J.. eli. Hence Z' maps both el+l,j+l and ej+lj to Hj+i Hj+2 in such a case, and hence, by (2.48), Y+, Y_: IIi÷i,j÷i —' Inductively, one has that the Lie algebra action leaves HA invariant, so the C-action must leave its closure invariant. If ir is irreducible, this is impossible, so the lemma is proved. Consequently, the inequality (2.29) can be strengthened to a strict inequality: 32(1 +

+ 812 + 161) —

>0

for 1 = + k, k = 0, 1,2,.

which in turn is equivalent to its special case (2.56)

32(lo + 1)2(iii +

+ l6lo) —

> 0.

Thus the action of Csl(2, C) on H defined by a given irreducible unitary representation of SL(2, C) is determined by the action of {Z,

Z', Y±} on elm

SL(2, C) AND for

I

been

MORE GENERAL LORENTZ GROUPS

217

to + k, k = 0,1,2,..., mE {—1,—1 + 1,...,1 — 1,1}. This in turn has specified in terms of {Io, lhi, 1h2} in the argument presented above. There

=

is one further analysis to be made, giving P2 in terms of io and pi. Before we get to this, we make some comments on specific formulas for the action of these Lie algebra elements on 51mrn Of course, we have (2.57)

ZC1m = 2imeim,

and, by (2.10), we can (2.58)

write

R+Cjm

=

R_eim =

with

=

(2.59) To start

the

+ 1) — m(m + 1),

recursion

for

=

+ 1) — m(m

—

1).

and Z', recall that

the action of

=

(2.60)

where

pj

> 0 satisfies

(2.37),

and, by (2.36), '7, Li CU

= alell + plel+l,I

where (2.62)

aj = —ip2/8(I + 1),

The analysis of Z' on )4,z... 1 (2.63)

= —i,71/'121 + 2.

above yields the formula

Z'e1,j_i =

for 1 Io + 1, where (2.64)

discussed

rn

+

+

is defined by (2.62), and

a1 = aj(I — 2)/I,

=

+ 2.

We have '21

Li e10,10_i

These formulas determine

(266)

= i[R_, Z']eu; we have

II =

,

—oi)el,1_i

Note that = —nj_i. Again, for I = to, the first term on the right is taken to be 0. If 1 = to = 0, then of course 00 and are to be replaced by 0, in both (2.65) and (2.66); one would have (2.67)

Y_e00 =

= 7loel,_1,

which parallels the identity = tloell, from (2.60). Note that, in the event that = 0, if we apply the identities —2iZ' = and 21Z' = [R_, Y+] to COO, we obtain 2iZ'e00 = R_Y+eoo = —2iZ'e00 = R+Y_eoo = in either case, Z'eoo =

and

SL(2, C) AND MORE GENERAL LOR.ENTZ GROUPS

218

Comparing the formula (2.61) for 1 = —ip2/8, we have established (2.68)

yields a0 =

0

0.

Since, by (2.62), ao =

P2 = 0.

10 = 0

This is the first case of a constraint on P2 implied by a specification of the constraint for general lo > 0 is given by the following result. Define for any real 1> —1, by the formula (2.35). LEMMA 2.4. If the lowest K-type of ir is the representation on

=

(2.69)

then

0.

PROOF. We will deduce this by examining the identity [Y÷, Y_] = —4iZ, applied to ejj, for I lo: (2.70)

For 1 >

[Y+,Y_]ejj = 8lejj.

the left side, (Y+K. — Y_Y+)ejj, is a sum of four terms:

(2.71)

component of eu in

(2.72)

component of eli in Y+(e,+i,,_i component of Y_eui),

(2.73)

component of eli in Y+(eij_i component of Y_e,z), component of e11 in Y+(ez_i,i_i component of V_eu).

(2.74)

All these coefficients are algebraic functions of 1, and they sum to 81 for each 1 = 1o +k, k E {1,2,3,...}. The fourth term is equal to Fort = to, the left side of (2.70) is the sum (2.71)—(2.73), with (2.74) omitted. Now, for I = to, the sum (2.71)—(2.74) must also continue to equal 81o if (2.74) is replaced by The only way this can happen is for the conclusion (2.60) to hold. The identity (2.69) is equivalent to (2.75)

=

+

—

8).

This is also consistent with the result (2.68) derived in case to = is convenient to set (2.76)

=

(P1

+

—

0.

If

> 0, it

8)/8,

which is 0. Then we can write P1 and P2 as (2.77)

P1 = 8(82 + 1 —

(2.78)

P2

l6los.

this produces members of the principal series of representations of SL(2, C). In case = 0, when P1,P2 satisfy (2.77)—(2.78), the associated representation is also a member of the principal series. But the inequality (2.56) only requires P1 > 0 in this case, whereas (2.77), with s real, requires P1 8. We also have irreducible unitary representations such that (2.77)—(2.78) hold, with 10 = P2 = 0, and 5 = it, t E (—1, 1), called representations of the complementary series.

As we will see shortly, for any choice of s E Rand o E

SL(2, C) AND MORE GENERAL LORENTZ GROUPS

219

Whenever this condition on 1L2 holds, the construction above based on (lo, Iii, 112) yields

an irreducible unitary representation of SL(2, C). The fact that it

yields a Lie algebra representation of sl(2, R) by skew-symmetric operators could be checked by explicitly solving the recursion formula (2.29) for Z'eim; such explicit formulas are given in [73]. Rather than produce such formulas here, we will be content with the realizations of the principal and supplementary series given

below, which establish the existence of all irreducible unitary representations described by the parametrization {lo,pi, above, subject to (2.77)—(2.78), for s E R or is E (—1, 1). We state the result on the classification of the irreducible unitary representations of SL(2, C), first derived by Gelfand et a!. [74].

THEOREM 2.5. Each nontrivial unitary irreducible representation of the group SL(2, C) 18 equivalent to one of the following 8ort: Principal series: irlc,,s,

(2.79)

where lo E (0,

1,

.}, 3 E R. This

determined by the condition that

(2.77)— (2.78) hold.

(2.80)

Complementary 8erzes:

(t

0),

where t E (—1, 1); = 0 and (2.77) holds with s = it. The3e repre3entatzon3 are mutually inequivalent, except lro,3 lro,_8 and lro,it and all irreducible.

The mutual inequivalence is clear since for different parameters the triples 111, } differ. Other than the realizations of these representations, which we wiil undertake shortly, the only point of the theorem which remains to be established is the irreducibility of each of these representations. This can be accomplished in the same fahion as the proof of irreducibility for SL(2, R), in Theorem 2.2 of Chapter 8. Namely, by (2.54), each C-invariant subspace of H1 must be a direct sum of certain Hg's, so if the representation ir = lrj0,3 of C = SL(2, C) were not irreducible, there must be an identity of the form (2.81)

(lr(g)ezm,elsmi) = 0

for all g E SL(2,C),

for some pair Cirn, el'm', which in turn implies (2.82)

(Teim,ei'm') =

0

for all T E

Setting T equal to an appropriate product of powers of and R_ produces a contradiction, thus proving irreducibility. We turn to the construction of realizations of the representations in the principal series. In parallel with the case for SL(2, R), we can construct these by decomposing the regular representation of SL(2, C) on L2(C2) = L2(R4): (2.83)

R(g)f(z) = f(gtz),

g E SL(2, C), z E C2.

In this case, R(g) commutes with the group of complex dilations: (2.84)

D(a)f(z) = IaI2f(az),

a E C = C \ {O}.

SL(2, C) AND MORE GENERAL LORENTZ GROUPS

220

We expect to decompose R into irreducibles via the spectral decomposition of D. Since CS = S' x this decomposition is accomplished by combining Fourier series and the Mellin transform. Thus, given 13(0, t) on S1 x set sf3(n, a)

(2.85)

=

f

dO dt,

f 13(0,

for n E Z, a E R. Then we get the inversion formula 13(0, t) = (2ir)_2

(2.86)

s$(n,

L

da,

and the Plancherel formula 113(9, t)l2tdOdt = (2ir)_2

(2.87)

Is/3(n, 3)12 da.

Thus, if we define

(2.88)

=

f f

d9dr,

we see that, for / E C8°(R4), P,,,,f belongs to the space

=

E L?OC(R4 \ 0): g(retOz) =

(2.89)

/

gI2 0 of the hyperboloid — — R", endowed with the Lorentz metric dx? +•

stereographic projection is produced by the following obvious variant of Figure 1. We set x= (x1,... E R".

FIGURE 2 We have

= (1 —

The image of We compose is the complement of the unit ball in with the inversion (2.4) to produce a map of onto the unit ball: (2.13)

=

(1

In this case, (2.14)

=

+ 1z12,

GROUPS OF CONFORMAL TRANSFORMATIONS

231

and we can rewrite (2.13) as

ir\ If we

I

set y =

(2.16)

— 11 —

5%.X,

s(x,

we

s*dy =

(1

and a straightforward

+

Z•

see that

+

dx, — (1 +

calculation gives

5*>2dy2

(2.17)

=

Note that from

+ 1x12 and

IxI = 21y1/(i — 1y12), which

(1 +

=

Thus we can rewrite

(2.17) as

(2.18)

1+

follows the identity

=

in turn gives

(1

+ 1Ji + 1x12)2 = 4(1

—

IyI2Y2.

s4(1 — 1y12)2 >dy,2 = Edx,2 —

(2.19)

is conformally equivalent to the unit ball Thus we see that hyperbolic space in and more, the left side of (2.19) gives an explicit hyperbolic metric on the unit ball. Since 1) is a transitive group of isometrics of (n, we see that SOe (n, 1) acts as a transitive group of conformal automorphisms of the unit ball in

Let us relate the ball to the upper half space again. We have the conformal map r, defined by (2.8), from the unit ball to the upper half space in We see what happens to the hyperbolic metric on the ball when it is pulled back to the half space. If y = r(x), then (2.8) gives (2.20)

in

analogy

= with (2.7). Now the hyperbolic metric on the ball is 4(1 — 1x12)2

and, using x + (2.21)

we see after a short calculation that + and hence (2.20) gives

+

= =

(1 — 1x12)21x +

r

= 4(1

—

as the hyperbolic metric on the upper half space Note that this metric is invariant under dilations, and also under rigid motions parallel to the boundary. It is easy to see that any metric on the upper half space with this property would have to be of the form Thus we obtain y;2

in

Cy;2

_________________________

GROUPS OF CONFORMAL TRANSFORMATIONS

232

and changing the variable we can make a = 1, so the hyperbolic metric on the upper half space could also be derived that way. What is not so clear when one looks at this metric alone is the appearance of a group of metric preserving rotations about any fixed point of the upper half space, which is manifest for the center of the ball. On the other hand, we can gain insight by puffing the dilations of the upper half space back to the ball. So we want to conjugate the dilation action

= ox,

(2.22)

u > 0,

on the upper half space by the map r, given by (2.8). Note that

r'(y) =

(2.23)

+

—

acting on the unit ball in

One obtains for

=

p224'

'I,—

Note

y+

lutz +

+

+

1

the formula —en.

—

that = [(ku + = [(1— u)/(1

(2.25)

—

on the boundary of the ball; as u 1

so as o j 0, b0(O) approaches

00,

approaches —es.

We can use the results of §1 to solve the Dirichiet problem for functions harmonic on the unit ball B1 C R":

u=

on B1,

uIsn_i = f. First recall that harmonic functions satisfy the mean value principle, so (2.26)

0

u(0) = Avg f(x). Sn-i Now, as a special case of the results of §1, we see that, if u(z) is harmonic in B1, (2.27)

sois =

(2.28)

preserves the hyperbolic metric

In this case, since (2.29)

=

4(1 —

we have

= =

(2.30)

(1

—

—

lzl2)2.

We will apply the mean value property to (2.28), in order to evaluate u at We obtain the formula Let r=(1—o)/(1+o), so (2.31)

(Q)_(fl_2)/4 Avg

=

=

(1

—

Avg 5n_i

(z)

(x))

,(0).

GROUPS OF CONFORMAL TRANSFORMATIONS

233

Note that

=

(2.32)

=

so

(2.33)

Avg

= (1 —

sn-i

In order to utilize the formula (2.30) for we have to pass to the limit xE from lxi < 1. A fairly straightforward calculation from the formula (2.24) gives (2.34) (1

lDa(X)12)/(1 — ixl2) = a[ 1

+

we get

2c[c2(1

(2.35)

lcy2x +

—

+ (1 + Xn)]',

XE

Since c= (1—r)/(1+r), we obtain (2.36)

yg(x) = (1

r2)2[1

—

+ r2 + 2rx.

xE

and hence (2.37)

xE

1i/g(X) = (1 —r2)2[1+r2

Thus (2.33) gives (2.38)

u(ren) = (1 — r2) Avg f(x)(1 + r2 — 5n-i

In light of rotational invariance, we immediately obtain the following formula for the solution to (2.26): (2.39)

u(rw) = (1

—

Sn-i

r2 —

2rx . w)—"/2 dx,

where is the volume of r < 1. This is Poisson's formula, wE which we used in Chapter 4, §1. Note that, for our argument to be complete, we need Yamabe's identity, which in this case is

=0, where is obtained by raising (2.34) to the power (n — 2)/2. Any derivation of the Poisson integral formula seems to have a grubby calculation somewhere, and

this is it. We leave it as a challenge to the reader to verify this identity. We have seen the action of SOe(n, 1) on the unit ball in R?, by conformal automorphisms. This action induces a group of conformal diffeomorphisms of the boundary The ball can be pictured as the lower hemisphere in S?*. Here we will sketch an action of SOe(n+ 1, 1), as a group of conformal automorphisms of S's, such that the natural subgroup SOe(fl, 1) leaves the equator, and hence each hemisphere, of Sri, invariant.

GROUPS OF CONFORMAL TRANSFORMATIONS

234

In fact, consider = forward light cone, Xn+2 > 0, + f+, and R+ action, by dilations,

with a Lorentz metric. Let r+ be the — = 0. There is a natural +

= Sn.

(2.40)

1,1, preserving the Lorentz metric, leaves

The group SOe (n +1, 1), acting on

r+ invariant, and acts as a group of "isometries" of r+. Now the Minkowski restricted to r+, is degenerate in the radial direction. However it induces a conformal structure on via (2.40), coinciding with the usual conformal structure given by the standard metric on S", and SOe(fl + 1, 1) is seen to act as a group of conformal transformations on = Similarly 1), acting on acts on the forward light cone and hence on S" Let us consider the subgroup of SOe (n, 1) fixing some point p E 5n1, which we will identify with the ray in through the point (0,.. . , 0, 1, 1). Clearly this is fixed by the subgroup M = SO(n — 1), acting only on the first n — 1 variables, and by the subgroup A = SOe(1, 1), acting on the last two variables. Consider now N, the connected subgroup of SOe(fl, 1) with Lie algebra metric on

a, given by (3.9)—(3.10) of Chapter 9. If such an element X of a as described by these formula acts on (0,.. . ,0,1,1) E one obtains 0, so exp tX fixes this point. Hence from the material developed in Chapter 9, §3, we see that MAN is precisely the subgroup of SOe (n, 1) fixing p E S"1. If we use a stereographic projection with p as the north pole, we see that MAN acts as a group of conformal automorphisms of Eudidean space In this representation MN is the Euclidean group and A is the group of dilations. = 5fl, which This last description of the action of + 1,1) on is the conformal compactification of R", can be generalized to the action of SOe(p + 1, q + 1) on the conformal compactification of with the metric = — — — as follows. If o: + IIxPI2 is the canonical map onto projective space, define (2.41)

J(x) =

Then J maps

o(1

—

11x112,

x, 1 +

11x112),

xE

p + q = n.

into the set

M = {c(e):

= 0). The set M gets a natural conformal structure, from the degenerate metric on 1,q+1: Cu2 = 0), making it a conformal compactification of the set (C e ii In particular, SOe(4, 2) acts as a group of conformal transformations on the conformal compactification of Minkowski space R3". For more on this, see (2.42)

[189].

E

lieu2

CHAPTER 11

The Symplectic Group and the Metaplectic Group The symplectic group Sp(n, R) arose in Chapter 1 as a group of automorphisms of Here we make a more detailed study of Sp(n, R). In Chapter 1 we produced a unitary representation of the universal covering group Sp(n, R), related to this group of automorphisms. It is an important and beautiful fact that one actually obtains a representation of the double cover, denoted Mp(n, R), and we present the details on this here.

1. Symplectlc vector spaces and the

group. A symplectic

structure on a real vector space is a bilinear form

u:VxV—'R

(1.1)

which is skew symmetric (c(u, v) =

u)) and nondegenerate, i.e., if a(u, v) = 0 for all V E V, then u = 0. If al is a nonzero element of V, there is a E V such that c(ai,$i) = 1. If W1 denotes the linear span of a1 and and if V1 is the subspace of V consisting of vectors orthogonal to W1 with respect to a (i.e., v V1 if and only if a(v, w) = 0 for all w E W1) then V1 is also a symplectic vector space with respect to xV1. We can repeat this argument on V1. Thus, inductively, we can choose a basis a1, . . , of V such that /3k,. . , —o-(v,

.

(1.2)

a(cx,,cxk) = c($3,$k) =

.

0,

= 63k•

Such a basis of V is called a symplectic basis. It follows that the dimension of V is even, and any two symplectic vector spaces of the same dimension are symplectically isomorphic. Thus we can consider the standard model, V = = with symplectic form

o(u,v)=u1.v2—u2.v1

(1.3)

where u = (u1,u2), v = (v1,v2), u3,v3 E and u3 vk is the standard inner product on Alternatively, with v representing the standard inner product on (1.4)

we can write

o(u,v)=u.Jv 235

236

THE SYMPLECTIC GROUP AND THE METAPLECTIC GROUP

where (1.5)

I denoting the n x n identity matrix. The group of linear transformations on preserving the symplectic form is called the symplectic group, and is denoted Sp(n, R): (1.6)

Sp(n,R) = {A E

=

If we denote "standard" coordinates on

by (z1,...

,

ti,...

,

then, as

an element of A2(R2n)l, the symplectic form can be written (1.7)

a volume element on

preserved by each A E Sp(n, R), so

Sp(n,R) C SL(2n,R).

(1.8)

Note that Sp(1,R) = SL(2,R). The Lie algebra sp(n, R) of Sp(n, R) is identified with the set of linear transformations T on which are skew symmetric with respect to a, i.e., such that

u(Tu,v) = —a(u,Tv),

(1.9)

or, equivalently, such that

TtJ =

(1.10)

—JT,

where J is given by (1.5) and Tt denotes the transpose of T:

Tu.v=u.Ttv.

(1.11)

We can give the following alternative characterization of sp(n, R). For any smooth function I on R2", there is associated the Hamiltonian vector field

H1 =

— (af/8x1)(.9/.9e,)].

The flow generated by Hj preserves the form a. Now this flow consists of linear transformations on R2" precisely when f(x, is a second order homogeneous Thus sp(n, R) is isomorphic to "P2, defined by polynomial in z and (1.12)

hp2 = {Q(x, e): Q homogeneous polynomial of degree 2}.

The Lie bracket on sp(n, R) corresponds to the Poisson bracket on hp2. The Poisson bracket of two smooth functions is given by (1.13)

{f,g} = H1g =

—

An explicit Lie algebra isomorphism from hp2 to the set of linear transformations on satisfying (1.9) is given by (1.14)

QI—IFQ

THE SYMPLECTIC GROUP AND THE METAPLECTIC GROUP

237

where FQ is defined by

Q(u, v) = O'(FQU, v)

(1.15)

polarizing Q, i.e., where Q(u, v) denotes the symmetric bilinear form on such that Q(u, ii) = Q(u). The map FQ is the Hamilton map associated with Q, already considered in Chapter 1, §6. One way to look at (1.13) is the following. Suppose (1.16)

with A and C symmetric. If we identify HQ, a vector field on with linear coefficients, in a natural fashion with a linear transformation XQ on we have

(B -A

(1.17)

_Bt

If Q denotes the bilinear form associated with Q(u),

since HQ =

Q=

(1.18)

(AB\ C

and these formulas clearly give

XQ=Q0J.

(1.19)

leave it to the reader to verify that the Lie algebra structures are preserved. To grasp the structure of the Lie algebra sp(n, R), it is useful to consider the

We

following subalgebras: (1.20)

nxnrealmatrix}i

m=

nxnrealmatrix}i

(1.21)

(1.22)

:Y is n x n real matrix}.

n

= If we identify sp(n, R) with hp2, we can write (1.23) (1.24)

m=linearspanofx,xk,

1j, kn),

(1.25)

Then sp(n,R) is the linear direct sum of m, iii, and n; (1.26)

sp(n,R)=m+1IH-n.

THE SYMPLECTIC GROUP AND THE METAPLECTIC GROUP

238

We see that m and iii are commutative subalgebras of sp(n, R), and n is a subalgebra of sp(n, R) isomorphic to gl(n, R). Furthermore, simple calculations give

c n,

(1.27)

[m,n]

cm,

C

For example,

(o x\ (o o\

L128

(xy

o

o)

o

(1.29)

o) '

o

+ Ytx 0

=

—

The action of gl(n, R) on m, identified with the linear space of symmetric real n x n matrices, devolving from (1.27), is the differential of the action of Gl(n, R) on this linear space given by A . X = AXAt, as is easy to verify. Another subalgebra of sp(n, R) is spanned by (1.30) In

)t,k(X,

= Z,Zk +

= x3ek —

1h,k(X,

the picture of sp(n, R) given by (1.10), this subalgebra is of the form

(1.31)

:

A, B real symmetric n x n matrices}.

—

This generates a subgroup of Sp(n, R) isomorphic to the unitary group U(n). with C's. Then the symplectic form Indeed, let (z, i— z + identify (1.3) becomes

c(zl,z2) = Imz1

(1.32)

The group generated by (1.31) leaves invariant the Hermitian form (z1, z2) = z2, and thus U(n) is exhibited as a subgroup of Sp(n, R). It is a maximal compact subgroup of Sp(n, R), as we will see in the next section. inThe map (x, i— —x), which preserves the symplectic form on duces an involutive automorphism 0 of Iip2 sp(n, R): •

(1.33)

(OQ)(x,

=

—x).

Note that 0 is the identity on the Lie algebra of U(n), spanned by (1.30), which we will denote t. Note also that 0 = —I on the linear space p, spanned by (1.34)

and

Note that

hp2=P+p

(1.35)

and that (1.36)

[P,t]

c

c p,

[p,p] Ct.

THE SYMPLECTIC GROUP AND THE METAPLECTIC GROUP

239

There is a further condition, which will be discussed in Chapter 13, which makes 0 a special case of a Cartan involution. As a consequence, we have the following

result, analogous to polar decomposition. A proof, in a more general setting, will be given in Chapter 13. THEOREM 1.1. Let G be any connected Lie group with Lie algebra sp(n, R). Denote the exponential map by exp: sp(n, R) —' C. Let K be the connected Lie group in G with Lie algebra t. Then K 1., closed and the map p x K to C given by

(X,k) '—' (expX)k

(1.37)

is a surjective diffeomorphism.

This result applies to the group Sp(n, R) in view of PROPOSITION 1.2. Sp(n, R) is connected.

We leave the proof of this to the reader. Note that it gives us topological information on Sp(n, R), as it follows that Sp(n, R) is homeomorphic to K" x U(n), N = n2 + n. Thus Sp(n, R) is homotopically equivalent to U(n). In particular, its fundamental group is isomorphic to Z. Thus, for each k, there is a unique connected Lie group which is a k-fold cover of Sp(n, R). The 2-fold cover is called the metaplectic group and denoted Mp(n, R).

2. Symplectic Inner product spaces and compact subgroups of the symplectic group. If a real vector space V is endowed with a symplectic form a(u, v) and a positive definite inner product Q(u, v), we will call V a symplectic inner product space. Note that any compact subgroup K of Sp(n, R) must leave invariant some such inner product, and hence K is contained in the compact group KQ defined by (2.1)

KQ = {g E Sp(n, R): Q(gu, gv) = Q(u, v) for all u, v

V}.

As before, we have the Hamilton map FQ on V, defined by

Q(u,v) =

(2.2)

We also introduce the linear map AQ =

(2.3) the unique square root of

mal basis E1, F3, 1 j (2.4)

with positive spectrum. If we choose an orthonor-

n, of V such that

FQE2 =

FQF3 = —ii3E,,

>0,

whose existence is proved in Lemma 6.2 of Chapter 1, then (2.5)

AqE1 =

AQF, =

Now consider the map J: V —' V defined by (2.6)

J=

240

THE SYMPLECTIC GROUP AND THE METAPLECTIC GROUP

Note that

J2 =

(2.7)

—I.

In fact, in the basis E,, F, above,

JE, = F3,

(2.8)

JF, = -E1.

Thus J defines a complex structure on V. Note that J and FQ commute and

Q(Ju,v) = —Q(u,Jv).

(2.9)

Define a new quadratic form Q by

v) =

(2.10)

v).

Note that

=

(2.11)

also defines a positive inner product on V. It is easy to see that and hence, if .1 is associated with by the process above, then i = J. so

LEMMA 2.1. We have (2.12)

PROOF. Clearly, from (2.1), we have

KQ = {g E Sp(n, R): g commutes with FQ}.

(2.13)

Also

g commutes with FQ

(2.14)

g

commutes with J.

This makes (2.12) apparent. Now define an R-bilinear form (, ):V x V —p C by (2.15)

(u, v) =

=

v) = Jv).

v) + v) +

v)

v) —

In the last identity, we have used (2.9) with Q replaced by

(Ju, v) =

(2.16)

We see that

v) + i()(u, v) = i(u, v)

and

(u, Jv) =

(2.17)

Also, since i(Ju, v) = (2.18)

Jv) —

v) = —i(u, v).

Jv),

(v, u) = r(Jv, u) —

io(u, v) =

and furthermore

(2.19)

(u,u) =

0.

= J,

THE SYMPLECTIC GROUP AND THE METAPLECTIC GROUP

Thus (, that

241

) is a Hermitian inner product on the complex vector space (V, J). Note

(2.20)

= {g E Sp(n, R): g commutes with J} = {g E Gl(V): g preserves

and commutes with J},

so (2.15)—(2.17) imply (2.21)

= {g E Gl(V): g commutes with J and g preserves (, )} =U(n).

In light of the observation of §1 that U(n) c Sp(n, R), we have THEOREM 2.2. U(n) is a maximal compact subgroup of Sp(n, R). Thus we see that any symplectic inner product space (V, a, Q) has a complex structure such that (by (2.9)—(2.11)) (2.22)

a(Ju, Jv) = c(u, v).

There is furthermore associated a new symplectic inner product space (V, a,

= U(n). In fact, (2.10) produces a one-to-one correspondence between symplectic inner product spaces (V1 a, Q) with maximal symmetry group = U(n) and complex symplectic spaces (V, J, a), satisfying (2.22), such that whose symmetry group is enlarged from KQ to

o(Ju, u) 0.

(2.23)

Then (2.15) produces a one-to-one correspondence between each of these classes and complex Hermitian spaces (V, J, (, )). We could generalize, discarding the positivity hypothesis (2.23). If J is any

complex structure on V satisfying (2.22), then the quadratic form (2.10) and the Hermitian form (2.15) are still nondegenerate. It is clear that the Hermitian form (, ) could have any signature (p, q) such that p + q = n. Such a form gives an inclusion (2.24)

U(p, q) C Sp(n, R),

p + q = n.

3. The metaplectic repreeentation. The metaplectic representation on the Lie algebra level is quite easily described. If Q (3.1)

E hp2

sp(n, R), we set

w(Q) = iQ(X,D),

where Q(X, D) is the second order differential operator associated to Q(x, the Weyl calculus. Explicitly, we have

w(x1xk)u(x) = (3.2)

= i.92U/ÔXkÔXI, + ä(x,u)/ôxk). =

via

THE SYMPLECTIC GROUP AND THE METAPLECTIC GROUP

242

It is easy to verify that w({Qi,Q2}) = erations, discussed in Appendix D, imply that

Now general consid-

exponentiates to a unitary representation of the universal covering group Sp(n, R). This representation and some of its important consequences were discussed in Chapter 1, on the Heisenberg group. A more subtle result is that w actually gives a representation of the metaplectic group Mp(n, R), the 2-fold cover of Sp(n, R). We will prove this result here. First, we see that cannot give rise to a representation of Sp(n, R) itself. Indeed, consider Q E hp2 sp(n, R) given by Q,(x, = Then + (3.3)

HQ,

= 2(e,8/ox,

—

x,8/oe,) =

—.28/89,

where 8/89, generates rotation in the x3 — plane. It is clear that the group of transformations on generated by the vector field HQ1 coincides with the group of symplectic transformations generated by Q, E sp(n, R) under the exponential map. In view of (3.3), exp(irQ,) =

(3.4)

the identity in Sp(n, R).

e,

However, as we have seen in Chapter 1, §6, the spectrum of Q,(X, D) =

consists of {1,3,5,7,...}. Hence

=

(3.5)

+

has the property

=

(3.6)

—I.

This of course leaves open the possibility of w existing on the double cover of Sp(n, R). We will see that this happens by looking at the Bargmann-Fok representation, introduced in §5 of Chapter 1. Here, Sp(n, R) is represented on the following Hubert space of entire functions on C": (3.7)

=

holomorphic on C":

ff

< oo}.

Let us denote by the representation KwK1 where K: L2(R") —i I is the unitary map given in (5.6) of Chapter 1. We have the following key result.

PROPOSITION 3. 1. The representation restricted to P, the Lie algebra of U(n), exponentiates to a unitary representation of MU(n), the double cover of U(n), to give (3.8)

w#(g)u(z) = (detg)112u(g' . z),

g E MU(n).

Here g.z is the action of MU(n) on C" defined by MU(n) —' U(n) and (det g)'/2 is the unique smooth square root equal to 1 at g = e, the identity element. PROOF.

It is clear that the right side of (3.8) defines a representation of

MU(n). That its derived representation of P is given by (3.1), conjugated by K, follows from the proof of Lemma 7.8 of Chapter 1.

THE SYMPLECTIC GROUP AND THE METAPLECTIC GROUP

243

We are now ready to define the metaplectic representation. First, if k e MU(n), set w(k)

(3.9)

= K'w#(k)K

where c,j#(k) is defined by (3.8). Then, if g e Mp(n,R) is given by (3.10)

g

= (expQ)k,

Q

e p,

k e MU(n),

set

w(g) =

(3.11)

By Theorem 1.1, (Q, k) '—p (exp Q)k defines a surjective (analytic) diffeomorphism of p x MU(n) onto Mp(n, R), 80 w is a well-defined strongly continuous function on Mp(n, R), taking values in the set of unitary operators on

THEOREM 3.2. The formula (3.11) defines a representation of Mp(n, R). PROOF. What we need to show is that (3.12)

w(gl)w(g2)f = w(glg2)f,

fE

g3 E Mp(n, R).

We will use the fact that (3.1) does exponentiate to a representation of Sp(n, R);

call it (see Appendix D). We can identify a small neighborhood U of the identity element in Sp(n, R) with a neighborhood U of the identity element in Mp(n, R), and view as defining a local representation of U. Thus, if V is a neighborhood of e e Mp(n, R) so small that V . V C U, we have (3.13)

= g

= (exp Q)k, then must be given by on U. Thus, in view of (3.13), we see that g

the right side of (3.11), so = w (3.12) holds for g, E V. Now if f is an analytic vector (with respect to w(hp2)), both sides of (3.12) are real analytic in (ga, ga), so the identity (3.12) must hold for all g, E Mp(n, R). Since the set of analytic vectors is dense in (see Appendix D), it follows that (3.12) holds in general. This completes the proof. The representation w of Mp(n, R) is called the metaplectic representation. As was remarked in Chapter 1, it is not irreducible. The subspaces of consisting of functions which are, respectively, even or odd under the involution 1(x) i—p f(—x), are invariant under w; w acts irreducibly on each of them. We leave these assertions as an exercise. For g belonging to a large subset of Mp(n, R), w(g) has an explicit integral representation, which we will now derive. Let g° denote the image of g in Sp(n, R). Suppose g° E Sp(n, R) has a generating function A(z, x') such that

=

(3.14)

is equivalent to (3.15)

= (ÔA/ôx)(x,x'),

= —(ÔA/e3x')(x,x').

THE SYMPLECTIC GROUP AND THE METAPLECTIC GROUP

244

A(x, x') is a homogeneous second order polynomial in (x, x'). We can hence write

A(x, x') =

(3.16)

P

Q are symmetric linear maps on R", L E End(R"). We will say

g° E SPreg(fl, R) if this holds with det L 0. Such g° has two preimages in Mp(n, R), and a choice of preimage coincides with a choice of (det L)1/2. We denote by MPreg(fl, R) the preimage of Spreg(n, R). Clearly SPreg(fl, R) is open and dense in Sp(n,R).

PROPOSITION 3.3. Let g E MPreg(n, R) determine A and (det L)1/2, ma (3.14)—(3.16). Then

w(g)u(x) =

(3.17)

dy.

L)"2 J

PROOF. We analyze the right side of (3.17) as a composition of several unitary

operators in the metaplectic representation, applied to u. Write xe_i

=

(3.18)

Ye(h/2)iQYVu(y)

Then the right side of (3.17) is equal to

PeQu(x)

(3.19)

where

Qu(x) =

(3.20)

Lu(x) =

(3.21)

dy,

L)1/2 J Pu(x) =

(3.22)

= the unitary operators (3.20) and (3.22) are in the metaplectic

Note that if Q(x, so

= Qx . x, then Q(X, D)u(z) = Qx . zu(x) and

representation. We can write (3.21) as

Lu =

(3.23)

where Y' is the Fourier transform: (3.24)

=

J

dy,

and

(3.25)

Lv(x) = (det L)'/2v(Lx).

The operator L is clearly unitary. it is also in the metaplectic representation. In fact, the last line of (3.2) defines a representation of the Lie algebra of

(via

A

0))

THE SYMPLECTIC GROUP AND THE METAPLECTIC GROUP

245

which gives rise to the representation of M1(n, R), the 2-fold cover of Gl(n, R) lying in Mp(n, R), defined by (3.25). Also the Fourier transform belongs to the

metaplectic representation. In fact, the map J E Sp(n, R) given by J(z,

=

defines an automorphism a of the Heisenberg group such that the representations In and In o a of are conjugate by w(g) if g is a preimage of J (i.e., ir(g) = J where in: Mp(n, R) Sp(n, R) is the natural projection). The Fourier transform accomplishes this and is so characterized to within a scalar. Indeed, a simple calculation, using J = Q(z, = 1z12 + gives —x)

= ±7,

(3.26)

if ir(g) = J.

Now the right side of equation (3.17) is consequently unitary, and is equal to w(glg2g3g4) where g3 E Mp(n, R) have just been described. A straightforward calculation shows that glg2g.3g4 E Mp(n, R) is equal to the g E Mp(n, R) corresponding to (A, (det L)1/2), and the proposition is proved. We make note of

PROPOSITION 3.4. Any g E Mp(n, R) can be written as a product of two terms

= g = gigs, We leave the proof to the reader. As an application of Proposition 3.3, we will derive formulas for where H= + 1z12 = Q(X, D) is the harmonic oscillator Hamiltornan. Formulas for have already been derived in Chapter 1, by other methods. In and (3.27)

view of (3.3), we have

HQ =

(3.28)

in this case, so the group of symplectic linear transformations generated by Q (3.29) = ((cos2s)x+ (sin28)e,—(sin2s)z+

is

= go(s)(z, We see

that generating functions A3(x, x')

(3.14), (3.15) are given by

A3(x,y) = (sin2s)' Thus, according to (3.17), (3.30) (3.31)

= (211)"/2(sin28)"/2 .fdYu(Y)exP{_i

+

1y12)

+ x y] /sin2s}.

This is valid whenever a is not an integer multiple of ir/2, where the correct square root in (sin continues to give (3.32)

must be taken for n odd. Note that (3.31) analytically

=

.Jdyu(y)exp{_

+ 1v12)

—z .y] /sinh2s}.

Compare this formula with formulas (7.14)—(7.15) from Chapter 1.

CHAPTER 12

Spinors In previous chapters, we have seen the double coverings SU(2) —, SO(3),

SU(2) x SU(2) — SO(4),

and SL(2, R)

SL(2, C)

SOe(2, 1),

SOe(3, 1).

Generally, the orthogonal groups SO(n) and their noncompact analogues SO(p, q) have double coverings, denoted Spin(n), and more generally Spin(p, q). We study these spin groups here. We also consider spinor bundles and Dirac operators on manifolds with a spin structure.

1. Clifford algebras and spinors. To a real vector space V (dim V = n) equipped with a quadratic form Q(v), with associated bilinear form Q(u, v), we associate a Clifford algebra Cl(V, Q), which is an algebra with unit, containing V, with the property that, for v E V,

vv=

(1.1)

—Q(v).

1

We can define Cl(V, Q) as the quotient of the tensor algebra 0 V by the twosided ideal J generated by v 0 v + Q(v) 1, v E Cl(V,Q) = ®V/J.

(1.2)

Note that, for u,v e

u•v+v•u= —2Q(u,v). 1.

(1.3)

By construction, C1(V, Q) has the following universal property. Let A0 be any associative algebra over R, with unit, containing V as a linear subset, generated

by V, and such that (1.1) holds in A0, for all v E V. Then there is a natural surjective homomorphism

a:Cl(V,Q) — Ao.

(1.4)

If , e,,} is a basis of V, any element of Cl(V, Q) can be written as . a polynomial in the Since e3ek = —eke2 — 2Q(e2, ek) 1 and in particular . .

246

SPINORS

247

1, we can, starting with terms of highest order, rearrange each monomial in such a polynomial so the e3 appear with j in ascending order, and no exponent greater than one occurs on any e3. In other words, each element w E C1(V, Q) can be written in the form e32 = —Q(e,)

(1.5)

w= i1=O or 1

We claim this representation is unique, i.e., if (1.5) is equal to 0 in Cl(V, Q), then all coefficients a1,.. .j,, vanish. This can be seen as follows. Denote by A the linear space of expressions of the form (1.5), so dim A = T'. We have a natural linear map /3Q:A—4Cl(V,Q)

(1.6)

is injective. and by the discussion above is surjective. We want to show First consider the (most degenerate) case Q = 0. In such a case, Cl(V, 0) is the exterior algebra A* V; (1.5) is customarily written (1.7)

w= i,=O or

1

A e3 defines an algebraic structure on A, in this case. By the universal property mentioned above, we have a surjective homomorphism of A Cl(V, 0), algebras a: Cl(V, 0) —+ A. To compose this with the linear map which we also see to be a homomorphism of algebras, we note that af30 and The rule ej A ek =

are clearly the identity on V, which generates each algebra, so a and /3o are inverses of each other. Hence (1.6) is seen to be an isomorphism of algebras for Q

=0:

(1.8)

13o:A Z Cl(V,0) =

AV.

Using this, we will identify A and K V as linear spaces. If we write

(1.9) where Ac V

is the linear span of (1.7) for i1 +... + i,, = k, we have a natural identification of V with the space of antisymmetric k-linear forms V' x ... x V' R (V' = dual space to V), via

(1.10) (V1A•.•Avk)(Xl,...,Xk)=(1/k!) CESk

for v3 E V, X1 E V'.

In order to show (1.6) is an isomorphism for general Q, we produce an algebraic structure on A V = A, for each quadratic form Q on V. Note that such Q defines a linear map (1.11)

Q:V—+V'

SPINORS

248

by (1.12)

(w, Q(v)) = Q(v,w),

v,w e V.

We will use both the exterior product on K V, i.e., the product yAw in Cl(V,0), and the interior product, defined as follows. First, for X E VI, we define Lx:AkV

(1.13) by

(1.14)

(txw)(Xi, .

.

= w(X,Xi,... ,Xk_1),

.

where we have identified w E V with an antisymmetric k-linear form, as described above. Then, for v E V, the interior product (1.15)

is defined by (1.16)

Jv

=

with Q given by (1.11)—(1.12). Let us consider the algebraic relations among these operators and the operators (1.17)

VEV,

defined by (1.18) The algebraic properties of these operators are the following. Clearly

=

(1.19)

— A,,, A,,

for all v, w e V, and also a simple calculation shows that X E V', hence j,,j,, = 0 for all v E V, and thus (1.20)

JvJw =

JwJv

for all v, w E V. Finally, a calculation yields (1.21)

(Lx A,,, + A,,, tx)w

= (X,w)w

and hence (1.22)

3v

+ A,,, 3,, = Q(v, w)I.

In particular, if we define linear operators M,, on K V by

vEV,

(1.23) we

have the anticommutation relations

(1.24)

M,,M,,, +

= —2Q(v,w)I,

for v,w E V, as a consequence of (1.19)—(1.22). Note also that (1.25)

M,,v = —Q(v). 1

=

0

for all

SPINORS

249

and (1.26)

Thus the algebra .M in End(A5 V) generated by v E V} naturally contams V, and by the anticommutation relations (1.24) and the universal property of Cl(V, Q), we conclude that there is a natural surjective homomorphism of algebras jz:Cl(V,Q) —,

(1.27) extending

v '—+

Define a linear map 'y:Cl(V,Q)

(1.28) by

'y(x) = p(x)(l).

If we use (1.8) to identify A and K V, we have a commutative diagram -Z

Cl(V,Q)

defining

a map K V

A5V

4t3Q1

171

A

A5V

A5 V. We see that —

if

A1V

13o(x)

E ACV.

1 0. Then we get it for A2 E C \ analytic continuation.

R, by

THE FOURIER TRANSFORM AND TEMPERED DISTRIBUTIONS

291

One could also tacide the resolvent by looking at the Fourier transform on the right side of (A.22). If we use (1.21) to express this Fourier transform as a Hankel transform, the formula (A.35) is seen to be equivalent to the special case when ii = — 1 of the classical identity

J(A2 +

ds

=

See Lebedev [154j, page 133, for a proof of this identity (in the case of absolute convergence) and further generalizations.

APPENDIX B

The Spectral Theorem Our purpose here is to give a brief discussion of the spectral theorem. We consider several forms. The best known form is THE SPECTRAL THEOREM. If A is a selfadjoint operator on a Hubert space H, then there is a strongly countably additive projection-valued measure dE such that (B.1)

Au

=

j

u E V(A).

...\

This result is a consequence of the STRONG SPECTRAL THEOREM. I/A is a selfadjoint operator on a separable Hubert space H, there is a o-compact space fl, a Borel measure p on fl, a unitary map (B.2)

W: L2(fl, dp) —, H,

and a real-valued measurable function a(x) on Il, such that (B.3)

W'AWJ(z) = a(x)f(x),

Wf E V(A).

In order to establish this result, we will work with the unitary group (B.4)

U(t) =

For a given E H, let He denote the closed linear span of for t E R. If He = H, we say is a cyclic vector. If He is not all of H, note that Ht

is invariant under U(t). Using this, it is not difficult to decompose H into a (possibly countably infinite) direct sum of spaces He,. It will suffice to establish the Strong Spectral Theorem on each factor, so we wili establish the following result. PROPOSITION B. 1. If H = (so there is a cyclic vector then we can take Il = R, and there exists a positive Borel measure p on R and a unitary map W: L2(R, dp) —* H so that (B.5)

W'AWI(x) = xf(x), 292

Wf E V(A),

THE SPECTRAL THEOREM

293

or equivalently,

W1U(t)Wf =

(B.6)

The measure

fE

on R will be the Fourier transform

(B.7)

where (B.8)

c(t) =

A priori, it is not at all clear that (B.7) defines a measure, though, since clearly ç E L°°(R), we see that is a tempered distribution. We will show that is indeed a positive measure during the course of our argument. We will in the meantime use notation anticipating that z is a measure: (B.9)

(ettAe,

=

1:

keeping in mind that the Fourier transform in (B.9) is to be interpreted in the sense of tempered distributions. As for the map W, we first define (B.1O)

where S(R) is the Schwartz space of rapidly decreasing functions, by

W(f) = f(A)e,

(B.11)

where we define the operator 1(A) by the formula (B.12)

/

f(A) = (2ir)'/2

dt.

The reason for this notation will become apparent shortly; see (B.21). Note that, using (B.9), we have

(f(A)e, g(A)e) =

(B.13)

(2ir)'

(f

= (2ir)1 // = (2ir)'

f(s)

ds, / (ei(8_t)Ae,

dt) d8dt

f/f

= f, g E S (R), the last "integral" is to be interpreted as (fe,,'). Now, if g = f, the left side of (B.13) is IIf(A)e112, which is 0. Hence,

For

(B.14)

0 for all I E S(R).

This is enough to imply that the tempered distribution z is actually a positive measure! Knowing this, we can now interpret (B.13) as implying that W has a unique continuous extension (B.15)

W: L2(R, d1i) —. H,

294

THE SPECTRAL THEOREM

and W is an isometry. Furthermore, since e is assumed to be cyclic, it is easy to see the range of W must be dense in H, so W must be unitary. Now from (B.12) it follows that, if I E S(R),

=

(B.16) where

(B.17)

co8(r) = et8Tf(r).

Hence

(B.18) Since

(B.19)

W_le28Af(A)e = W1ca8(A)e =

f E S} is dense in H, we obtain

W_lesA = ei8TW_l,

which proves (B.6), and hence Proposition B.1. Note that since W_leitAW = the formula (B. 12) implies (B.20)

W'f(A)W = 1(r),

which justifies the notation f(A) in (B.12). In the language of (B.1), we have (B.21)

f(A)u

=

f

f()t) dEAU.

If a selfadjoint operator A has the representation (B.5), one says A has simple spectrum. It follows from Proposition B.1 that a selfadjoint operator has simple spectrum if and only if it has a cyclic vector. We remark that a more typical method of showing there is a positive measure such that (B.9) holds is to invoke Bochner's theorem, characterizing the Fourier transform of positive measures as positive definite functions. It is well known that, using the theory of tempered distributions, one can give a simple proof of Bochner's theorem. Our proof of the spectral theorem has avoided this detour entirely, working directly with tempered distributions. In general, the spectral theorem is considered to be nonconstructive in nature. However, especially if A has simple spectrum, or spectrum of low multiplicity,

the method of proof of the spectral theorem can sometimes be implemented to give an explicit spectral representation of A. Doing this involves knowing explicitly the unitary group U(t) = and particularly knowing explicitly the function ç(t) = (ettAe, for some cyclic vector and then finding explicitly the Fourier transform of this function, which is known generally to be some positive measure. We can illustrate this program in the case (B.22)

A=D=id/dx,

so

(B.23)

ettDf(x) = f(x — t).

THE SPECTRAL THEOREM

E S (R) such that ç(t) =

If we pick (B.24) where

=

never vanishes, then

= * =

so we have

295

i.e.,

onR.

(B.25)

In this case the unitary map (B.26) W: L2(R,

dr)

L2(R,dx)

is given by

Wf = = /* = eitr.f(r). If we take to be an approximate delta

(B.27) and we have

function, then in the limit W becomes the Fourier transform. Of course, as has been shown several times in this monograph, an explicit knowledge of the unitary group eitA is a very convenient tool for studying the spectral resolution of A. In fact, the spectral measure is given by (B.28)

dEA =

(2ir)'

Lao

dt,

where the right side is interpreted a priori as an operator valued tempered distribution. Compare the spectral analysis of the Laplace operator on spheres and hyperbolic space, deduced from a knowledge of the fundamental solution of the wave equation, in Chapters 4 and 8.

One can generalize the notion of a cyclic vector to the case of a k-tuple of commuting selfadjoint operators (A1,.. . , A,). More precisely, suppose the unitary groups Uj (t) = eutAi all commute. We say E H is cyclic if the closed linear span of U1 (t1) . is all of H. In that case, the proof of Proposition B.! generalizes to show that there is a positive measure p on and a unitary map W: —. H such that W'A1WI(x) = x31(x), WI E V(A3). We would say (A1,.. . , has simple spectrum. If a single operator A = A1 does not have simple spectrum, it would elucidate its spectral behavior to find has simple spectrum. a k-tuple of commuting operators such that (A1,.. . More generally, given a selfadjoint operator B, without simple spectrum, we might try to find (A1,... ,Ak), commuting and with simple spectrum, such that For example, for the Laplace operator = /3(x)I(x), WI E on R", one takes (A1,.. . , = (D1, .. ,Dk), D3 = it9/t9x3. Of course, given such B, one would like such A1,... ,Ak to arise in some "natural" fashion. In many cases, such as B = on the sphere S't, n > 2, it is natural to write B as a function of some selfadjoint operators which commute with B, but not with each other, and which generate a noncommutative group of transformations acting irreducibly on each eigenspace of B. Thus noncommutative harmonic analysis arises naturally in the search for a "simple spectrum." As an application of the spectral theorem, we prove the following result, a version of Schur's lemma, characterizing irreducible group actions, which is used from time to time in this monograph (see Chapter 0, Theorem 1.6). .

.

THE SPECTRAL THEOREM

296

PROPOSITION B.2. A group G of unitary operators on a Hubert space H is irreducible if and only if, for any bounded linear operator A on H, (B.29)

UA = AU for all U E G

A = Al.

PROOF. First, if H0 c H is a closed invariant subspace and P is the orthogonal projection on H0, then (B.29) implies H0 is 0 or H, so clearly (B.29) implies irreducibility. Conversely, suppose G is irreducible, and let A E £(H) be such

that

UA=AU forallUEG.

(B.30)

The unitarity of U implies the same identity also holds for A*, so considering A + A* and A — A*, we can assume without loss of generality that A is selfadjoint in (B.30). Now (B.30) continues to hold with A replaced by any polynomial in A, and then, by the spectral theorem, we can deduce that, if A = fA dEA, then, for any Borel set B, E(B)U = UE(B) for all U E C. But then the irreducibility implies E(B) = 0 or I, since E(B) is a projection on an invariant subspace. Since this holds for any Borel set B, we have A = Al for some A. This completes the proof.

We say a few words about the general justification of the assertion that for any

selfadjoint operator A, the operator iA generates a unitary group (B.4), which is part of Stone's theorem. First, if A is bounded, the operator etA can be defined by a convergent power series expansion, and power series manipulation gives ez8AeztA = e%(8+t)A and (eitA)* = For unbounded A, there are direct demonstrations that iA generates a unitary group, but it is perhaps simpler to use von Neumann's trick, and consider the unitary operator V = (A+i) (A—i)'. Then B1 = (V + V*)/2, B2 = (V — V*)/2i form a pair of commuting bounded selfadjoint operators, to which the spectral theorem proved above applies. Then the spectral theorem for A is a corollary of that for (B1, B2). Of course, granted the spectral theorem for A, the fact that iA generates a unitary group is a simple consequence.

We conclude this appendix by mentioning one important connection between the unitary group and the notion of selfadjointness. A symmetric operator A0 with domain V is said to be essentially selfadjoint if there exists a unique selfadjoint operator A such that V C V(A), the domain of A, and A0u = Au for U E V. For symmetric operators, this extends Proposition 2.2 of Chapter 0. PROPOSITION B.3. Let A0 be an operator on a Hilbert space H, with domain

V. Assume V is dense in H. Let U(t) be a unitary group, with infinitesimal generator iA, 80 A is selfadjoint, and

U(t) =

(B.31)

Suppose P C V(A) and A0u = Au for u E V, or equivalently, (B.32)

lim

t—.o

—

u)

= A0u for all u E V.

THE SPECTRAL THEOREM

297

Also suppose V is invariant under U(t):

c

(B.33)

U(t)V

Then A0 is essentially furthermore that

A is its unique selfadjoint extension. Suppose

(B.34)

A0:V-+V.

Then

V.

with domain V, is essentially selfadjoint, for each positive integer k.

PROOF. We use the following well known criterion for essential selfadjointness

(see, e.g., [202]). A0 is essentially selfadjoint if and only if the range of i ± A0 is dense in H. So suppose v E H and (B.35)

((i±Ao)u,v)=O forallueV.

Using (B.33) together with the fact that A0 = A on V, we have (B.36)

((i ± Ao)u, U(t)v) =

0

for all t

R, u e

V.

Consequently f p(t)U(t)v dt is orthogonal to the range of i ± A0, for any p E an approximate identity, we approximate v by L'(R). Choosing p(t) E elements of V(A), indeed of V(Ak) for all k. Thus we can suppose in (B.35) that V E V(A). Hence, taking adjoints, we have (B.37)

(u, (i ± A)v) =

0

for all u e

V.

Since (i ± A)v E H, and V is dense in H, it follows that (B.37) holds for all u E H, hence for u = v. But then the imaginary part is IIvD2, so (B.35) implies v = 0. This proves the first part of the proposition. Granted (B.34), the same proof works with A0 replaced by (but U(t) unaltered), so the proposition is proved. An example of an application of Proposition B.3 is Chernoff's proof [38] that

all powers of the on a complete Riemannian manifold M, with V = Laplace operator are essentially selfadjoint on V. This is because, with V(B0)=C000eC000,

(B.38)

the group (B.39)

U(t) =

etB0

is the fundamental solution to the wave equation (B.40)

82U/0t2

—

=0,

and its unitarity is equivalent to the standard energy conservation law for the wave equation, while finite propagation speed for solutions to the wave equation implies V(B0) is preserved by U(t). Note that (B.41) so

essential selfadjointness of .A and its powers follows.

APPENDIX C

The Radon Transform on Euclidean Space In this appendix we give a brief discussion of the Radon transform on Euclidean space defined as follows. If H is any (n — 1)-dimensional linear subspace of w E Rn/H, set H)

(C.1)

L f(y + w) dy,

= where H gets induced Lebesgue measure. This Radon transform is useful in commutative harmonic analysis, and also played a role in Chapter 6, in deriving the Plancherel formula on a nilpotent Lie group. Other variants of the Radon transform include transforms on spheres (see Chapter 4) and also on hyperbolic space and more general symmetric spaces (see Helgason [102, 103]). Our goal will be to obtain a Plancherel formula for (C.1), since that was used in Chapter 6. This will be a simple consequence of the Plancherel formula for the Fourier transform. In fact, if is a unit vector orthogonal to H, note that (C.2)

f(sw) =

provided Rn/H is made isomorphic to R via w + H formula Il/ 1IL2 = II!IIL2 on

(C.3)

Of

=

1.

Now the Plancherel

gives

2

f

Meanwhile, the Plancherel formula for the Fourier transform on R gives, in view of (C.2), (C.4)

J/(sw)12 ds = (21r)_(n_1)12

J

H)12 dw.

In order to combine (C.3) and (C.4), note that (C.5)

f

00

00

ds = f 298

ds,

THE RADON TRANSFORM ON EUCLIDEAN SPACE so

299

we get

(C.6)

Ill

=

H) 12 dw d vol(H),

ff00

2

as our Plancherel formula for the Radon transform. Here d vol(H) is the invariant volume element on the Grassmannian manifold of hyperplanes in R", normalized so that vol(,g) = We can polarize (C.6) and get, for f,g E

(1,

(C.7)

=ff =ff Q

ft

I

iRg(w, H) dw dvol(H)

H)

H)dw dvol(H).

Note that but if n is even,

near a point p e

=

nodd

(C.8)

is not a local operator. Taking g to be highly concentrated we see that

H) =

(C.9)

.p

—

w)

where WH is a unit normal to H, determining an isomorphism Rn/H we get the inversion formula (C.1o)

1(P) =

cnf

R. Thus

.p,H)dvol(H).

The Radon transform provides a nice tool for proving the Huygens principle for constant coefficient hyperbolic systems when n is odd; this arises from the fact that (C.8) is a local operator in that case. It also is a tool for scattering theory. See Lax and Phillips [150] for more about these uses of the Radon transform.

APPENDIX D

Analytic Vectors, and Exponentiation of Lie Algebra Representations We discuss briefly the role of analytic vectors in passing from representations of a Lie algebra to representations of a Lie group. Let X1, . , be a set of (unbounded) skew adjoint opertors on a Hubert space H. We say u E H is an analytic vector for (X1,... ,Xk) if u E for and, for some C Psi.

We get (D.27)

= (2/ir)'/2 f

f(s) cos

ds,

if dist(p, x) > r.

ANALYTIC VECTORS

304

satisfies the following estimates:

Now, suppose (D.28)

for some çOEL'.

If we set

(D.29)

=

then

f

(D.30)

f

ds

p(s) ds,

Ci(k/A)klI,(r).

Since we have the elementary L2 operator norm bound cos easily deduce the following. Let dist(p, q) = r + 2a. Then

1, we

C((2k +

(D.31)

provided u E L2 has support in the ball Ba (p) of radius a, centered at p. Thus, for such u, and for A/2, the power series v(x, y) = (D.32)

=

(formally)

converges to an element of L2(Ba(X) x [—A/2, A/2]). Moreover,

+ 82/8y2)v =

(D.33)

0,

and, by (D.31), (D.34)

lvii L2(Ba(X)) x f—A/2,A/2J CiP(r) hull L2(Ba(p))

Since C is a homogeneous Riemannian manifold, we can apply regularity estimates for solutions to the elliptic equation (D.33), uniformly, to get 1/2

(D.35)

f

Ba(p)

if

dp'

(Most of the effort in [36] was devoted to the study of nonhomogeneous manifolds, with semibounded Ricci tensor, where this last piece of reasoning must be replaced by very much more elaborate arguments.) Similarly we obtain (D.36)

(x) hiL2 (Ba(p))

so for each x we can set (D.37)

w(x,p',y) =

and get a harmonic function of (p', y) on Ba(p) x [—A/2, A/2] whose L2 norm

satisfies (D.38)

hiw(x,, )hiL2(Ba(p)XE_A/2,A/21)