Dynamical Systems and Semisimple Groups: An Introduction [1 ed.] 0521142164, 9780521142168

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126 DYNAMICAL SYSTEMS AND SEMISIMPLE GROUPS AN INTRODUCTION RENATO PERES

126

Dynamical Systems and Semisimple Groups

The theory of dynamical systems can be described as the study of the global propertiesof groups of transformations.The historical roots of the subject lie in celestialand statisticalmechanics,for which the group is the time parameter. In some of its recent developments,the theory is concernedwith the dynamics of more general, bigger groups than the additivegroup of real numbers,particularly semisimpleLie groups and their discrete subgroups. Some of the most fundamentaldiscoveriesin this area are due to the work of G. A. Margulis and R. Zimmer. This book comprisesa systematic, self-containedintroductionto the Margulis-Zimmertheory and provides an entry into current research. Takingas prerequisitesonly the standardfirst-yeargraduatecoursesin mathematics,the authordevelopsin a detailedand self-containedway the main results on Lie groups, Lie algebras, and semisimple groups, including basic facts normallycoveredin first courses on manifoldsand Lie groups plus topics such as integrationof infinitesimalactions of Lie groups. He then derives the basic structuretheorems for the real semisimpleLie groups, such as the Cartan and Iwasawadecompositions,and gives an extensiveexpositionof the generalfacts and conceptsfrom topologicaldynamicsand ergodictheory,includingdetailed proofs of the multiplicativeergodic theorem and Moore's ergodicitytheorem. This book should appeal to anyone interested in Lie theory, differential geometry,and dynamicalsystems.

CAMBRIDGE TRACTS IN MATHEMATICS

GeneralEditors B. BOLLOBAS, F. KIRWAN, P. SARNAK, C. T. C. WALL

126

Dynamical Systems and Semisimple Groups

REN A TO PERES WashingtonUniversityin St. Louis

Dynamical Systems and Semisimple Groups: An Introduction

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, ¥elboume, Madrid, Cape Town, Singapore, Sio Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521142168 0 Cambridge University Press 1998 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1998 This digitally printed version 2010

A catalogue record/or this publication is available from the British Library ISBN 978-0-521-59162-l Hardback ISBN 978-0-521-14216-8 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

-,

To my parents

Contents

page xiii

Preface 1

1

TopologicalDynamics

I

G-Spaces 1.2 The Orbit Space 1.3 Suspensions 1.4 Dynamical Invariants 1.5 Anosov Diffeomorphisms

1.1

2

Review of Measure Theory Recurrence Ergodicity Measure-Theoretic Entropy

Smooth Actions and Lie Theory

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4

17 17 22 26 29

ErgodicTheory - Part I

2.1 2.2 2.3 2.4 3

3 8 9 12

Vector Fields and Flows Foliations Lie Groups and Lie Algebras Infinitesimal and Local Actions The Exponential Map Lie Subgroups and Homogeneous Spaces Continuous Homomorphisms Are Smooth Representations of Compact Groups Integrating Infinitesimal Actions

AlgebraicActions

4.1 Affine Varieties 4.2 Projective Varieties 4.3 The Descending Chain Condition ix

32 32 38 41 43 45 46 49 50 52 59 59 61 63

Contents

X

4.4 4.5 4.6 4.7 4.8 4.9 4.10

s

Functionsand Morphisms NonsingularPoints Real Points of ComplexVarieties The OrdinaryTopologyof a Variety HomogeneousSpaces Rosenlicht'sTheorem The Rough Structureof AlgebraicGroups

64

The Classical Groups 5.1 GL(n, R), GL(n, C), and GL(n, lllI) 5.2 AutomorphismGroups of Bilinear Forms 5.3 The Lie Algebras

77

65 67 67 69 71

74 77 80 86

6.1 6.2 6.3 6.4 6.5 6.6 6.7

GeometricStructures PrincipalBundles GeometricA-Structures InvariantGeometricStructures The ReductionLemmas The AlgebraicHull Applicationsto Simple Lie Groups Measuresas GeometricStructures

91 95 96 100 107 112

7

SemisimpleLie Groups 7.1 Reductiveand SemisimpleLie Algebras 7.2 The Adjoint Group 7.3 The Cartan Decomposition 7.4 The RestrictedRoot Space Decomposition 7.5 Root Spaces for the Classical Groups 7.6 The IwasawaDecomposition 7.7 Representationsof sl(2, C) 7.8 The WeylGroup 7.9 Generationby Centralizers

119 119 122 125 129 132 136 139 144 146

8

ErgodicTheory - Part n 8.1 InvariantMeasureson Coset Spaces 8.2 Ergodicityand Unitary Representations 8.3 Moore's ErgodicityTheorem 8.4 Birkhoff's ErgodicTheorem 8.5 Ergodicityof Anosov Systems 8.6 Amenabilityand Kazhdan's Property T

149 149 154 156 162 165 167

6

88 88

Contents

9 Oseledec's Theorem 9.1 Lyapunov Exponents 9.2 Products of Triangular Matrices 9.3 Proof of Oseledec's Theorem 9.4 Nonuniform Hyperbolicity and Entropy

10 Rigidity Theorems 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

Straightening Sections H-Pairs H-Pairs for Centralizers The C' Rigidity Theorem Contracting Subbundles The Theorems of Zimmer and Margulis The Lyapunov Spectrum and Entropy Lattice Actions

Appendix A: Lattices in SL(n, JR) A.I A.2

SL(n, Z) Cocompact Lattices

References Index

xi 174 174 179 187 192 196 196 199 206 212 215 221 224 226 231 231 236 241 243

Preface

An action of a group G on a set Mis a homomorphism, g ~ 8 , from G into the group of invertible transfonnations of M. Traditionally, actions of R and Z have been the main objects of concern of the theory of dynamical systems. For example, if X is a smooth vector field on a compact manifold M, its flow defines an action of R on M. One thinks of the group as parametrizing "time," so that the orbit

G.

X

:= {g(X)I g E G}

describes the evolution of the system, starting from an initial state represented byx EM. For actions of a more general group G, the "time evolutions" associated to its various one-parameter subgroups are "interlocked" according to the algebraic structure of G. Thus, suppose that instead of a single smooth vector field on M we have a family of them, X 1, ••• , Xm, whose Lie brackets satisfy m

[X;, Xi]=

'Eatxk, k=I

afi

where the are constants. This means that these fields span a finite-dimensional Lie algebra, associated to a Lie group G. By Lie's second fundamental theorem (essentially theorem 3.9.8), these fields integrate to an action of (the universal covering group of) G. If, for example, the constants vanish, G is an abelian group, and the flows of X 1, .•• , Xm may be thought to define "noninteracting" evolutions. On the opposite extreme to abelian groups are the semisimple groups. A not very revealing - definition of semisimplicity is that the matrix (c;j), with entries given by

at

m

m

c;i= 'E'Eataii' k=I l=l

xiii

xiv

Preface

is nonsingular. The various subgroups of a semisimple group are tightly interwoven, and one might expect the dynamical properties of actions of G to be accordingly constrained. The actions considered in this book will, for the most part, be assumed to possess a finite invariant measure. Invariance of a measure µ, means that for each measurable subset A c M we have µ,( µ,(A) for each g e G. 8 (A)) The existence of a finite invariant measure forces upon the long-term evolution of the system a kind of statistical regularity, given by the ergodic theorems of chapters 8 and 9, most of which are for actions of JRor Z. This is the basic setup of ergodic theory. Actions ofR. and Z very commonly admit (possibly singular) invariant measures - this is always the case if the action is continuous and M is compact. For nonabelian groups, however, this is a strong requirement. One can obtain much useful information about a group action on M (assuming that the action is differentiable) by studying its linearization along the orbits. A good understanding of this linearization can yield information about the global properties of the system. (For Z-actions, a modest attempt to justify this claim will be made at the end of chapter 9, in a brief discussion on Pesin theory.) One of the main results of the book can be formulated as follows: If the Z-action is part of a differentiable action by a noncompact semisimple Lie group G on an n-dimensional manifold M, preserving a finite measure µ,, then it is possible to give a very precise description of the linearization, along almost every orbit (relative to µ,), in terms of the representations of G in dimension n. The key result behind this vague assertion is Zimmer's cocycle superrigidity theorem, which will be studied in chapter 10. The ergodic theory of actions of semisimple Lie groups and their discrete subgroups has grown in the past decade into a large and very active chapter of the general theory of dynamical systems. It is also a subject with deep foundations; most notably, the work by G. A. Margulis concerning rigidity and arithmeticity of lattices in semisimple groups and the work of R. Zimmer, in particular his cocycle superrigidity theorem. The main purpose of this book is to serve as a relatively gentle introduction to the rigidity theorems of Margulis and Zimmer. Passing from some knowledge of the linearization of the system along orbits to an understanding of its global topological structure is a hard and wide-open problem. R. Zimmer conjectured in [37] that the actions of Lie groups and lattices considered in this book should be, in some sense, "classifiable" on the basis of a few classes of well-understood examples. This classification program gained momentum with the introduction of ideas from hyperbolic dynamics and the theory of Anosov diffeomorphisms, by S. Hurder, A. Katok, and J. Lewis, about six or seven years ago, and continues today with great vitality. In spite of

=

Preface

xv

all the recent progress, it is clear that this is not an area of research approaching exhaustion any time soon. Although I make no attempt to survey the current research activity, the book does contain a few new ideas. The proof of the cocycle superrigidity theorem given in chapter 10 is new, and is due to F. Labourie and myself [11). The presentation is also different from that in [36] in that it uses throughout a differential-geometric language that some readers may find more natural, or at least more congenial, than the language of cocycles over a group action. I hope that the experts will see some novelty here. In any event, the book was written having in mind primarily the nonexpert, especially the graduate student interested in Lie theory and dynamical systems. The reader is assumed to have a good working knowledge of measure and integration and the basic theory of differentiable manifolds. Even though first courses on manifolds usually provide some acquaintance with Lie groups and Lie algebras, I have chosen to develop this subject in detail and from the beginning, up to those results that are needed for chapters 8, 9, and 10. The text, in effect, integrates two courses in one. First, it contains an introduction to part of the modem theory of dynamical systems. This comprises chapters I, 2, 8, and 9. The dynamics "subcourse" culminates with a detailed proof of the multiplicative ergodic theorem of Oseledec and a hurried discussion of Pesin theory. Taken in isolation, this is necessarily a lopsided account of dynamics. For an even-handed and thorough introduction to dynamical systems, the reader cannot presently do better than to go to [16]. The second "subcourse" is on Lie groups, Lie algebras, and semisimple groups, and comprises chapters 3, 5, and 7. The reader who wishes to pursue this topic further will have no difficulty in continuing on with, say [17], from the point where chapter 7 ends. There is a large degree of independence among chapters I through 9, although almost everything that is developed in them is put to use in chapter I 0. The main exceptions are chapter 6, which relies on facts about algebraic actions discussed in chapter4; section 9.3, which uses some of the language introduced in section 6.1; and section 8.3 on Moore's ergodicity theorem, which uses some of the structure theorems for semisimple Lie groups. Chapter 4 is independent of the first three, although it is also the least self-contained in the book. It consists of a very pedestrian introduction to algebraic geometry and algebraic actions. The shortest path to arrive at the main results of the book, namely, theorems 10.4.1, 10.6.1, and 10.6.2, is to read chapters 4, 6, and 7, sections 8.2 and 8.3, and the first sections of 10, referring to the earlier chapters for the definition of some occasional unfamiliar term. All the other chapters are there to provide a dynamical systems "context" for the results of chapter I 0.

xvi

Preface

It seemed to me appropriate from the point of view of the "narrative" to place chapter 6, on geometric structures, immediately after a discussion on the classical groups, even though the results from that chapter are not needed until chapter 10. The reader should keep this in mind in case the discussion in chapter 6 seems at first too formal and unmotivated. The exercises are an integral part of the text. Their main purpose is to expand or illustrate an idea under discussion. Occasionally, an exercise may also be referred to in proofs. They should always be read, even if not always worked on. The book began as a short series of lectures given at Penn State University in 1996, and was later expanded after a one-semester course I taught at Washington University, in St. Louis, in 1997. I am grateful to Anatoly Katok for inviting me to give the lectures at Penn State and for bringing my notes to the attention of Cambridge University Press. Had I pursued the subject just a little further the impact of his own work would also become apparent in the text. I am also especially indebted to Robert Zimmer, from whom I learned much of this subject firsthand; to Fran~ois Labourie, my c~llaborator in [11], the reference on which much of chapter 10 is based; to Scot Adams, for many enlightening conversations and, in particular, for explaining to me the proof of Moore's ergodicity theorem given in chapter 8; and to Mohan Kumar, Mark de Cataldo, and Vladimir Masek, for lending their expertise on algebraic geometry. Finally, I would like to thank Peter Lampe, Michelle Penner, Holly Lowy, Lawrence Roberts, Mark de Cataldo, and Meeyoung Kim for suggesting many improvements and corrections to the original manuscript. Thanks to them, many - but certainly not all - embarrassing mistakes, obscure phrases, wrong signs, and barbarisms of the first draft will not appear in print.

RenatoFeres Washington University, St. Louis August, 1997

1 Topological Dynamics

The theory of dynamical systems, loosely speaking, studies those properties of group actions that are asymptotic in nature, that is, that become apparent as we "go to infinity" in the group. We call a set X equipped with an action of a group Ga dynamical system with group G or, alternatively, a G-space. After introducing a few notions that apply to general group actions, we focus our attention on some of the basic properties of topological G-spaces. Other aspects of dynamical systems, relating, for example, to their measurable or smooth properties, will be discussed in later chapters.

1.1 G-Spaces Let G be a group and X a set. A G-action on X is a map ct>: G x X satisfies the following two properties:

~

X that

1. (e,x) = x for all x e X, where e is the identity of G. 2. cl>(g2,(g1,x)) = (g2g1,x) for all 81, 82 E G and x e X.

For each g e G, let 4>8 : X ~ X be defined by ct> 8 is 8 (x) := (g,x). Then 4> a bijection from X onto itself, with inverse ct> 8 from 8 -1, and the map g ~ ct> G into the group of bijective self-maps of X is a group homomorphism. We often write g · x or g(x), or simply gx, instead of (g,x). The definition of G-action just given is usually called a left action of G. By a right action of G on M we mean a map ct>: M x G ~ M such that property 2 is replaced with

For each x e X, we define the orbit of x by Gx := { 8 (x)

I

Ig

e G}.

2

1

Topological Dynamics

The orbits of a G-action partition X into disjoint sets; namely, the Gx are the equivalence classes of the relation x "' y if and only if there exists g e G such that x

= gy.

The orbit space is the set of equivalence classes, denoted G \ X. The action is called transitive if the G-space has only one orbit, that is, X = G x for some x. Typically, the G-action will leave invariant, or preserve, some structure on X such as a topology, a measurable structure, a smooth manifold structure, or an algebraic variety structure. Of course, these structures are not independent. For example, when studying a smooth group action on a compact Riemannian manifold whose volume fonn is invariant under the action, one could find it useful at times to focus, say, on the underlying measure-space structure determined by the Borel-measurable sets and the measure obtained by integrating the volume fonn. The group G, however, will always be regarded here as being, at least, a topological group, that is, a Hausdorff space that is also an abstract group for which multiplication and inversion are defined by continuous maps. A topological G-space consists of a topological space X and a continuous acof G on X. In this case each tion 8 , g e G, is a homeomorphism of X. Some of the basic properties of topological G-spaces are discussed in this chapter. A smooth G-space consists of a smooth (C 00 ) manifold X and a smooth 00 diffeomorphism of action of a Lie group G. In this case, each 8 is a C X. We discuss smooth actions in chapter 3. A measurable G-space consists of a measurable space (X, B), where Bis a u -algebra of subsets of X, and a measurable action : G x X --+ X. We will often be interested in actions that preserve a finite measure µ, on ( X, B). In that case,µ, can be nonnalized so that µ,(X) = 1, andµ, is then called a probability measure. The study of group actions on measurable spaces is the subject of ergodic theory, to which we return later, beginning in the next chapter. An algebraic G-space consists of an algebraic variety X defined over some field k (which, in this book, will almost always be JRor C) and an algebraic group G and an algebraic action ,both defined over k. Algebraic actions will play an important role in some of the results described in this book. Definitions and general properties concerning them are discussed in chapter 4. The remainder of the chapter concentrates on the elementary properties of topological dynamical systems, that is, topological G-spaces. Let H be a closed subgroup of G. Then the coset space

GIH = {gH I g

E G}

has the quotient topology induced by the natural projection rr : G --+ G/ H, rr(g) = gH; namely, the open subsets of G/ Hare rr(U) = {gH I g e U} for

1.2

3

The Orbit Space

all open sets U C G. With respect to the quotient topology, 1r is continuous and open and G/ H is a Hausdorff space. A discrete subgroup of G is a subgroupthat is a discrete subset in the topology of G. If G is connected and locally arcwise connected and H is a closed subgroup, it can be shown that H is a discrete subgroup if and only if 1r is a coveringmap. If H is discrete and G / H is compact, we say that H is a uniform lattice of G. The kernel of an action ,denoted Ker(),is the kernel of the homomorphism g 1-+ is trivial, the 8 , which is a normal subgroup of G. When Ker() action is said to be effective. If the action is not effective, induces an effective action of G /Ker( ) on X. The action is called locally effective if Ker()is a discrete subgroup of G. For each x e X, the isotropy group of x is defined by Gx := {g E G I gx = x}.

=

Gx is a subgroup of G and it is immediate that Ggx gGxg- 1 for each g E G and X E X. Moreover, Ker()= nxeX Gx. If Gx = {e}for all X E G, we say that the G-action isfree. The actiori is called locally free if Gx is a discrete subgroup of G for all x in X. Recall that a topologicalspace X is said to be T 1 if each point x e X is closed. It is an easy consequence of the definitions that, whenever X is a T 1 G-space, each isotropygroup Gx as well as the kernel of are closed subgroupsof G and that G /Ker( )is a topological group in a natural way. Moreover,the induced (effective)action of G/Ker()makes X a topological G/Ker()-space.

1.2 The Orbit Space

Until we impose any further requirements G will be a locally compact second countable topologicalgroup and X a complete second countable metrizable Gspace. We give G\X the quotient topology induced by the natural projection that to each x e X associates its orbit. It will be apparent from some of the examples described later that the orbit space G\X can easily fail to have good separation properties, due to the existence of orbits that wander about in X in a complicated way. This is not the case, however,when is a proper action. By definition, is a proper action if for each x, y e X there exist neighborhoods U of x and V of y such that {g e G

I v n gU =F0}

is relatively compact. Clearly, the action is proper whenever G is compact.

4

1 Topological Dynamics

Exercise 1.2.J Show that the orbit space G\X of a proper action is Hausdorff. In particular, each orbit Gx is closed in X. Also show that, for each x e X, the map Gx, gGx 1-+ gx, is a homeomorphism.

'Px:G/G,,-+

A somewhat more complicated situation, but still rather simple from the viewpoint of the general theory of dynamical systems, corresponds to the case in which the u-algebra B of Borel sets, that is, the u-algebra generated by the open sets in G\X, is countably separating. This means that there is a sequence B; e B such that for each pair of points in X one can find a B; that contains exactly one of the two points. In this case, the G-action will be called tame. Notice that a proper action is tame. In fact, the quotient topology of G\X is second countable, since X is second countable, and Hausdorff, so points can already be separated by open sets. The next theorem gives a useful characterization of tame actions. It is taken from [36), where a tame action is called smooth. The result is due to Glimm and Effros. The orbit Gx of a topological G-space Xis locally closed if it is open in its closure Gx c X.

Theorem 1.2.2 Suppose that is a continuous action of a locally compact second countable group G on a complete second countable metrizable space X. Then the following are equivalent:

1. All orbits are locally closed. 2. The action is tame. 3. For every x e X, the natural map G/Gx -+ Gx is a homeomorphism, where Gx has the relative topology as a subset of X. Proof. The implication 2 =} 1 is the hardest to prove and will not be discussed here. We refer the reader to [36, 2.1.14] for a proof. We begin with the assertion 1 =} 2. Since the topology of X has a countable basis and the projection 1r : X -+ G\X is open, the topology of G\X also has a countable basis. To prove that the Borel-measurable structure is countably separating it suffices to show that G\X is a T 0 -space, that is, that we can separate any two. points by an open set that contains only one of the points. Let x, y e X. If 1r(x) and 1r(x) are not separated by an open set, Gy C Gx and Gx C Gy. Therefore Gy is dense in Gx. But by assumption Gx is open in its closure, so Gy n Gx =/:0. This implies that 1r(x) = 1r(y). We now show that 3 and 1 are equivalent. We may assume without loss of generality that Gx is dense in X. If this is not the case, simply let X denote the closure of that orbit. We begin with 3 =} 1 and assume that G / Gx -+ Gx is a homeomorphism. Then Gx with the subspace topology satisfies the Baire

1.2

The Orbit Space

5

category theorem, because G / G x satisfies it. (G is locally compact, hence a Baire space. It follows that the quotient is also a Baire space.) Now, G is a-compact, being second countable and locally compact. Therefore, by Baire's theorem, some compact set A c Gx contains a nonempty open set, that is, for some nonempty open set Uc X, Uc Un Gx c A. Thus Gx = GU, which is open. For the converse, suppose that Gx is open in X. Notice that G/GJC -. Gx is continuous, so it suffices to prove that it is also open. We call U C G a symmetric set if g e U implies g- 1 e U. Any open neighborhood V of e contains a symmetric neighborhood: V n v- 1• We claim that it suffices to show that for any compact symmetric set U c G whose interior is an open neighborhood of e e G, U x contains a nonempty open set. Namely, let N be any neighborhood of e and choose a compact symmetric U with U 2 c N. If Ux contains a neighborhood of some ux, u e U, then u- 1ux contains a neighborhood of x, and hence so does Nx. Therefore G/GJC -. Gx is an open map. To show that U x contains an open set, choose a countable dense g;Ux, a union of compact sets, so by the Baire set {g;} c G. Then Gx = LJ; category theorem, we have that one g; Ux contains an open set, and hence so D does Ux. Exercise 1.2.3 Show that the R-action on R 2 defined by ,(x,y) := (e' x, e-'y), where t e R and (x, y) e R 2, is tame but not proper. Verify that the orbits are locally closed, but the orbit space is not Hausdorff.

The following definitions are concerned with different orbit types. An element x in a G-space X is said to be a fixed point if G JC = G. It is a periodic point if G / GJC is compact. A (topological) G-space X is said to be topologically transitive if some G-orbit is dense in X. If all orbits are dense, the action is called minimal. A subset A c X is called G-invariant if for each x e A and g e G, g x e A. An equivalent definition of minimal action is that X does not have a proper closed G-invariant set, since the closure of a G-invariant set is a G-invariant set. A point x of a topological G-space X will be called recurrent if for each neighborhood U of x and each compact K c G, there is g in the complement of K such that gx e U. It is immediate from the definitions that periodic points are recurrent. Furthermore, if both the orbit of x and its complement are dense in X, then x is a recurrent point. We leave the verification of this last claim as an exercise to the reader. Notice that the action of G on itself by translations is topologically transitive - in fact, transitive - but not recurrent. The preceding notions can all be illustrated with actions defined on the ntorus. Let 'll'" = R" /Zn denote then-dimensional torus, defined as the quotient

1 TopologicalDynamics

6

of the abelian group ]Rn by its integer lattice subgroup zn. The element x = v + zn will also be denoted[v]. 'fn can alternativelybe describedas the product of n copies of the circle S 1 = {z e C I lzl = l }; namely,

(a,, ...

r,n , an) + 1u

i-+-

( e2,ria1 , ...

, e2,ria.)

is a diffeomorphismbetween'fn and S 1 x · · · x S 1• JR.nacts transitivelyon 'fn via the smoothaction : ]Rn x 'fn -+ :(u, [v])

i-+-

[u

'fn

such that

+ v].

For each u = (u1, .•. , Un) e JR.ndefine the translation t'u = (u,·). Then t'u generatesa Z-action on -rnby (m, [v]) 1-+ t':'([v]). If the componentsof u are rational numbers, the orbit of each x is periodic and all orbits are finite with same cardinality,as one can easily check. Real numbersx 1, ••• , Xs are called rationallyindependentif given integers k; such that E:=Ik;x; = 0, then k; = 0 for all i.

Proposition1.2.4 Fix a vector u = (u1, ... , Un) e ]Rn and consider the Zaction on 'fn generatedby t'u. Then the followingstatementsare equivalent: 1. The action is topologicallytransitive. 2. The action is minimal. 3. The numbers 1, u 1, ••• , Un are rationallyindependent.

Proof. The proof is taken from [16]. It is clear that 2 =>1. On the other hand, if some x e 'fn has a dense Z-orbit, then all points have dense orbits since we can get from x to any other point by a translationand all translationscommute with the Z-action. Therefore, 1 and 2 are equivalent. We now show I =>3. Noticethat if 1, u 1, ••• , Un are not rationallyindependent, we can find integers k;, not all 0, such that I:7= 1 k;u; = ko. Therefore, the function q,(v) := sin(21r tk;v;)

r=I

is a continuousZ-invariantfunction on 'fn, that is, q, o t":'= q, for all m e Z. But q, is not a constant function, so there exists c e JR such that the sets U q,- 1({t e JR.It > c}) and V q,- 1({t e JR.It < c}) are nonempty and disjoint. Furthermore, U and V are invariant sets since q, is Z-invariant. It followsthat the action cannot be topologicallytransitive.

=

=

7

1.2 The Orbit Space

If the action is not topologicallytransitive, there is a nonempty Z-invariant open set U such that [/ # 'll'n.In fact, if no such U exists, one obtains a dense orbit as follows. Let U 1, U2 , ••• be a countablebase of open setsfor the topology of 'll'n.By assumption,there exists an integer N1 such that -c:• (U1) n U2 # 0. Let Vibe a nonemptyopen set such that V1 c U1n -c;;N•(U2 ). There exists an integer N2 such that 2 (Vi) n U3 # 0. Again, take an open set V2 such that V2 C Vi n "'u-N 2 (U3). By induction, we construct a nested sequence of open Vn = sets Vnsuch that Vn+IC Vnn -c;;N•+l(Un+2).The intersectionV = 1 (x) Vnis nonemptysince the Vn are compact. If X V, then Un for each n E N. This shows that the orbit of any x E V intersects each open set in a basis for the topology of 'Jl'R.Therefore,the Z-orbit of x is dense. The previous claim can now be used to show 3 =>1. Thus, suppose that the action is not topologicallytransitive,which implies by the claim that there exists an open nonempty Z-invariantset U whose closure is not 'Jl'R.Let x be the characteristicfunctionof U. In what follows,we think of x as a functionon !Rnthat is periodic in each variable. Since U is invariant,we have x o "Cu = x. Take the Fourier expansion

-c:

n:1

E

n:1 -c:•- E

_ ~ 21Cik•X X (Xt, ••• , Xn) - L.J Cke , keZ•

where k · x denotes the ordinary dot product k · x x(-cu(x))

= k1x1 + · · · + knXn. Then

= L Cke27Cik·ue27Cik•x. keZ•

Invarianceof x and uniquenessof the Fourier expansionimply ck = cke2"ik·u for each k E zn. If Ck = 0 for all nonzero k E zn, it would follow that X is constant almost everywhere with respect to the Lebesgue measure on !Rn. Therefore,the measureof either U or its complementwould be 0, which is not the case. Therefore, for some nonzero k E zn, ck # 0, whence e2"ik·u = 1. 0 This shows that the numbers 1, u 1, •.• , u n are rationally dependent. Exercise 1.2.5 Show that if Xis a locally compact second countable space and :G x X --,. X is a topological action, then is topologically transitive if and only if any two nonempty open G-invariant sets intersect. (The argument is essentially contained in the preceding proof.) Also show that if Xis a Baire space (e.g., locally compact) and the action is topologically transitive, then the set of points with a dense orbit is a dense G ,-set, that is, it is a countable intersection of open dense sets.

Orbits of differentpoints of a G-space can be very different,as the next example will show. Let SL(n, '71,)be the group of n-by-n matricesof determinant

1 TopologicalDynamics

8

1 with integer entries. Since the linear action of SL (n, Z) on Rn leaves invariant we obtain an action on the torus the integer lattice

zn,

cl>:SL(n, Z) x 'Ir" -+ Tn

by setting cl>(A,[v]) := [Av], where Av denotes matrix multiplication of A and v e Rn, the latter now viewed as a column vector. The next exercise shows that cl>is topologically transitive but not minimal. Exercise1.2.6 Show that each point [u,, ... , u2 ] e 'IL"'with rational componentsis a periodicpoint for the above actionof SL(n, Z) on 'IL"'.Using the argumentemployedin the last proposition,show that the action is topologicallytransitive.The same l!fgllment 2 generatedby the single matrix ( ~ : ) is also can be used to showthat the Z-actionon 11' topologicallytransitive.

The example introduced in the previous exercise can be generalized as follows. Let G be a topological group and r a discrete subgroup of G such that the quotient X = G/ r is compact or, more generally, such that X admits a G-invariant probability measure. (Invariant measures will be defined and discussed in detail in a later chapter.) Let H be a closed noncompact subgroup of G and define an H-action on X by c1>(h, gr) := hgr.

It will follow from results of chapter 8 that if G = SL(n, JR.)(or any other connected, noncompact, simple Lie group) then for any noncompact closed subgroup H of G, the H -action on X is topologically transitive.

1.3 Suspensions

Starting with an action cl>: r x X -+ X, where r is a discrete subgroup of a connected topological group G, it is possible to define in a canonical way a locally free action of G that "looks transverselylike" cl>,called the suspension of cl>,or the inducedaction from cl>.It is defined as follows. First notice that r acts diagonally on the product G x X by y • (g, x)

= (gy-

1,

cl>(y,x)).

We denote the orbit space by S = (G x X) / r and the element of S represented by (g, x) will be written [g, x]. S is the total space of a fiber bundle p : S -+ G / r, (g, x) r 1-+ gr (see the beginning of chapter 6 for the definition of a fiber bundle), whose fibers are

1.4 Dynamical Invariants

9

homeomorphic to X. In fact, let 1r: G -+- G/ r be the natural projection (a covering map) and, for each z e G/ r, let U be a sufficiently small connected neighborhood of z such that 1r- 1 ( U) is a disjoint union of open sets in G, each homeomorphic to U via 1r. Choose one component of the preimage, say Uo, and set u := (1rlu0 1 : U-+- Uo, Then

r

(z,x)~

[u(z),x]

is a homeomorphism between U x X and p- 1(U), showing local triviality. Notice that by making a different choice of connected component of 1r - 1 ( U), say U 1 = U0 y, then the change of trivialization is given by the map from U x X onto itself that sends (z,x) to (z, (y,x)) for some ye r independent of z. The group G acts on S by 4>(h, [g, x]) := [hg, x].

Notice that [hg, x] = [hgy- 1, (y,x)], so the action is indeed well defined and is clearly continuous. For example, let G = JR and r = Z, and consider the Z-action on Tn generated by a diffeomorphism t of the n-torus. Then G / r = JR/Z = T 1, so the suspension of the Z-action is an JR-action on Tn+1• Similarly, one obtains an SL(n, JR)-action on a fiber-bundle with typical fiber Tn over SL(n, JR)/SL(n, Z), by suspending the SL(n, Z)-actionon Tn defined earlier. Exercise1.3.J Show that the r -action on X is topologically transitive if and only if the suspension cf>is topologically transitive.

1.4 Dynamical Invariants 1\vo topological G-spaces X and Y are said to be (topologically) equivalent if there exists a homeomorphism F : X -+- Y that intertwines the respective Gactions. More precisely, F(gx) = gF(y) for each x e X, ye Y, and g e G. Equivalent G-spaces have the same (topological) dynamical properties; for example, the G-action on X is topologically transitive if and only if the action on Y is, and periodic orbits of X correspond under F to periodic orbits of Y. Any attempt to classify topological G-spaces in terms of their global dynamical properties immediately calls for characteristic "quantities" that have the potential to distinguish inequivalent systems. An analogy can be made with linear maps of vector spaces. Linear maps 1i : V; -+- V;, i = 1, 2, are equivalent if there exists a linear isomorphism F : V1 -+- V2 such that FT1 = T2 F.

1 TopologicalDynamics

10

Equivalentlinear maps must have the same spectrum of eigenvalues, so the spectrumis an "invariant"in this case. (There is no need to define the concept of an invariantfonnally, since we will be concerned only with some specific examplesof them.) An interestingexampleof an invariantfor G-spacescan be definedas follows. Let Hq(X;, R) denote the real qth singular homology spaces of G-spaces X;, i = l, 2. (For the basic definitionsin algebraictopologysee, for example,[14]. This example,however,will not be needed later.) Then G acts in a natural way on Hq(X;, R) by vector space isomorphisms. If the actions on X 1 and X 2 are equivalent,via a homeomorphismF: X 1 -+ X2, then the linear actionsof G on Hq(X;, R) are linearly equivalentvia the map induced by F on the homology spaces,which we denote Hq(F). Therefore,the linear invariantsof Hq(/;) are also invariantsof the G-spaces. Wenow restrictour attentionto Z-actions. Thus, let/ : X -+ X be a homeomorphism of a topological space X, generating a Z-action on X. Denote by Pn(/) the number of periodic points of / of period n, that is, the cardinality of the set of fixedpoints of Definethe exponentialgrowth rate of periodic points by

r.

p(/)

:= limsup n-.oo

!n log(max{Pn{/), l}).

It is immediate that p{/) is also an example of a topological invariant for Z-actions. Exercise1.4.1 Let f : T" ~ 'Il'nbe a homeomorphismof then-torus obtainedfrom an integermatrix A of determinant l. Supposethat none of the eigenvaluesof A lie on the Show that unit circle {Ae C I IAI= l }. (For example, A = (~

!).)

p(f)

= L log IAI, IAl>l

wherethe sum ranges over the eigenvaluesof A of modulusgreater than 1. In particular, the numberof periodic points of period m grows exponentiallywith m. (For the details, see [16]. The key point here is to use the Lefschetzfixedpoint formula, from which one derives Pm(/)=

where H1(/)

ldet(l- H,(fr)I,

= A is the linear map induced by f on !Rn= H1(T").)

An important topologicalinvariantof Z-actions is the topologicalentropy. Roughly speaking, it captures the exponential growth rate, as m -+ oo, of the number of orbit segments of length m that can be distinguished with a

1.4 Dynamical Invariants

11

specified, but arbitrarily fine, precision. To quote Katok and Hasselblatt [16], "In a sense, the topological entropy describes in a crude but suggestive way the total exponential complexity of the orbit structure with a single number." The precise definition is as follows. Let f : X -+ X be a homeomorphism of a compact metric space X with metric d. We define an increasing sequence of metrics d~, m = 1, 2, ... , such that d{ = d and for each x, ye X,

d!,(x, y) :=

~ax

0!,1!,n-l

d(/(x),

/(y)).

The open ball relative to the metric d~ with center at x and radius a will be denoted B1(x, a, m). A set E C X is said to be (m, E)-spanning if X C UxeE B1(x, E, m). Let Sd(f, E, m) be the minimum cardinality of an (m, E)-spanning set. Again to quote [16], "One can verbally express the meaning of Sd(f, E, m) by saying that it is equal to the minimal number of initial conditions whose behavior up to time m approximates the behavior of any initial condition up to E." Define now the exponential growth rate for Sd(A· 11 'N-1

)

for all I :::: io ::::· · · ::::iN-t :::: n. Notice that a point x e X belongs to the foregoing element of ~N if the orbit segment fk(x), 0 ::::k ::::N - l, falls into A;t at time k. Introduce now the entropy of the panition ~N H(~N) := -

L µ,(A) logµ,(A). Ae~N

2 Ergodic Theory - PartI

30

=

(If µ(A) 0, µ(A) log µ(A) is naturally defined to be 0.) It is clear that H(;N) ~ 0.

We define the entropy of T with respect to the partition ; by the limit

h(T, µ, ;) := limsup ..!..H(;N) N-+oo N

~ 0.

It can be shown, in fact, that the limit exists, so it is not necessary to use the limsup. Finally, define the measure-theoretic entropy of T relative to µ by

h(T, µ) := sup{h(T, µ, ;) I ; is a finite measurable partition of X}. For the details, see (16]. It can be shown, for example, that if T u is the shift map for a Markov chain with matrix A= (a;j) and probability vector p (as defined earlier in the discussion of the um problem) then

=

n

h(u, µ)

=-

L Piaii logali· i,j=l

A similar but simpler example is provided by Bernoulli shifts. Here, the measurable space (0 1, ... , n }z, 8) and the transformation u are the same ones defined before for the Markov chains, but the measure µ is defined as follows. l, ... , n, be the probabilities of occurrence of events labeled Let 0 < Pi, i by i in a complete family of mutually exclusive events, so Pi l. Define µ by extending to B the measure

={

=

µ({w E O I w(i1)

:E7=t =

=it,···,

w(ik)

= ik})

=Pia··• Pit·

Exercise2.4.1 Letµ be the Bernoulli measure on '2 just defined, associated to the

probabilityvector (p 1, ••• , Pn), and let t be the measurablepartition of '2 by the sets A; = (w e '2 lw(O) = i}, i = 1, •.. , n. Show that h(u, µ, t) = -E7=1p; logp 1• (It can be shown that in this case h(u, µ) = h(u, µ, t). See [16] for the details.) Measure-preserving transformations T1 and T2 of probability spaces (X 1, B 1, µ 1) and (X 1, 8 1, µt), respectively, are said to be isomorphic if there exist conull sets Mi e Bi, i = l, 2, such that M; is T;-invariant and there is an invertible measure-preserving transformation F : X 1 -+ X 2 with ( F o T1) (x) (T2 o F)(x) for all x e M 1• It is not difficult to see that the measure-theoretic entropy is the same for isomorphic measure transformations. The question of finding conditions for two Bernoulli shifts to be isomorphic is one of the oldest questions in ergodic theory. It should be pointed out that

=

2.4

Measure-Theoretic Entropy

31

a large class of measure-preservingdiffeomorphismsof smooth manifoldscan be shown to be (measurably)isomorphic to Bernoulli shifts, so the question

is important for smooth systems as well. For example, it can be shown that an Anosov diffeomorphismof a compact manifold M preserving a smooth measure(i.e., a measurecoming from a volume form on M) is isomorphicto a Bernoullishift. For example,given two Bernoulli shifts T1and T2defined by the probabil¼,¼,¼>and ,how can one tell whether they are ity vectors 0 such that

jj(DB!tll ~

K

for all k ~ m2 and all m. (If e, denotes the flow of [X, Y], we can write B! = E>rk/m2, so that K = max{ll(DE>,)xllIO~ s ~ I}.) Let f be an arbitrary smooth function on Rn with compact support, let x, y e Rn be arbitrary points, and let y : (0, I] ~ Rn denote the straight line segmentfrom x to y. Then, writing lldfll := sup{lldfxvll/llvllIx, v E Rn, v # 0},

we have

j(f

o

B!)(y) - (f

o

11(f ~1

B!)(x)j =

1

1

:,

o B! o y)(t) dt

I

Klldfll lly'(t)IIdt

= Klldflllly -xii. Observethat f (B': 2 z)- f (A';,2z) can be writtenas a telescopingsum as follows: m2-J

L

k=O

2 2 [f(B! o Bmo A;:: -k-lz) - J(B! o AmoA;:: -k-lz)).

38

3

Smooth Actions and Lie Theory

Since A;;;2-k-l is a diffeomorphism, we can bound the absolute value of the sum by the expression m2-1

L suplf(B!

o Bmz)-

f(B!

o Amz)I~ m 2 KlldfllllBmz-Amzll

k=O zelR•

l ~ m2 Klldfllc(m) 2

m

= Klldfllc(m), which goes to Oas m goes to +oo. Therefore,

,.ji exp(t[X, Y])(x)goes to Oas m goes to

(exp(:

X),exp(

,.ji mt

m2

Y))

(x)

+oo,which is the assertion of the theorem.

D

The following notation will be used later, in the context of left-invariant vector fields on Lie groups. If f is a diffeomorphism of M and Y e X(M), we define Ad(f) Y :=

f. Y.

If X, Y e X(M) are complete vector fields, it is immediate that

exp(t Ad(exp X)Y)

= exp X o exp(tf)

o exp(-X).

3.2 Foliations The orbits of a locally free smooth action partition the manifold into a family ofsubmanifolds forming afoliation of M. We now discuss some of the basic notions regarding foliated spaces. Let M be a smooth n-dimensional manifold and let E be a smooth subbundle of the tangent bundle TM. More precisely, for each y E M, we have an mdimensional subspace E(y) of TyM and, for each x E M, there are smooth vector fields X 1, ••• , X m defined on some neighborhood U of x such that X1 (y), ... , Xm(Y) form a basis of E(y) for each y e U. We say that a vector field X belongs to E if X (x) E E (x) for each x e M. The subbundle E is called involutive if whenever X, Y are differentiable vector fields that belong to E, the bracket [X, Y] also belongs to E. A submanifold L of M is an integral manifold of a subbundle E of TM if for each x EL, TxL = E(x). A coordinate chart (x,, ... , Xn, U) on Mis said to be adapted to E ifthecoordinatevectorfields a!,, i = 1, ... , m, belong to Elu-

3.2

Foliations

39

Having a coordinate chart adapted to E clearly implies that the m-dimensional slices of U obtained by fixing the values of the remaining n - m coordinate functions are integral manifolds of Elu, and that Elu is involutive. Theorem 3.2.1 (Frobenius) Given an m-dimensional smooth subbundle E of TM, E is involutive if and only if for each x e M there is a coordinate chart adapted to E defined on a neighborhood of x. Proof. We assume that E is involutive and show the existence of the adapted coordinate system. The other direction has already been noted. With an appropriate choice of smooth coordinate chart (z 1, .•. , Zn, W) centered at x, we may assume that E is a smooth subbundle of the tangent bundle and denote by n of IR.nand x is the origin of IR.n.We write IR.n= IR.mx JR.n-m the projection onto the first factor, which we refer to as the horizontal subspace of the product. We can choose the coordinates so that E is tangent to the horizontal subspace at the origin. Fix a neighborhood of the origin in IR.nof the (resp., form V x U, where V (resp., U) is a neighborhood of the origin in IR.m JR.n-m).By choosing these neighborhoods sufficiently small, we may assume for that the restriction of (Dil)x to E(x) is a linear isomorphism onto Tncx>IR.m allxeVxU. Define vector fields Z 1 , ••• , Zm on V x U that belong to E and project onto on V (i.e., Z; is n-related to for the coordinate vector fields i = I, ... , m). Notice that [Z;, Zj] also belongs to E since E is involutive, and is n-related to the zero vector field O = la~,, a~1 ]. Therefore Z1, ... , Zm are commuting vector fields that span the subbundle E on V x U. Denote by ! the local flow associated to Z; near the origin. By making V and U sufficiently small, we may assume that there is an E > 0 such that the domain of ci,i, where ci,i (x, t) := :(x), contains V x U x (-E, E). We define a smooth map : ( -E, E )ffl X U ~ JR.n given by

a~., ..., a:m

(t1,•.• , tm, y)

a!,,

= (: o • · · o ~)((0, y)). 1

(Notice thatfixes the origin.) A simple computation shows that (Dc!>)0 is the identity map of IR.n.Therefore, by the inverse function theorem, has a smooth inverse on some neighborhood of the origin. The coordinate system we seek can now be obtained by restricting the original coordinates z; to a sufficiently small open subset of W and composing (x, y)

+ higher-order

terms int)) - f(e))

y)f.

= [x, y].

D

Exercise3.5.3 Let X 1, ... , Xn be a basis for the Lie algebra g of G. Show that the map cp: (t1, ... , tn)

~

exp(t1 XI)•··

exp(tnXn)

is a smooth diffeomorphism from some neighborhood of O e Rn onto some neighborhood of e e G.

3.6 Lie Subgroups and Homogeneous Spaces For each Lie subgroup Hof a Lie group G, the subspace fJ:= TeH clearly is a Lie subalgebra of g = TeG. The converse is given by the next proposition. Proposition 3.6.1 Let g be the Lie algebra of a Lie group G, and let fJbe a Lie subalgebra of g. Then I) is the Lie algebra of a unique connected (not necessarily closed) Lie subgroup H of G.

3.6

Lie Subgroups and Homogeneous Spaces

47

Proof. Since the Lie subalgebra ~ can be viewed as a left-invariant involutive subbundle of T G, we get a foliation 1-l in G whose leaves are the maximal

connected integral manifolds of 1-l. Denote by H the leaf of 1-lcontaining e. Notice that left translation by g e G maps any leaf of 1-ldiffeomorphically onto another leaf. In particular, if h e H, then hH = H, since hH is another leaf of 1-lthat intersects H at h (since he = h) so the two leaves must coincide.· Similarly, if h e H we have h- 1 H = H, since the two leaves must contain e = h- 1h. Therefore, His an abstract subgroup of G and a smooth submanifold. The proposition is now a consequence of the next lemma. 0

Lemma 3.6.2 Let G be a Lie group, X a differentiable manifold, and f a oneto-one immersion of X into G such that f(x)f(y)

e f(X)

and f(x)-

1

e f(X)

for all x, ye M. Introduce a multiplication on X by xy := J- 1(f(x)f(y)) and suppose that this map is continuous. Then X is a Lie group under this multiplication. Proof. It is clear that f(X) is a subgroup of G. Thus, the map from X x X into G given by (x, y) i--+ f(x)f(y) is differentiable. Define a map u from X x X into X by u(x, y) := f- 1(f(x)f(y)). We wish to show that u is also

differentiable. Since u is assumed to be continuous, it suffices to show that there is a neighborhood U of u (x, y) such that for every differentiable function 0, so that W' is written as an open cube of the form (-a, a)m. Notice that an arbitrary union of cubical neighborhoods of e is also a cubical neighborhood of e. Cubical neighborhoods can easily be constructed using the exponential map. Following [21), we call a smooth map v,:V x U -+ G x Man auxiliary map if V is a cubical neighborhood of e in G, U is an open set in M, 1/1is a diffeomorphism onto its image of the form v,(g, x) = (g, q,(g, x)) for some smooth function q, : V x U -+ M, and the following conditions hold: 1. For each y e U, g i-+ v,(g, y) maps V diffeomorphically onto the component of (e, y) in Lyn n 01(V), where Ly is the leaf of E that contains (e, y).

2. For each y e U, no maps the component of (e, y) in Ly bijectively onto V2 •

n n01(V 2 )

It is a simple exercise to show that for each x e M, one can always find a neighborhood U of x and a sufficiently small cubical neighborhood V of e e G

3 Smooth Actions and Lie Theory

54

such that there exists a unique auxiliary map of V x U into G x M. Moreover, ifwe define v,' := v,(·, y): V ~ G x M, then

for all X e g and y e U. In fact, (Dv,')eX(e) is a vector in E(e, y) that projects onto X(e) under (Dila)ce,y), but 0*(X)(e, y) = (X(e), 0(X)(y)) is the unique such vector. Lemma3.9.1 Letv,:VxU~ iliary maps, and write q, =

GxMandv,':

V'xU'~ GxMbetwoauxThen the following hold.

nMo v, and q,' = nMo v,'.

1. v, and v,' agree on the intersection of their domains. 2. If (h, x) e V x U, then Rh-• maps Lx diffeomorphically onto L•and is given by 0 : X ~ - X*. E>is indeed a group homomorphism and an involution, that is, 8 2 is the identity map, as one can easily check. Let G be a connected Lie subgroup of G L(n, C). We say that G is a reductive group if it is conjugate to a subgroup that is stable under the Cartan involution 119

120

7 Semisimple Lie Groups

e.

In other words, G is reductive if there is g E GL(n, C) such that gGg- 1 is mapped into itself by e. A Lie algebra g C gin (C) is reductive if it is conjugate by an element in G L(n, C) to a 0-stable subalgebra. In particular, G is reductive if and only if g is. We recall that the center of a group G is the subgroup Z(G) = {a E G I ag = ga for all g

E

G}.

The center is clearly a nonnal subgroup of G. The center of a Lie algebra g is the subalgebra J(g) ={XE

g I [X, Y] = 0 for all YE g}

and is an ideal of g. (A subalgebran C g is an ideal if [X, Y] E n for all X E n and all Y E g.) A Lie algebra g c gln (C) is semisimple if it is reductive and has trivial center. G c GL(n, C) is a semisimple Lie group if its Lie algebra is semisimple. As an example, we show that SU (p, q ), p + q ~ 2, is a semisimple group. Recall that A belongs to SU(p, q) if, by definition,A* lp,qA = lp,q anddetA = 1 (see chapter 5). It is immediate from the definition that if A E SU (p, q) then 0(A) E SU(p, q), so the group is reductive. We show by direct computation that its centeris finite. First observe that the diagonal matrix diag[).1, ••• , Ap+q] is in SU (p, q) if and only if l).k I = 1 for each k and the product of the diagonal entries is 1. We can choose one such matrix, call it B, such that its diagonal entries are all mutually distinct. If A = (a;i) is in the center, AB = BA implies that a;i).i = ).;a;i (no summation involved), so all off-diagonalentries of A must be zero. Moreover, we clearly have that AXA- 1 = X for all X E .su(p, q). Thus, we look for a diagonal element diag[a 1, ..• , ap+q] of 1 xx 12 ) of trace O such that SU (p, q) that commutes with all matrices X = (xx~ 12 22 Xj 1 = -X11 (a p-by-p matrix) and Xii = -X22 (a q-by-q matrix). By inspecting the fonn of X, it is clear that for every index (i, j) with i =I-j, we can always find a matrix X = (Xij) in .su(p, q) such that Xii =I-0. Writing out the product we obtain a;X;j = xiiai so a; = ai for all i and j. Consequently, A = ).J for some complex number). such that ).P+q= 1. Therefore, the center is the finite subgroup of U ( l) of (p + q )th roots of 1, so the center of g is trivial and SU (p, q) is semisimple. A similar argument shows that the classical groups of chapter 5 are all semisimple. Exercise 7.1.1 Show that SL(n, JR) is semisimple.

Let g C gln(C> be a reductive Lie algebra, and assume, possibly after having to conjugate g by some matrix in GL(n, C), that it is stable under the involution

7.1 Reductive and Semisimple Lie Algebras

121

0. One defines an inner product on g as follows. Given, X, Y e g, set {X, Y} := Re(Tr(XY*)) = -Re(Tr(X0(Y))). Exercise 7.1.2 Show that(·,·) is a positive-definite symmetric bilinear form.

We denote by At the adjoint of a linear transformation A of g with respect to the inner product {·, ·} and by ad(Z) the linear transformation defined by ad(Z)(X) = [Z, X], X e g. Observe that the Jacobi identity can be written in the following form: ad([X, YJ) = [ad(X), ad(Y)], for all X, Y e g. Exercise 7.1.3 Verify the following statements. I. ad(O(Z)) = -ad(Z)t. 2. (0(X), 0(Y)} = (X, Y) for all X, Y e g. 3. If a is an ideal of g, then 0(a) is also an ideal.

A direct sum decomposition of a Lie algebra ~ is a direct sum (in the sense of vector spaces)~= ~ 1 $ · · · $ ~k such that~ ••... , ~k are ideals of~- In particular, [X, Y] = 0 whenever Xe~;, Ye ~j• i 1-j. Exercise 7.1.4 Given an arbitrary ideal a of the 0-stable Lie algebra g, define the subspace a.L := {X e g

I (X, Y) =

0 for all Y e a}.

I. Show that a.Lis also an ideal of g and g = a EBa.L. 2. ShowthatO(a.L) =0(a).L.

Exercise 7.1.5 For an arbitrary Lie algebra (), denote by [(), ()] the subspace of() spanned by all elements [X, Y] for X, Y e (). I. Show that [(), ()] is an ideal of(). 2. If g is a 0-stable subalgebra of gln(C), show that [g, g] also is 0-stable.

Exercise 7.1.6 If g is 0-stable, show that [g, g].L = 3(.9). In particular, g

= [g, g] EBJ(g).

Conclude that, if g is a semisimple Lie algebra (not necessarily 0-stable), then g = [g, g].

A Lie algebra is said to be simple if its only ideals are {O} and itself. Proposition 7.1.7 Let g be a 0-stable semisimple Lie subalgebra of gln(C). Then g decomposes as an{·, ·}-orthogonal direct sum g 1 $ · · · $ gk of0-stable nonabelian simple ideals.

7 Semisimple Lie Groups

122

Proof. Let a be a nonzero ideal of g of minimal dimension. The Jacobi identity implies that [a, a] is also an ideal of g. Moreover, [a, a] =/.{0} since, otherwise, a would be contained in the center of g, which is zero. Therefore, [a, a]= a. We claim that a is 0-stable. Once this is shown, the proposition follows by considering a.L and proceeding by induction. Observe that [0(a), a] is an ideal of g contained in a, so either it coincides with a or it must be zero by the minimality of a. In the latter case, we would conclude that for any Z = I:;[X;, Y;] Ea= [a, a], (Z, Z) = L([Xi,

Y;], Z) = - L(Y;,

i

[0(X;), Z]) e (a, [0(a), a]),

i

so that Z, hence a, would be zero, which is not the case. Therefore, [0(a), a]= a. It follows from this that the ideal an 0(a), which contains [0(a), a], must coincide with a, so a= 0(a) as claimed. D

7.2 The Adjoint Group If g is a real Lie algebra of dimension N, the vector space gc = g ®ntC has a natural structure of a Lie algebra over C such that

[X ® 1, Y ® l]

= [X, Y] ® 1.

It is called the complexification of g. The group of Lie algebra automorphisms of g, defined as the subgroup of all A E GL(gc) such that A[u, v] = [Au, Av] for all u, v e 9c, is in a natural way an algebraic group defined over ll. Namely, let {e1 , ••• , eN} be a basis for g and define real numbers ct by [e;, ej] = I:k cfjek. Then the group Aut(gc) ofC-linearautomorphisms of 9c becomes identified with the subgroup of GL(N, C) consisting of all complex matrices A = (aij) that satisfy the polynomial equations with real coefficients I:cfjak1 k

- 'Ec~sairajs

=0

(-# A[e;, ej] - [Ae;, Aej]

= 0).

r,s

Let G be a connected Lie group and let g be its Lie algebra. Then G has a canonical representation into the group of real automorphisms of g, the adjoint representation, defined as Ad: G -

GL(g)

such that Ad(g)(X) = gXg- 1• Although the adjoint representation is not in general faithful, its kernel is precisely the center of G, as one can easily check. If G is semisimple, its center Z is a closed subgroup with trivial Lie algebra,

7.2

The Adjoint Group

123

and therefore Z is discrete. Therefore, for a connected semisimple G, G/ Z is isomorphic to Ad(G).

Exercise 7.2.1 If G is a connected semisimple Lie group, show that G/ Z is a connected semisimpleLie group with trivial center. (Hint: The main point is to show that Ad(G) has trivial center. Show that for any A in the center of Ad(G) and all X, Y E g, A[X, Y] = [X, Y]. But g semisimple implies [g, g] = g so A is the identity transformation.) Exercise 7.2.2 Show that the Lie algebra of Aut(gc) is the linear space of all C-linear derivations of gc, that is, of all complex linear maps 8 from gc into gc such that 8[X, Y] = [8X, Y] + [X, 8Y] for all X, Y E gc, A derivation 8 of 9c is called an inner derivation if it is of the form 8X ad(Z)X := [Z, X] for some Z E De· The Killing form of g is the symmetric bilinear form

=

K(X, Y) := Tr(ad(X) o ad(Y)),

for X, Y e g. It is ad-invariant, in the following sense: K(ad(H)X, f)

+ K(X,

ad(H)Y)

=0

for each H, X, Y e g.

Exercise 7.2.3 Prove that K is ad-invariant. (Use ad([X, Y]) = [ad(X), ad(Y)], X, Y E g, and the fact that Tr(AB) = Tr(BA) for linear transformations A, B.) Also show that K(0X, 0Y) = K(X, Y), X, Y E g. (Express Tr(ad(0X) o ad(0Y)) in terms of an (·, .)-orthonormalbasis and use that 0 is an isometry for (·, ·).) The Killing form is clearly a symmetric bilinear form. If g is semisimple, K is also nondegenerate. To see this, let {e; I i 1, ... , n} be an orthonormal basis for g with respect to the inner product (·, ·) and recall that the adjoint -ad(0H). Then, if of ad(H) with respect to this inner product is ad(H)t

=

K(X, ·)

=

= 0, 0

= K(-0(X),

X)

n

= L((-ad(0X)

o ad(X)e;, e;)

i=l n

= L (ad(0 X) t o ad(X)e;, i=l n

= L(ad(X)e;, i=l

ad(0X)e;),

e;)

7 Semisimple Lie Groups

124

=

so ad(X) = 0. But if g is semisimple, this also implies that X 0 since the kernel of ad is the center of g, which is trivial. It is useful to introduce another inner product on g, defined by the equation (X, Y}K := -Re(K(X, 0Y)). Exercise 7.2.4 Using the facts already established for the Killing form, show that if g is a 0-stable semisimple Lie algebra, then 1. (·, ·}K is a symmetric nondegenerate bilinear form; 2. with respectto (·, ·}K,the adjoint of ad(X) is ad(X)t 3. (0(X), 0(Y)}K = (X, Y}K for all X, Y E g.

= -ad(0X);

Theorem 7.2.5 Let G be a connected semisimple Lie group and Z its center. Then the adjoint representation defines an isomorphism between G/ Z and the connected component of the identity of the group of real points of a linear algebraic group defined over R The algebraic group of the previous theorem is called the adjoint group of G. In particular, if G is a connected semisimple Lie group with trivial center, then G is naturally isomorphic to the identity component of a real algebraic group. To prove the theorem it suffices to show that the Lie algebra of the group of real points of Aut(gc) is isomorphic tog. Since that Lie algebra is the space of derivations of g, the theorem will be proved once we show the next lemma.

Lemma 7.2.6 Any derivation of a semisimple Lie algebra is inner. Proof Let 8 be a derivation of g and define a linear functional f on g such that f (X) = Tr(ad(X) o 8) for each X e g. Since K is nondegenerate, we can find Y e g such that f (X) = K(X, Y) for all X e g. Defining the derivation D := 8 - ad(Y), we show that D = 0, as follows. First notice that Tr(D o ad(Z)) = 0 for all Z, by definition. Therefore, for all Z 1, Z 2 e g, 0 = Tr(D o ad[Zt, Z2])

= Tr(D = Tr(D

o [ad(Z1), ad(Z2)]) o ad(Z1) o ad(Z2) - Do ad(Z2) o ad(Z 1))

= Tr(D o ad(Z1) o ad(Z2) = Tr([D, ad(Z1)] o ad(Z2))

= Tr(ad(DZ1) = K(DZ1,

ad(Z1) o Do ad(Z 2))

o ad(Z2))

Z2),

which shows that DZ 1 = 0 for all Z1. Therefore D = 0.

D

7.3

The Cartan Decomposition

125

Table 7 .1. The maximal compact

subgroups of the classical groups K

G SL(n, 0. Multiplying by e- th N and changing notation, we may assume that bN = 0. We pass to the limit in the expression Ef=tCje1h1 as t approaches

+oothrough integer values and find CN = 0. But this is a contradiction.

Proposition7.3.4 Set K

D

= U (n) and let .J1be the space of all n x n Hermitian

matrices. Then the map F: K x .J1--+GL(n, C),

(k, X)

H-

kexpX

is a smooth diffeomorphism. F restricted to {e} x .J1is a smooth diffeomorphism onto the space S of positive Hermitian n x n matrices.

Proof. Given g e GL(n, C), g*g is a positive Hermitian matrix, so there are A e U(n) and positive numbers A; such that g*g = A diag[)..1, ... , An]A- 1. Define C = Adiag[,J>:i', ... , .Jf,;]A- 1• Then C 2 = g*g and C* = C, so C*C g*g. It follows that (Cg- 1)- 1 (Cg- 1)*, so cg- 1 e K. Therefore g = kC, for some k e K, showing that the map is surjective. We wish to show that the map F is also injective. First notice that if kC = k' C' fork, k' e K and positive Hermitian matrices C, C', then Ck- 1 = (kC)* = (k'C')* = C'k'- 1, which implies that

=

=

C 2 = (Ck- 1)(kC) = (C'k'- 1)(k'C') = C' 2 • The previous lemma implies that C belongs to the subgroup H c G L(n, C) consisting of the Zariski closure of the set of all powers c2nfor n e Z. (Indeed, let Q be any polynomial that vanishes on H. Since C2

= Adiag[.i..1,...

, An]A-

1,

and Q(C2m) = 0 for all m e N, we obtain a polynomial P(x 1, ••• , Xn) such that P(.i..f, ... , .i..;:')= 0 for each m. The lemma now implies that Q(C) = I I P(Af, ... , .i..J) = 0.) On the other hand, also by the previous lemma, C' must commute with each element of H since C' commutes with all c2m(commuting with C' can, of course, be characterized by polynomial equations). Therefore C and C' commute. It follows that D = C' c- 1 is a positive Hermitian matrix such that D 2 = I; therefore D = I and C = C'. This shows that F is injective, hence bijective.

7.3 The Cartan Decomposition To show that diffeomorphism the target space injective. Write

127

F is a diffeomorphism, it suffices to prove that it is a local at each (k, Zo) e K x p. In fact, since the dimensions of and domain are the same, it suffices to show that D Fck.Zo> is y(t) := exp(Zo + tZ), Z e p. Then

DFck,Zo)(X,Z)

= -ddi t

Suppose that this equation is 0, so that X show that X = Z = 0. Notice that 0(y'(O)y(O)-I)

+ y'(O)y(0)- 1 = 0. Our goal is to

= ddI

E>(y(t)y(0)-1)

= dd I

y(t)- 1y(O)

t

= kXy(O) + ky /(0).

(kexp(tX)y(t))

t=O

t=O

t

(since y(t) is Hermitian)

t=O

= -y(0)-

1y'(O).

Therefore, recalling that 0 (X) = X, we obtain X -y (Or I y' (0) = 0. Therefore y'(O)y(0)

2

= y(0) 2 y'(O).

Since y(O) is Hermitian symmetric, we can apply the same argument used already to show commutativity of C and C' to conclude, here, that y (0) and y'(O) commute. We get at once y'(O) = 0 and X = 0. 0. Since y'(O) commutes with y(O) exp Zo, We still need to show Z we also conclude that y'(O) commutes with Zo, (Use the lemma once again to prove that y'(O) commutes with (exp(¾Zo) - /)/m for each m and then pass to the limit as m ~ oo to conclude that y' (0) and Zo commute.) Expanding exp(Zo + t Z) in a Taylor series and taking the derivative term by term yields

=

r'(O)

=

f

n=O

=

< 1 1)' n+

·

(t z~zz:;-k). k=O

Therefore 0

= Zoy'(O) -

y'(O)Zo

=~ 1 (zn+IZ - zzn+I) ~ (n + 1)! o o n=O

= (exp Zo)Z -

Z exp Zo,

7 SemisimpleLie Groups

128

It follows that Zo and Z commute. Hence 0

= y' (0) = (exp Z0)Z and, finally,

Z=O. To conclude the proof of the proposition, observe that A diag[>..t,... , An]A- 1 e

.p .-. A diag[exp >..,,... , exp An]A- 1 e S D

is a bijection. The inverse of the map F is called the polar decomposition.

Theorem 7.3.5 (Cartan decomposition) Let G be a subgroup of GL(n, C) defined by polynomial equations, with real coefficients, in the real and imaginary parts of the matrix entries. Let g be the Lie algebra of G. Suppose that G is stable under the Cartan involution. Let K = G n U (n) and .pbe the subspace of Hermitian matrices in g. Then the map T :K x

.p ~ G, (k, X) .-. k exp X

is a surjective diffeomorphism.

Proof Since T is the restriction to K x .p of the smooth diffeomorphism of the previous proposition, T will also be a diffeomorphism once we show it is surjective. Letg e G and writeg kexp X, the polar decomposition of gin GL(n, R). By hypothesis,

=

0(g)-

1

= g* = (exp X)k- 1

lies in G, so 0(g)- 1g = exp(2X) also does. Since Xis Hermitian, we can write 2X = Adiag[a 1, ••• , an]A- 1, for some A e U(n), where a 1, ••• , an are real. For each integer k, (exp(2X))k = A diag[ea 1k, ••• , ea•k]A- 1 e G, so an application of the previous lemma shows that exp X e G. Therefore both exp X and k lie in G, showing surjectivity of T and concluding the proof. D

Corollary7.3.6 K is a maximal compact subgroup of G. Proof If KI is another compact subgroup of G that properly contains K, then there is k 1 = kexpX e Kt such that X =f:.0. But then expX = k-tk, e Kt so exp(nX) e K 1 for all n. But this is impossible since the eigenvalues of exp(nX) are not bounded.

D

7.4

The Restricted Root Space Decomposition

129

Proposition 7.3.7 Let G be a subgroup of GL(n, C) defined by polynomial

equations, with real coefficients, in the real and imaginary parts of the matrix entries. Then the center of G is finite. Proof Denote the center by Z. We know that Z is discrete in G since G is semisimple. In order to show that Z is finite, it suffices to prove it is contained

in a maximal compact subgroup. We may assume that G is 0-stable. Let K be as in the previous theorem. Let z e Z and write z = k exp X. Since G is stable under e and e is a homomorphism from G into itself, it follows that Z is also 0-stable. Therefore 0(z)- 1z = (expX)k- 1kexpX = exp(2X) also belongs to Z. But Z is an algebraic subgroup of G, so another application of the previous lemma gives exp(t X) e Z for all t e R Therefore, if X were not 0, Z would have dimension greater than 0, a contradiction. Therefore X = 0 and Z CK. D It is also immediate from the theorem that the number of connected components of G is the same as the number of connected components of K, which is finite.

7.4 The Restricted Root Space Decomposition

For each X e .p, the operator ad(X) on g is self-adjoint with respect to the inner product(·, ·}K. Therefore ad(X) is diagonalizable with real eigenvalues. Let a be a maximal abelian algebra in .p. More precisely, a is abelian and is not properly contained in a subspace of .p consisting of commuting elements. The operators ad(X), X e a, commute since 0

= ad([X, Y]) = ad(X) o ad(Y) -

ad(Y) o ad(X)

for X, Y e a. Therefore, it is possible to find a basis for g that simultaneously diagonalizes all the operators ad(X), X E a. Denote by a* the space of real linear functionals on a. We now define g,_ := {X e g I [H, X] = A(H)X, for all H e a}.

If A e a* is nonzero and g,. is nonzero, we say that A is a root of (a, g) (or a restricted root of g), with associated root space g,_.The set of all such roots is denoted (a,g). We denote by g0 the centralizer of a in g, that is, g0 is the subspace of all X in g such that [X, H] = 0 for all H e a. Therefore, we have

130

7 Semisimple Lie Groups

the direct sum decomposition of vector spaces 1l = !lo EB

EBg,.

AE(a,g)

called the restricted root space decomposition of g. Exercise 7.4.1 If 1, µ, e (a,g), show that [g,., g,.] C DJ.+µif 1 [g,., g,.] is zero otherwise. Moreover, [g 0 , g,.] Cg,..

+ µ, e (a,g) and

Exercise 7.4.2 Show that 1 e (a,g) if and only if -1 e (a,g) and that 0 restricts to an isomorphism between g,. and g_,.. Exercise 7.4.3 If 1 and µ, are distinct roots, show that g,. and g,,. as well as g,. and g 0 , are orthogonal subspaces with respect to(·, ·}K. Exercise 7.4.4 Show that g0 is stable under0 and that we have a(·, ·}K-orthogonal direct sum g0 = a EBm, where m ={Xe t I [X, Y] = 0 for all Ye a}.

The subalgebra a will be called an IR-split Cartan subalgebra of g. A more descriptive name is "maximal abelian "JR-diagonalizablesubalgebra." = (a,g). ChooseH E ainthe Wedefineanorderingonthesetofroots complement of U,.e ker A. Since A(H) is nonzero for all>.. E ,we can write as the disjoint union of the sets + and -of roots >..such that ')..(H) > 0 and>..(H) < 0,respectively. Duetoexercise7.4.2, -= -ct>+. A Lie algebra n is said to be nilpotent if the sequence

nco>= n, n(I) = [n, n], ... , ncn+O= [n, ncn>1,... eventually terminates at 0, that is, ncn>= {0} for some n. The Lie algebra is said to be solvable if the sequence n = n, n°> = [n, n], ... , n be a 0stable semisimple Lie algebra. Then, as a vector space, g has the direct sum decomposition g = t EBa EBn. Moreover, there exists a basis X 1, ••• , Xn of g such that the matrix representing ad(X), X E g, has the following properties: 1. It is skew-Hermitian if X E t.

7.4 The Restricted Root Space Decomposition 2. It is diagonal with real entries if X e a. 3. It is upper triangular with Os on the diagonal, if X

131

e n.

=

Proof. We first show that the sum is direct. It is clear that an n {O},so we need to check that any X e t n (a EBn) is 0. Such an X must be fixed by 0 and must have the form X Xo + LAec1>+ XA, Xo e g0 and XA e 9A· Since 0(X) X, we can write 2Xo + LAec1>+ XA- LAec1>+ 0(XA) 0. Therefore, as the root space decomposition is a direct sum, we conclude that each component must be zero, so X 0. All that is left to do is show that E0Aec1>gA is contained in t EBa EBn. But this is a consequence of the fact that any X e 9-A, >..e 4>+,can be represented as X X + 0(X) - 0(X), where X + 0(X) e t, 0(X) e 9A· Take an(·, ·)K-orthonormal basis X1, ... , Xn of g adapted to the restricted root space decomposition, having the property that if X; e gA and Xi e g/L with i < j then >..:::::µ,. For X e t, we have by exercise 7 .2.4

=

=

=

=

=

ad(X)t

= -ad(0(X)) = -ad(X),

which implies that the matrix representing ad(X) is skew-Hermitian. The remaining properties are immediate consequences of the restricted root space 0 decomposition. Unless specified otherwise, we assume from now on that g is semisimple and 0-stable, and that a is an JR-split Cartan subalgebra of a. The dimension of a is called the real rank of g. This definition seems to depend on the choice of a in .p. It turns out, however, that any two such subalgebras are conjugate by an element of K, so their dimensions are the same. This is proved by the next proposition.

Proposition7.4.7 If a 1 is another choice of maximal abelian subspace of .p, we can find k e K (the subgroup of G pointwise fixed by E>)such that a 1 = k a k- 1• Since any X e .pis contained in a maximal abelian subspace, it follows that

.p=

LJkak-

1•

keK

Proof. We choose H e a (resp., H 1 e a 1) such that no root in Cl>(a,g) (resp., Cl>(a1,g)) vanishes on H (resp., H1), We now define a smooth function on K by f(k) := (Ad(k)H, H 1)K. Since K is compact, there must be some ko that minimizes/. We claim that ko1a 1ko=a.To see this, it suffices to show that

132

7 Semisimple Lie Groups

k01a I ko c g0 n p = a, and equality will be a consequence of the maximality of a 1• Inclusion in pis clear since K normalizes p. We now show inclusion in g 0 • Since ko is a critical point off, for any X E t we have 0

= dd I t

t=O

f(e'xko)

= dd I

(Ad(e'xko)H, H1)1e

= dd I

(Ad(e'x)Ad(ko)H, H 1)"

t t

t=O

t=O

= (ad(X)(Ad(ko)H),

H1)1e

= (-ad(Ad(ko)H)X, H1)1e = (X, ad(0(Ad(k 0)H))H 1)" = (X, -ad(Ad(ko)H)H1)1e = (X, [H1, Ad(ko)H]),c. Therefore, as t and p are orthogonal and [H 1, Ad(ko) H] c [p, p] c t, we have (X, [Hi, Ad(ko)H])" = 0 for all X E g, so [H 1, Ad(ko)H] = 0. But 0 or, equivalently, no root of (a 1, g) vanishes on Hi, so [a 1, Ad(ko)H] 1 [k0 a 1k0 , H] = 0. By the same reason, asnorootof(a, g) vanishes on H, we 0 conclude that k01a1ko C 9o·

=

Theorem 7.4.8 (K AK decomposition) Let G be a closed subgroup of GL(n, C) and suppose that it is the common set of zeros of some set of polynomials, with real coefficients, in the real and imaginary parts of the matrix entries, and let g be its Lie algebra. Suppose that G is semisimple and stable under the Cartan involution. Then G = KA K, where A is the connected abelian subgroup of G with Lie algebra a. More precisely, for each g E G we can find (not necessarily unique) a E A and k1, k2 E K such that g = k1ak2.

=

Proof. The Cartan decomposition gives g k exp X and by the previous proposition, expX = k21exp Hk2, so g = kk21exp Hk2. 0

7.5 Root Spaces for the Classical Groups We describe now the restricted root space decomposition in some particular cases. As a first class of examples let us talce g .sl(n, IF), where IF IR.,C, or ]fl[ (recall that .sl(n, llil) is isomorphic to .su*(2n)). In this case t is the subalgebra of skew-Hermitian n x n matrices (skew-symmetric if IF = JR.)and

=

=

7.5 Root Spacesfor the ClassicalGroups

133

p is the subspace consisting of Hermitian n x n matrices of trace O. Denote by a the abelian algebra consisting of real diagonal matrices of trace 0. Then a C p, and by a simple computation we see that the subalgebra consisting of all matrices in p that commute with each element of a is a itself. Therefore, a is an JR-splitCartan subalgebra of sl(n, IF) and the real rank of sl(n, IF)is n - 1. For each i, 1 ::: i ::::n, define /; e a* by /; (diag[a1, ... , anD= a; and set a;i := /j - /;. Defineg;i := IFE;i, where E;i is the matrix with 1 at the (i, j) entry and O at the other positions. Notice that dim gii = 1, 2, 4 for R, C, Ill. One can easily check that

EB gii•

sl(n, IF)= go EB

i,t.j

where g0 is the subalgebra of all diagonal matrices of trace 0. We can write = m EBa, where m is trivial if IF= R, m is the subalgebraof all diagonal matriceswith imaginaryentries for IF= C, and m is the direct sum of n copies of su(2) if IF= Ill. (Noticethatsu(2) is isomorphicto the algebraof imaginary quaternions.) Before considering other examples, we make some general remarks about the way the root space decompositionof a subalgebra is placed inside a larger Lie algebra. g0

Proposition7.5.1 Let g 1be a semisimplesubalgebraof a semisimpleLie algebra g, and let a I be a Cartan subalgebraof g 1contained in a Cartan subalgebra a of g. Then each root of (g 1, a1) is the restriction to a1 of a root of (g, a). The root space (g 1)a is the intersectionof g 1 with the direct sum of the gp for all /J such that /Jla,= a. Proof. For each y e aj we denote by Vy the direct sum of root spaces gp for a such that /J1111 = y. Then g is the direct sum of a finite number of Vy. For each root a of (g 1, a1) and each X in the root space (g 1)a we can uniquely write X = Er Xy, Xy e Vy. Therefore, for all H e a1,

L a(H)Xy y

= a(H)X = ad(H)X =

L ad(H)Xy = L y(H)Xy, y

y

and we conclude that wheneverXy is not 0, y(H) = a(H) for all He a 1. In other words, (g 1)a is contained in Va, so (g 1)a = Van g 1• D We now use the previouslemma to compute the restrictedroot space decomposition for glF = su(p, q),, p ::::q. Recall that glF = so(p, q), su(p, q), or

7 SemisimpleLie Groups

134

s.p(p, q) for lF = JR,C, or IHI,respectively. It was shown in an earlier exercise that

9JF={Xe

sl(p

+ q, JF) I X*L + LX

= O},

where

0 0

L := ( 0 F

/

F) 0

,

O 0

F:=G

J

The middle block is the identity matrix of size p - q and F is a square block of size q with ls on the SW-NE diagonal and Os everywhere else. The point of writing the algebra in this form, rather than taking the representation for which L is diagonal, is that we can now show that the intersection of g, with the diagonal Cartan subalgebra of sl(p + q, JF) is a Cartan subalgebra for g,. In fact, a simple computation shows the following. The algebra g, is still 0-stable, and elements of .p (X e g, I X* X} have the form

=

=

X

=(

A B* C*

B 0 -FB

C ) -B*F , -FAF

where A* = A and C* = -FC F. Ifwe set a= .pn {diagonal matrices}, then a is the set of diagonal matrices of the form diag[x1, ... , Xq, 0, ... , 0, -xq, ... , -xi], for x; e JR. Another simple computation shows that any element of g, that commutes with all elements of a must take the form diag[>..1,... , >..q, E,

-iq, ... , -ii]

where>..; e lF and E is a square block of size q such that E* = - E. Of all such matrices, the only ones that lie in .pare the ones already in a, so a is indeed an JR-split Cartan subalgebra. In particular, the real rank is q. The roots and root spaces can now be obtained with the help of the previous proposition. Define as before /; to be the linear map on the space of diagonal matrices in s l(p + q, lF) that evaluates the i th diagonal entry. The roots of g, are then the restrictions of /; - fJ to a. Notice that if a is the restriction of /; - / 1 for 1 ~ i < j ~ p + q, then Sa is contained in the subspace of upper0(ga) is contained in triangular matrices with Os on the diagonal, and 9-a the subspace of lower-triangular matrices with Os on the diagonal.

=

7.5 Root Spacesfor the Classical Groups

135

Any root of 91Fis, therefore, one of the following: ±f;, ±2f;, ±(f;+ /j), 1 ~ i, j

~

q, i =fij, and ±(f;-/j),

1~ i < j

~

q.

In order to get the corresponding roots spaces, it is convenient to observe that if C is a square matrix of size q, then F C1F is the matrix obtained from C by "flipping" it about the SW-NE diagonal. The root space 9 /. consists of all matrices of the form 0 B ( 0 0 0 0

O ) -B*F , 0

where B has entries Oin all places except the ith row, the entries of which are arbitrary elements of IF.The dimension of 91, is (p - q) dima IF.The elements of 9 21, have the form 0 0 ( 0 0 0 0

C) 0 0

,

C

= F diag[O,...

, 0, x;, 0 ... , 0JF,

where x; is real. Therefore 9 21, has real dimension 1. Elements of 9 /. +11take the form 0 0 ( 0 0 0 0

MF) 0 , 0

where M = (mrs)t:,;r,s:,;qhas all entries Oexcept for mij (which is an arbitrary element of IF)and m ii = -mii · The dimension of 9 /.+ 11is dimIRIF. Elements of g /.- Ii have the form A ( 0

0

O O ) 0 0 , 0 -FA*F

where the square block A of size q has all entries Oexcept for a;i, which can be any element in IF.Thus, dima 9 /.- Ii = dima IF. Exercise7.5.2 Obtain the real ranks and the restricted root space decompositions for si,(2n, IR) and si,(2n, C). It may be convenient to use the representation si,(2n, lF) {Xe sl(2n, lF) I X'L + LX O}, where

=

L=

( ~F

~)-

=

7 Semisimple Lie Groups

136

Table 7 .2. Real ranks of the classical noncompactreal Lie algebras Real rank

9 .GU(p, q), p ~ q .GO(p, q), p ~ q

.G,P(p, q), p

~

q q q n

q

.s.p(2n,JR)

[i]

.G0*(2n) .sl(n, IR)

n- 1 n-1

.su*(2n)

Table 7.2 shows the real ranks of the classical noncompactreal Lie algebras.

7.6 The IwasawaDecomposition We have seen in the previous section that if g is a real semisimple Lie algebra stable under the Cartan involution0 and t is the eigenspaceof 0 associatedto the eigenvalue 1, then g decomposesas a direct sum of subspaces g = t EBa EBn, and each subspaceis a subalgebra. Recallthat a is a maximalabelian subalgebra of .p (the eigenspace of 0 associated to the eigenvalue -1) and n is a nilpotent subalgebra. We prove in this section a correspondingdecompositionfor a Lie group with Lie algebra g, called the /wasawa decomposition. Exercise 7.6.J Set G = SL(n, C), K = SU(n). Let A be the abelian subgroup of positive diagonal matrices, and let N be the upper-triangular group with ls on the diagonal. Use the Gram-Schmidt orthogonalization process to prove that the map

K x Ax N--+ G, (k,a,n)

i-+

kan

is a bijection. (Note: If {ei, ... , en) denotes the standard basis of en and g e G, apply the orthogonalization process to {ge1, ... , gen). You'll get an orthonormal basis {u1, ... , Un) such that the linear span of {ge1, ... , get) is equal to the linear span of {u1, ... , u;) and u; e JR+(ge;) + span{v1, ... , v;_i) for i = 1, ... , n. Define k by k- 1u; =e;,foreachi.)

Lemma 7.6.2 Let L be a Lie group with Lie algebra l, and suppose that I has a vector space direct sum decomposition l = u EBt>, where u and t> are Lie subalgebras. Let U and V denote subgroups of L with Lie algebras u and t>, respectively.Then the map M :U x V

~

L, (u, v)

~

uv

is everywhereregular(i.e., its differentialis everywherea surjectivelinear map).

7.6 The lwasawa Decomposition

137

Proof. Let (X, Y) E Tcu,v)(Ux V) be such that X = f,l,=oexp(tXo)u and Y=

f, l,=oexp(t Yo)v, where Xo E u and Yo E l1. Then

DMcu.vJ(X,Y) = dd t

I

exp(tXo)u exp(tYo)v = (Xo + Ad(u)(Yo))uv.

t=O

If DMcu,v)(X, Y) = 0, then Xo + Ad(u)(Yo) = 0, and hence Yo= -Ad(u- 1) (Xo) lies in u. Therefore Yo = 0, Xo = 0, proving that Mis everywhere an immersion. Since the dimensions of the domain and range are the same, M is everywhere regular.

D

Exercise7.6.3 Showthat the exponentialmap is a diffeomorphismfrom the subalgebra of g l(n, R) of upper-triangularmatriceswith Oson the diagonal(resp.,the real diagonal matrices)onto the subgroup of G L(n, R) of upper-triangularmatrices with ls on the diagonal(resp., the positivediagonalmatrices). (Hint: For n = I the result is clear. For n ;:: 2, write an arbitraryupper-triangularmatrix with Is on the diagonal in the form

where U1 is an upper-triangularsquareblock of size n - 1 with Is on the diagonaland x is a row vectorof dimensionn - 1. Use inductionto argue that U 1 = exp N 1, for some nilpotentupper-triangularN1 of size n - 1, and show that U = exp N, where N=

(~

x~~I)

Theorem 7.6.4 (lwasawa decomposition) Let G C GL(n, C) be a E>-stable semisimple Lie group, let g = t EBa EBn be the Iwasawa decomposition of the Lie algebra g of G, and let A and N be the connected subgroups of G with Lie algebras a andn, respectively. Then the multiplication map K x Ax N

~

G, (k,a,n)

1-+

kan

is a diffeomorphism onto G. The groups A and N are simply connected.

Proof. We first show the result for G = Ad( G), regarded as the closed subgroup Aut(g)° C GL(g), and then lift the decomposition to G. Let K = Ad(K), A= Ad(A), and N = Ad(N). Proposition 7.4.6 implies that for some basis of g, the matrices representing elements of K are rotation matrices, those representing elements of A are positive diagonal matrices, and those for N are upper triangular with ls on the diagonal. The group K, hence also K, is compact.

138

7 SemisimpleLie Groups

The groups A and N are closed subgroups of G and A and /1/are closed subgroups of (;. In fact, let A and N be the closures in G, which are connected Lie groups with Lie algebras denoted a and ft. Since Ad(A) (resp., Ad(N)) is contained in the space of diagonal (resp., upper-triangular, with ls on the diagonal) matrices, so is Ad(.A) (resp., Ad(N)). It follows that ad(a) (resp., ad(ft)) is contained in the space of diagonal (resp., upper-triangular, with Os on the diagonal) matrices. Therefore, ad(ii) n ad(ft) = 0, whence an ft= 0. Moreover, t n (a EBft) = 0, since for any X in the intersection, ad(X) is upper triangular with real diagonal entries such that ad(X)t = -ad(X). Therefore, g is the direct sum of t, a,and ft. It follows that a = a and n = ft and A = A, N = N. Since G modulo the finite center Z is diffeomorphic to its (closed) image in GL(g), the groups A, N, Kare also closed. The map from Ax ii into(; given by (a, ii) 1-+ anis clearly one-to-one since a can be recovered from the diagonal entries of the product. Moreover, Aii is a group, since /1/is normalized by A. It is, in fact, a closed subgroup of G: if amnm - x and ais the diagonal matrix with the same diagonal entries as f, then ammust converge to a E A and nmmust converge ton=a-1.xE /1/, so x lies in Aii. Note that the Lie algebra of Aii is ll EBn and that the multiplication map A x ii - AN is a bijection, so by lemma 7.6.2, A x ii and AN are diffeomorphic. The image of the multiplication map K x AN - G is the product of a compact set and a closed set, so it is closed. It is also open since by the previous lemma the map is everywhere regular. But G is connected, so the multiplication map

KxAxil-(; is surjective. The map is also injective since any rotation matrix with positive eigenvalues must be the identity. But a smooth everywhere regular bijection must be a diffeomorphism. The map Ad : G - (; is a finite covering sending A onto A and N onto ii, and the kernel of Ad is contained in K. A is simply connected by the Cartan decomposition, since it is diffeomorphic to its Lie algebra via the exponential map. /1/is also simply connected, due to exercise 7 .6.3 and the fact that it is a subgroup of the group of upper-triangular matrices with Os on the diagonal. Therefore, Ad : A - A and Ad : N - ii are diffeomorphisms. The multiplication map K x A x N - G is a smooth regular map, due to the previous lemma. It is also surjective: Given g e G, write g kanand let a and n be the unique elements in A and N, respectively, such that Ad(a) = a

=

7.7 Representations of.sl(2, C)

139

and Ad(n) = n. Then Ad(g(kanr'> = I, so g(kan)- 1 belongs to the center of G. But the center is contained in K, so g = k1an for some k1 e K. The proof will be complete if we show that the foregoing multiplicationmap is injective. Ifkan = k 1a 1n 1, thenk' := (k1 1k = a,a- 1(an 1n- 1a- 1) e AN, so Ad(k') is upper triangular with positive diagonal entries. Since it is also a rotation matrix we must have k' in the kernel of Ad. But Ad is injective on AN, so k' = e. Therefore k = k 1. It also follows that a = a 1 and n = n 1, concluding the proof. D

r

7.7 Representationsof .sl(2, C) Before we can derive further information about the structure of semisimpleLie algebras,it is necessaryto describe the finite-dimensionallinear representations of ..sl(2,C). We begin with a couple of results about linear representations of general semisimple Lie algebras. Recall from section 3.8 that a representation is said to be completely reducible if it decomposes as a direct sum of irreducible subrepresentations. Theorem 7.7.1 (Complete reducibility) Let g be a semisimpleLie algebra, and let p : g ~ gl(W) be a linear representation on a finite-dimensionalcomplex vector space W. Then p is completely reducible. Proof. We give the proof only for ..sl(2,JR.).It will be apparent, however,that the argument is much more general. The main argument that will be used is known as Weyl's unitary trick. Before specializing to ..sl(2,JR.)assume more generally that g is a real, 0stable subalgebra of the general linear algebra. Denote by gc = g ® C the complexificationof g (for example, if g = ..sl(2,JR.),then gc = ..sl(2,C)) and extend p to gc by linearity. Lett EB.pbe the Cartan decompositionof g. Then u := t EBi.p is also a Lie subalgebraof gc, as one easily sees by using the bracket relations

[t, tl

C t, [t,

.Pl C .P, (.p,Pl C t.

(If g = ..sl(2,JR.),then u = ..su(2).) Notice that each vector in u is fixedby 0, so

u is the Lie algebra of a subgroup U of the unitary group. In the general case, it can be shown that U is a compact group; in our special case, U which is seen to be compact as follows. Notice that

= SU (2),

7 Semisimple Lie Groups

140

is homeomorphic to the 3-sphere. It is also simply connected, so the restriction of p to au(2) exponentiates to a representation of SU (2) on W. (We are using here corollary 3.9.9.) Suppose now that W has a subspace S that is sl(2, R)-invariant. By linearity, it is also sl(2, C)-invariant, and by restriction, su(2)-invariant. By exponentiating the representation of the latter algebra, we obtain that S is SU(2)stable. Applying proposition 3.8.2, we obtain an SU (2)-invariant complement S'. Reversing the argument yields that S' is now .&u(2)-invariant, since we can differentiate the group representation to obtain a representation of the algebra. By linearity, S' is invariant under the complexification of su(2), which is .&1(2,C) su(2, R) EBi.&u(2,R), whence it is also invariant under .&1(2,R), by restriction. D

=

Corollary 7.7.2 Let R: G ~ GL(W) be a continuous representation ofaconnected semisimple Lie group on a finite-dimensional complex vector space W. Then R is completely reducible. Proof. This is immediate from the fact that p pletely reducible.

= DRe:

g ~ gl(W) is comD

We choose a basis of .&1(2,R) consisting of

x=(~~).

y

= (~ ~)-

Their bracket relations are: [h, x] = 2x, [h, y] = -2y,

[x, y] = h.

~ gl(W) be a linear representation on a finitedimensional complex vector space W. Then p(h) is diagonalizable.

Lemma 7.7.3 Let p: sl(2, R)

Proof. Let t EBp denote the Cartan decomposition of sl(2, R). Then h spans the one-dimensional Cartan subalgebra a C p. We now apply the unitary trick once again. Notice that i a C su(2) is the Lie algebra of a compact abelian subgroup T of SL(2, C). By corollary 3.8.4, the representation of T obtained from the original representation of sl(2, R), using the same method applied in the proof of the previous theorem, decomposes as a direct sum of one-dimensional representations. Each one-dimensional subrepresentation is also ia-invariant, therefore also a-invariant. D Let p : .&1(2,C) ~ gl(W) be a linear representation, where W is a finite-dimensional complex vector space. By the previous lemma, p (h) is diagonalizable.

7.7 Representations of .sl(2, C)

141

Therefore, W decomposes as a direct sum of eigenspaces

w,.={we WI p(h)w

= Aw}.

Lemma 7.7.4 If we W,.,then p(x)w e W>.+2and p(y)w e W>.-2· Proof. By the bracket relations, we have

= p([h,

p(h)p(x)w

x])w

+ p(x)p(h)w

= 2p(x)w + Ap(x)w = (A+ 2)p(x)w. D

The claim for y is shown similarly.

Theorem 7.7.5 Let p be an irreducible complex-linear representation of .s1(2, (C) on a complex vector space W of dimension m. Then there is in W a basis {wo, ... , Wm-d such that, using n = m - 1, we have

I. 2. 3. 4.

= = =

p(h)w; (n - 2i)w;; p(x)wo 0; p(y)w; W;+(, with Wn+I O; p(x)w; = i(n - i + l)w;-1, with W-1 = 0.

=

Proof. Since dim W < oo, there must exist w,.=f,0 such that W>.+2= 0. By the lemma, p(x)w = 0 for each w e W,.. Let wo be a nonzero vector in that

w,..

Define w; = p(yiw 0 • Then p(h)w; = (A - 2i)w;, by the lemma, so there is a minimum integer n with p(yt+ 1wo = 0. Then wo, •.. , Wn are linearly independent and the first three of the following properties hold: 1. p(h)w;

2. p(x)wo 3. p(y)w; 4. p(x)w;

= (A -

2i)w;;

= O; = W;+1,with Wn+I = O; = i(A - i + l)w;-1, with w_1 =

0.

The last equation can be shown by induction, the case i = 0 being equation 2. To prove the case i + I assuming that the equation is true for i, we write p(x)w;

= p(x)p(y)w;

= p([x,

y])w;

+ p(y)p(x)w;

= p(h)w; + p(y)p(x)w; = (A - 2i)w; + p(y)(i(A = (i + 1)(). - i)w;.

- i

+ l))w;-1

7 SemisimpleLie Groups

142

By the previous equations, the subspace spanned by the vectors w; is stable under p, and hence must be all of W by irreducibility. It remains to show that).. n. We write

=

Tr p(h) Therefore

"E:= 0 ().. -

2i)

= Tr(p(x)p(y) - p(y)p(x)) = 0. = 0, whence)..= n as claimed.

D

Corollary7.7.6 Let p: sl(2, IR) -+ gl(V) be a finite-dimensional linear representation, where V is now a real vector space. Then V has a basis consisting of eigenvectors of p(h), and each eigenvalue is an integer. Proof. Let W = V ® C and decompose W into irreducible subspaces. By the theorem, the restriction of p(h) to each irreducible subspace is diagonalizable with integral eigenvalues. Let {w,, ... , wt} be a basis of W consisting of eigenvectors of p(h). Since the eigenvalues are real and p(h) maps V into itself, it follows that the real and imaginary parts of each w i = u i + iv i are eigenvectors associated to the same eigenvalue of w i · Clearly, {ui, vi I j = 1, ... , !} spans V, since {uj + ivi I j = l, ... , !} spans W = V ® C. Pick D now a basis for V from among the vectors u i, vi. Exercise7.7. 7 Use the corollary and the proof of the theorem to show that a finitedimensional real representation p : sl(2, IR) ~ gl(V) decomposes as a direct sum of irreducible representations, and in each irreducible subspace there is a basis with respect to which p(h), p(x), and p(y) take the form described in the theorem.

It turns out that the preceding theorem imposes severe constraints on the structure of general semisimple Lie algebras and is a key ingredient used in their classification. We finish this section by indicating how representations of sl(2, C) bear on the problem of understanding the structure of a general semisimple Lie algebra. The key remark is that to each restricted root of ( a, g) and each nonzero X E 9;,, there is a copy of sl(2, IR) in g containing X, hence a representation of sl(2, IR) by restriction of the adjoint representation of g to that copy of sl(2, JR). In what follows, g will denote a 0-stable semisimple Lie subalgebra of gl(n, C). Lett EB.pbe the Cartan decomposition associated to 0 and let a c .p be an IR-split Cartan subalgebra in .p. Recall that a = .pn g 0 • Given anyµ, Ea*, let Hµ E a be the element such that µ,(H) = {H, Hµ)n for each H e a. By duality, a* acquires an inner product defined by

for all µ,, ).. E a*. The norm associated to {·, ·).- will be denoted simply by II• 11.

7.7

Representationsof .&((2, C)

143

Proposition7.7.8 Let J,. e cl>(a,g) be any root and choose X). e g). such that llx).11= J2/IIAII. Define y). := -0x). e 9-). and h). := 2H)./IIAll2 e a. Then [X).,y).] = h)., [h)., X).] = 2x)., [h)., y).] = -2y)..

Therefore, the linear span of {x)., y)., h).} is a Lie subalgebra isomorphic to .&((2, JR).

Proof First notice that 0[X)., y).] = [0X)., 0y).] = [-y)., -X).] = -[X)., y).];

therefore [x)., y).] e [g)., g_).] n .pc g0 n .p= a. Moreover, for each H e a, ([x)., y).], H)" = (ad(0x).)X).,H)" = -(ad(x).)tx)., H)" = -(x)., ad(x).)H)" 2 = (2H)./IIAll2, H)"; = (x)., [H, X).])" = J,.(H)llx).11 therefore [x)., y).]

= h).. We also have

A similar computation gives the bracket relation involving y). and h)..

D

It follows from the definition of h). that if Xis any element of 9µ, then [h

).,

X]

= 2 (J,.,µ)" X

(J,.,J,.)" .

By theorem 7. 7 .5 and exercise 7. 7. 7 we conclude, in particular, that 2 (J,.,µ)" / (J,.,J,.)" e Zand it can be shown that only a small number of integer values can occur. This observation points to the fact that there are severe constraints on the way the set of roots can lie inside a*. It will be seen in the next section, in fact, that the set of roots has a high degree of symmetry. The classification of semisimple Lie algebras hinges on the study of the geometry of the set of roots relative to the inner product(·,·)". For details, we refer the reader to [17]. Exercise 7.7.9 Draw a diagram describing the vectors Ha for each root a of .sl(3, JR) inside the subalgebra 11of diagonal matrices with trace 0. (Identify 11with JR2 and (·, ·)" with the Euclidean metric. There are six roots, which are the vertices of an hexagon.)

144

7 Semisimple Ue Groups

7.8 The WeylGroup We continue to use the notation of the previous section. Let a e a• be a root of (a, g) and let Ha e a be the dual vector, so that a = (Ha, ·)K. Denote by ra ; a ~ a the orthogonal reflectionin the hyperplane perpendicularto Ha, that is, ra is the identity map on H;- and sends Ha to -Ha. It is easy to check that ra(H) := H - 2 (Ha, H)K Ha (Ha, Ha)K

for each H e a. The set of reflections ra, a e (a,g), generates a group of orthogonal transformationsof a, which we denote W'. Proposition 7.8.1 Let G be a semisimple0-stable subgroupof GL(n, C), and let K be the compact group consistingof fixedpoints of 0. Denote by NK(a) the subgroup of K that stabilizes a. Then for each root a e (a,g), there exists ka e NK(a) such that ra = Ad(ka)l11• Proof. DefineZa := !(xa - Ya) and notice that Oza = Za, so that Za e t. Set ka := exp(za) e K. We claim that ra coincides with conjugating by ka. In order to prove the claim it is sufficientto check that conjugationby ka sends Ha to - Ha and fixeseach H in the orthogonalcomplementof Ha. It will follow, in particular,that kais in the normalizerof a in K. If H e H;-, then ').(H) = 0 and ad(za)mH = 0 for each positiveinteger m. Therefore,Ad(ka)H = exp(ad(za))H = H. Set ta = !(xa + Ya). Then,

Ad(ka)Ha

= exp(ad(za))Ha 00

=L

n=O

1

1 ad(zat Ha n.

= (1-1r22! +,r4 4! = COSHHa-

-···)H- llall2(1r-1r3 +···)t 1r 3! a

a

sin1r

llall2 --ta 7r

D

=-Ha.

By duality,the reflectionra is also definedon a•, and takes the form raµ

=µ

(µ,et)K

- 2---a.

(a, a)K

7.8

The Wey/ Group

145

Using the same symbol for the reflectionson a and on a*, we have immediately that raHµ = Hr andµ, o ra = ra(µ,) for eachµ, e a*. 0

Lemma 7.8.2 Given any root a, ra permutes the elements of 4>(a, g). More-

over, Ad(k,;1) : g-+ g maps the root space 9p isomorphicallyonto 9ra

• for each fJ e 4>(a, g). Proof. For each H

e a and each X e 9p we have

[H, Ad(k,; 1)X]

= Ad(k,;')[Ad(ka)H, X] = {J(Ad(ka)H)Ad(k,;')X = {J(raH)Ad(k,; 1)X

= ra(/J)Ad(k,; 1) X. Therefore, ra(/J) is also a root and Ad(k,; 1) maps 9p onto 9ro(/J) as claimed. D We denote W = W(a, g) = NK(a)/ZK(a), where ZK(a) is the subgroup of NK(a) that centralizes a. By the proposition, W contains the group W' of orthogonal transformations of a generated by the reflections ra. It turns out that W = W', although this fact won't be needed later. Notice that the proof of the previous lemma shows that W permutes the roots. We call W the Wey/ group of (a, g). It is shown next that Wis a finite group. Lemma 7.8.3 NK(a)/ZK(a)

is a finite group.

Proof. Since N K (a) is a closed subgroup of K, it is a compact group. In order

to show that the quotient is a finite group, it sufficesto prove that the Lie algebra of NK(a) coincides with the Lie algebra of ZK(a), since this will imply that the dimension of the quotient is 0. The latter Lie algebra is m = g 0 n t. The former is the normalizer of a int, which we denote Ne(a). To see that Ne(a) = m, let X e Ne(a) and write X=Z+H+

L

Xa, ae(a,g)

where Z e m, H e a, and Xa e 9a· Since X e t, it must be fixed by 0, so X = Z + Eae+(Xa + 0Xa), where 4>+is the set of positive roots. Acting on X with ad(H), for H E a, we obtain [H, X] = Eae+ a(H)(Xa - 0Xa), which is also in a since X normalizes a. It follows that [H, X] = 0 for each H e a, so Xa = 0, whence X = Z. D We summarizethe main results of this section in the next theorem.

146

7 SemisimpleLie Groups

Theorem7.8.4 The Weyl group W = W (4, g) of a 0-stable semisirnplelinear Lie algebra g is a finitegroup of orthogonaltransformationsof 4*, or 4, permuting the roots. Each element of W is uniquely determinedby the permutation it induces on (4,g). In particular, W acts on 4* and on 4 with no nonzero fixed points. Moreover,there are elements ka e NK(a.), a e (a., g), such that W is generated by (the duals of) the maps Ad(ka)laProof The only claim that remains to be checked is that each element of W is uniquely determined by the permutation it induces on (a.,g). As pointed out already, W actually permutes the roots, so it sufficesto check that a.*is the linear span of the set of roots. Let V denote the linear span of cl>( a.,g) in a.*. Then the orthogonalcomplement y1. of Vin a.* is fixed by all the reflections ra. Any vector H e a. that is dual to some µ, e V must then be in the kernel of a. Any such vector must commute with all the root spaces, so it is in the center of g, which is trivial since g is semisimple. Therefore, V = 0, proving the claim. D Exercise7.8.5 Let{ei, ... , en}be thestandardbasisof Rn anddenoteby Fi.1 thelinear automorphism of Rn that sendse; to e1 and e1 to -e 1 and fixesall the otherek. Show

that Fi.1 is an elementof Nso(nl(a), wherea is the subalgebraof diagonalmatricesof trace0. Therefore,each F1•1 producesan elementof W(a, sl(n, R)). Describehow Ad(Fi.1) : sl(n, R) -+ sl(n, R) actson therootsof sl(n, R). (Recallthateachrootcan be writtenas a;1 := / 1 - /;, where/;(diag[a,, ... , an]) = a1.)

7.9 Generationby Centralizers We prove in this section a technical but important result about semisirnpleLie groups of real rank at least 2, which will be needed in chapter 10. The result is that the group is generated by the centralizers of elements in a.. The following assumptions and notation will be in place throughout the section. Let G c G L (n, C) be a connected, semisimple, E>-stablelinear group with finite center. Let g be the Lie algebra of G and let g = t EB.pbe the Cartan decompositiondefined by 0. Let a.be a maximal abelian subalgebra of .p. We order the set ofroots (a.,g) as before, and denote by +(resp., -)the set of positive (resp., negative)roots. Recall that the ordering can be defined as follows. Choose X e a. such that a(X) # 0 for each root a and define a ::::{J if and only if a(X) :::: {J(X). We label the elements of += {a 1, ••• , as} accordingto this ordering, so that

Let n = EBAe4>+ 9A and n- = On. Let K be, as before, the group consisting of all E>-fixedpoints of G, and let M = Z K (a) the subgroup of K centralizing

7.9 Generation by Centralizers

147

=

a. We have seen before that g n- EBm e a EBn. Let N and A be the connected subgroups of G with Lie algebras n and a, respectively. Lemma 7.9.1 Let W be the image of the multiplication map m : N-

Then G

= Ui=I oo

x MAN -+ G.

.

W'.

Proof. By lemma 7.6.2, Wis an open set containing e. Therefore it generates G by lemma 3.7.3. D A much stronger statement than the previous lemma holds, namely, that G W 2• This is due to the fact that W is actually dense in G. (Notice that if W is open and dense, then W n g is nonempty for each g e G, so each g is the product of two elements in W.) The weaker statement given in the lemma will suffice for us. Recall that the exponential map of G restricts to a diffeomorphism between n and N. In particular, for each a e + (resp., -),there is a closed subgroup Ua C N (resp., Ua C N-) with Lie algebra 9a such that exp: 9a -+ Ua is a U-a· diffeomorphism. It is clear that 0Ua

=

w-•

=

Lemma 7.9.2 With the foregoing notation, the multiplication map

is a diffeomorphism onto a subgroup W; of N, for each i normal subgroup of W;+1and Ws N.

=

= l, ...

, s. W; is a

:= 9a, EB· · · e Sa,. Then tl1; is a Lie subalgebra of n and IV; is an ideal of tl1;+1 for each i, l ::: i =::s - l. This is due to the fact that if [Sat, Sa,] is not 0, then ak + a, is a positive root strictly greater than ak and a,, and the fact that [Sat• 9a,1 C 9at+a,· Let W; be the connected subgroup of N with Lie algebra tl1;. Since Ua,+1 normalizes W;, the set W;Ua,+1 is a subgroup of W;+1, which is open in W;+i due to lemma 7.6.2, hence a Lie subgroup. Therefore, as W;+i is connected, W;+1 W;Ua,+1·Notice that W1 W1 Ua,· Therefore, we obtain by induction that W; = W; for each i and Ws = N. That the multiplication map is a diffeomorphism is also easily shown by induction and by the fact that exp : 9a -+ Ua is a diffeomorphism. D

Proof. Define

=

tl1;

=

=

We now introduce the assumption that dim a ::::2; in other words, G has real rank at least 2. For each a e (a,g), let Ha e a be the dual vector to a. The

7 SemisimpleLie Groups

148

orthogonal complement in a (relative to the inner product {·, ·)K) of the line RHa is the hyperplane denoted H;-. Notice that H;- is nonzero by the rank assumption. Since a(H) = (Ha, H)K, we have H;- = ker(a). The centralizer of H;- in G will be denoted

Za = { g

E

I

G Ad(g) H = H for all H e H;-}.

Notice that Za is E>-stable(hence, it is reductive). This is a consequenceof the identity 0 o Ad(g) = Ad(E>(g))o 0 and of the fact that OH = -H for each H ea.

The Lie algebraof Za isJa = Z9 (H;-). Noticethatga iscontainedinJa since [H, X] = a(H)X = 0 for each X E Da and H e H;- = ker(a). Since Ja is 0stable,we also have 9-a C Ja· Therefore,by proposition7 .7.8, Za containsa 0stable copy of .sl(2, R) whose (one-dimensional)Cartan subalgebrais spanned by Ha, Each Za contains Ua and U-a, as well as the group MA. The following propositionis now an immediateconsequenceof the previous two lemmas.

Proposition7.9.3 Each g e G can be written as a product g where,foreachi, I~ i ~ l,g; e Za for some a e cl>(a,g).

= g 1g2 • • • g1,

8 Ergodic Theory - Part II

This chapter is mainly concerned with proving that certain measure-preserving actions are ergodic. Section 1 contains a general discussion of invariant measures on Lie groups and homogeneous spaces. We then introduce, in section 2, a characterization of ergodicity in terms of unitary representations, which is used in section 3 to prove the central result of the chapter, namely, Moore's ergodicity theorem. Moore's theorem asserts that if a G-space X is ergodic with respect to a finite G-invariant measure, where G is a simple noncompact Lie group with finite center, then for any closed noncompact subgroup H of G, the H -action on X is also ergodic. After that, we derive Birkhoff's ergodic theorem and use it in section 5 to indicate, by means of a simple homogeneous example, a method for proving ergodicity of Anosov systems. Finally, we introduce the notions of amenability and Kazhdan's property T.

8.1 InvariantMeasureson Coset Spaces Most of the examples of ergodic transformations given in chapter 2 were actions defined on the n-torus Tn. The simple nature of the space allowed us in those examples to prove ergodicity (with respect to the Lebesgue measure) using the elementary theory of Fourier series. A more general class of actions can be defined as follows. Let H be a topological group and let G and A be closed subgroups of H. We form the coset space H / A, which we regard as a G-space with G-action given by left translations. For example, if H = Rn, A = zn, and G is the group Z corresponding to translations in Rn by the vector (u 1, ••• , Un) e Rn, then the Lebesgue measure on Rn clearly induces a finite invariant measure on the compact space Tn Rn/Zn. If the numbers 1, u 1, •.• , Un are rationally independent, then the proof of proposition 1.2.4 essentially shows that the Z-action is ergodic. See also exercise 2.3.3.

=

149

150

8

Ergodic Theory - Part II

Not all actions admit invariant measures. One example is given in the next exercise. Exercise8.1.l The linear action of SL(2, JR) on IR2 induces a transitive action on the circle S 1, regarded as the space of rays in IR2 issuing from the origin. Show that this action does not preserve any Borel probability measure. (Hint: The Z-action generated by ( ~ : ) "squishes" all points toward the two horizontal rays. This forces a measure invariant under such an element to be supported in the union of these two points. It is possible to choose other elements for which invariant measures would be supported at other points. Therefore, there cannot exist a probability measure invariant under the entire group.)

Before discussing more general actions of G, we consider the action of G on itself by right or left translations, and invariant measures associated to these actions. Denote by L 8 (resp., R8 )thediffeomorphismofG suchthatL 8 g' = gg' (resp., R8 g' = g'g). Haar's theorem asserts that a locally compact group G always admits left-invariant regular Borel (resp., right-invariant) measures and that these measures are unique up to a multiplicative constant. They are called left (resp., right) Haar measures. Exercise8.1.2 Let G be an n-dimensional Lie group and fix a nonzero alternating nform w at T,G and denote by a the unique left- (resp., right-) invariant n-form on G that agrees with w at e. (The existence of O shows, in particular, that G is orientable. We take the orientation compatible with Oas the positive orientation on G.) Define a w. measure µ, on G by setting, for any relatively compact open set A C G, µ(A) := Show thatµ, is a left (resp., right) Haar measure on G.

JA

Exercise8.1.3 Show that for any two nonzero left (or right) Haar measures µ 1 and µ 2 on G there is a constant c such that µ 1 = cµ,2• (Hint: First note that µ 1 is absolutely

continuous with respect to µ 1 +µ 2 , and that the Radon-Nikodym derivative h; is a leftinvariant nonnegative measurable function. Use Fubini's theorem and the fact that G acts on itself transitively to conclude that h1 is constant (µ 1 + µ,2 )-almost everywhere.)

Exercise8.1.4 Show that any Haar measure is positive on nonempty open sets.

Since right and left multiplication commute, given any g e G and left Haar measureµ on G, the measure (R 8 )*µ is also a left Haar measure; therefore, there exists a positive function of g, Aa(g), such that (R 8 ).µ

= Aa(g)µ.

Ao is independent ofµ. The function Ao is called the modular function of the locally compact group G. If Ao = 1, G is called a unimodular group. Exercise8.I .5 Show that l:lGis a continuous homomorphism of G into the multiplicative group JR+of positive real numbers. As a consequence, if G is a connected semisimple

8.1

Invariant Measures on Coset Spaces

151

Lie group, show that G is unimodular. Show that compact, discrete, and abelian locally compact groups are also unimodular groups. (Hint: For the continuity of a, use that for every compactly supported function f on G and all g0 E G we have d(go)

fa J (g)

dµ(g)

= fa f(ggo)

dµ(g).)

Exercise8.1.6 If G is a Lie group, show that its modular function is a(g) = idet(Ad(g))I, where Ad: G-+ 9 is the adjoint representation of G. (Hint: Show that if Xis the leftinvariant vector field on G that corresponds to v E 9, then (R8 ) 0 X is the left-invariant vector field that corresponds to Ad(g)v. Use this to show that if w is a left-invariant n-formon G, withn =dimG, then R;w = (detAd(g))w, foreachg E G.)

Lemma 8.I. 7 If µ 0 is a left Haar measure on G and i : G associates to each g e G its inverse, then

~

G is the map that

d(i.µo) ---=l!!.. dµo

Inotherwords,foreach/

e Cc(G),f f(g)d(i.µo)(g)=

fo f(g)l!!.(g)dµo(g).

Proof If h is the Radon-Nikodym derivative of a measure v with respect to another measureµ, we write v = h · µ. We first claim that i 0 µ0 and the measure l!!. · µ 0 are both right-invariant. This is clear for i 0 µ0 since i o Lg = Rg-1 o i. To show it for l!!. · µ 0 , let/ be any continuous function on G with compact support. Then for each go e G,

la

f(ggo)d(l!!. · µo)(g)

:=

la

f (ggo)l!!.(g) dµo(g)

=[

=

la

=[

=

f (ggo)l!!.(ggo)l!!.(go)- 1 dµo(g) f(g)l!!.(g)l!!.(go)-

1d((Rg 0 ).µo)(g)

f(g)l!!.(g)l!!.(go)-

1l!!.(go)

dµo(g)

la

=[

f(g)l!!.(g)dµo(g) f (g)d(l!!. · µo)(g).

The Radon-Nikodym derivative of one measure with respect to the other must = c > 0. To see be a positive constant, by exercise 8.1.3. Therefore,

fc';;.t~>)

8 Ergodic Theory - Part II

152 that c

= I observe that /J,G

= (i 2).µ,o = i.(ct. · f.l,o) = c(t. = c2 (t.- 1) • (t. · f.l,o) = c2 µ,o.

o i) · (i.µ,o)

D

Theorem 8.1.8 (A. Weil) Let G be a locally compact group and H a closed subgroup. If t.o(h) = t. 8 (h) for all h e H, then G / H admits a nonzero Ginvariant measure. The measure, if it exists, is unique up to a scalar multiple. In particular, if G is a connected semisimple Lie group and H is a discrete subgroup, G/ H admits a nonzero G-invariant measure. Proof. (From [28].) To obtain a nontrivial invariant measure on G/ H it

will

suffice to obtain a nonzero continuous linear functional :Fon Cc(G/ H), taking nonnegative functions into [O,oo), such that :F[f o Lg] = :F[f] for all / e Cc(G/ H). In fact, by the Riesz representation theorem, there is then an (invariant) measureµ, on G/ H such that :F[f] = f018 f dµ,. Define a map I:

e Cc(G)

~

/()e Cc(G/H),

such that /()(1r(g)) := J8 (gh)dµ,H(h). l()(x), x e G/H, is indeed well defined and / ()has compact support as one easily checks. Moreover, given f e Cc(G/ H) ande Cc(G), we have l(f

O

Jr)=

l()f.

We first note that/ is a surjective map. Namely, given f e Cc(G/ H), we can find a relatively compact open set U C G such that its image 1r( U) contains the support off. Choose a nonnegative (real-valued) function>.. e Cc(G) such that>..lu = 1. It is clear that/ (>..)(x) > 0 for all x e 1r(U). Define a function h e Cc(G/ H) as follows: h(x) := { /(>..){x)- 1 f(x),

0,

if x e rr(U), if x ~ 1t(U).

Therefore, f = I (>..)h= I (>..ho1r), where >..ho1r e Cc(G). We claim that for anye Cc(G), I()= 0

la

=>

(g)dµ,o(g)

= 0.

Once the claim is proved we will be able to define :F by

:F[f]

=

la

(g)dµ,o(g),

where is any element in Cc( G) such that / ()= f.

8.1

Invariant Measures on Coset Spaces

153

To provethe claim, choose).. e Cc(G) such that /()..)(1r(g)) = 1 for all g contained in the support of , and suppose that/ ()(x) = 0 for all x e G/ H. Therefore, O = la )..(g)l()(1r(g)) dµ

l = la l = l =l = l la = l = la

la

(g)

0

A(g)

(gh)dµH(h) dµo(g)

A(g)

(gh-1)l!i.H(h) dµH(h) dµo(g)

)..(g)(gh-1)l!i.o(h) dµH(h) dµo(g)

la )..(g)(gh-1)l!i.o(h) dµo(g) A(gh)(g) dµo(g)

la

= la

dµH(h)

(by Fubini's theorem) (by the definition of l!i.0 )

dµH(h)

(by Fubini's theorem)

)..(gh)(g)dµH(h) dµo(g)

/()..)(1r(g))(g) dµo(g)

(since / ()..) o 1r = 1 on supp ).

=la() dµo(g)

Therefore the linear functional F can indeed be defined and we obtain the desired measure. The measure on G / H is G-invariant and is unique up to a multiplicative constant, since that is true for µo. D It follows from the preceding proof that the invariant measure µ on G/ H, when it exists, can be chosen so that { f(g) lo

dµo(g)

=

1{

f(gh)

dµH(h) dµ(gH)

G/HjH

for all f E Cc(G). A corollary of the previous theorem is that for any connected semisimple Lie group G and discrete subgroup r of G, the space G / r always admits nontrivial G-invariant measures. We will later show that the invariant measures of SL(n, IR.)/SL(n, Z) are finite. If a discrete subgroup r of a Lie group G has the property that G / r admits a finite nonzero G-invariant measure we say that r is a lattice of G.

154

8

Ergodic Theory - Part II

The fact that SL(n, Z) is a lattice of SL(n, JR.)is a special case of the following hard theorem. Theorem 8.1.9 (Borel-Barish-Chandra [61) LetG C GL(n, C) beasemisimple algebraic group defined over Q. Then G(Z) = G n G L(n, Z) is a lattice subgroup of G(IR) = G n G L(n, JR.).

8.2 Ergodicityand Unitary Representations Ergodicity can be characterized in terms of the unitary representations of G. We assume that X is a G-space with a finite G-invariant measureµ, and consider H := L 2 (X, µ,), with inner product

Then ( H, (·, ·)) is a separable Hilbert space. (Recall that a Hilbert space is said to be separable if it has a countable orthonormal basis.) For each g e G, denote by rr (g) the linear operator on H defined as follows. To each element in H, represented by a square-integrable function f, the (class determined by the) function rr(g)f satisfies (rr(g)f)(x) := f(g- 1x). As the measure µ, is preserved by G, each rr (g) satisfies the identity (rr(g)f1, rr(g)h)

= U1,Ji).

Therefore, rr(g) belongs to the group ofunitary operators U(H) and we have a homomorphism rr : G - U(H). More generally, let H be a separable Hilbert space and U(H) the group of all unitary operators on H. We give U(H) the strong operator topology, that is, the smallest topology that makes all the maps U t-+- Uf, f e H, U e U(H), continuous. With respect to the strong operator topology, if lUnl is a sequence of elements in U(H), we have limn--->oo Un = U e U(H) if and only if IIUnf - Ufll - 0 for all fin some set D whose linear span is dense in H. U(H) can also be given the weak operator topology. A sequence {Un}converges to U in the weak operator topology if each matrix coefficient converges, that is, limn---> f EH. 00 (Une, f) = (Ue, f) foralle, Exercise 8.2.1 Show that a sequence Un E U(H) converges to U E U(H) in the strong operator topology if and only if it converges to U in the weak operator topology. Also show that, with respect to either topology, U(H) is a metrizable second countable topological space.

8.2

Ergodicity and Unitary Representations

155

Exercise8.2.2 Use Fubini's theorem to show that the homomorphism 1r :

G -* U(l 2(X, µ))

is measurable.

We saw earlier that any continuous homomorphism between Lie groups must be smooth. The next theorem, due to Mackey, shows that it is sufficient that the homomorphism be measurable. Theorem 8.2.3 Suppose that L is a second countable topological group, G is a second countable locally compact group, and p : G -+ L is a measurable homomorphism, that is, a measurable map that is also a homomorphism of groups. Then p is continuous. Lemma 8.2.4 If A C G is a compact set with positive Haar measure, where G is as in the theorem, then AA- 1 contains a neighborhood of e E G. Proof We denote a Haar measure on G by µ. Since A is a compact set of positive measure, we can find an open set W containing A such that µ(W) < 2µ(A). Compactness of A also implies that there is a neighborhood N of e such that Iv is symmetric (i.e., if g E N then g- 1 E N) and gA c W for all g EN. Since µ(gA) = µ(A) and µ(W) < 2µ(A), we have that (gA) n A is nonempty for each g E N, so N is contained in AA - I. D

To prove the theorem, we may assume without loss of generality that p is surjective. Let U be an open neighborhood of the identity in L and V C U a symmetric open neighborhood with V2 C U. Let {ln} be a countable dense set in Land gn E G such that p(gn) = ln. Then Lis the union of the ln V, so G is the union of the 8nP-• (V). It follows that for some n, µ(gnp- 1(V)) > 0 so µ(p- 1(V)) > 0. Since p- 1(U) contains p- 1(V)p- 1(V), it contains in particular a subset K K- 1, where K is compact with positive Haar measure. The theorem now follows from the previous lemma. It follows from the theorem that the representation T( : G -+ U ( L 2 ( X, µ)) is continuous. Thus, we have a unitary representation of G into U(L 2 (X, µ)). (A unitary representation of G is defined in general as a continuous homomorphism of G into the unitary group of a separable Hilbert space.) Let Ho be the (G-invariant) orthogonal complement in L 2 (X, µ) to the subspace C of constant functions. If the G-space Xis not ergodic, then the characteristic function of a G-invariant measurable set that is neither null nor conull defines a nonzero vector in Ho fixed by G. The converse is given in the next proposition.

156

8

Ergodic Theory - Part II

Proposition 8.2.5 If X is a G-space with finite invariant measure µ, and 1r is the unitary representation of G on the orthogonal complement Ho of C in L2 (X, µ), then Xis ergodic if and only if there are no nonzero 1r(G)-invariant vectors in Ho. Proof. We need to show that if the G-space is ergodic then its unitary representation does not have nonzero fixed vectors in H0 • Let / be a fixed vector in Ho. It then corresponds to a (square-integrable)measurable G-invariant function on X relative toµ. By proposition 2.3.1, there is a strictly G-invariant

measurablefunction j that is µ-almost everywhereequal to /. Therefore j is constant µ-almost everywhere,by the definition of ergodicity. It follows that f represents an element in Hon C = {O}. D

8.3 Moore'sErgodicityTheorem Wediscuss now a result due to C. C. Moore concerningergodicityfor subgroups of semisimple Lie groups. The proof of Moore's theorem given here is due to R. Ellis and M. Nerurkar [10). Let X be a G-space with a finite invariantmeasure. X is called irreducible if every normal subgroup of G not contained in the center acts ergodicallyon X. Theorem 8.3.1 (Moore's ergodicity theorem [361) Suppose that G is a semisimple Lie group with finite center and no compact simple factors, and that X is an irreducible G-space with finite G-invariant measure. If H is a closed noncompact subgroup of G, then H also acts ergodically on X.

In view of the characterizationof ergodicity in terms of the unitary representation of G on L2 (X, µ), in order to prove Moore's theorem it sufficesto show the following: For any connected noncompact simple Lie group G with finite center, and unitary representation1r of G with no nonzero invariant vectors, a closed subgroup L of G such that 1rIL has nonzero invariant vectors must be compact. Observethat givena nontrivialL-invariantvector v e H, the function /(g) := (1r(g)v, w) is constant on L. Therefore, the proof will be complete once we show that for all v, we H, (1r(g)v, w) approachesOas g - oo in G. This is the content of the next theorem. Let Gi be a connected noncompact simple Lie group with finite center, and let G = ILGi be a finite direct product. Let H be a separable Hilbert space with inner product (·,·),and 1r : G - U(H) a unitary representationsuch that for each i, 1rlo,has no invariant vectors. Then Theorem 8.3.2 [36, 2.2.20)

8.3

Moore's Ergodicity Theorem

157

for all u, v E H,

Jim (:,r(g)u, v} = 0.

g-+oo

In other words, :,r(g) tends to O in the weak operator topology as g tends to infinity in G. Recall (theorem 7.4.8) that a connected semisimple Lie group G with finite center admits a KA K decomposition, where K is a maximal compact subgroup, A is abelian, and Ad(A) is diagonalizable over JR..(In that theorem, we assumed that G was linear algebraic. We have seen in theorem 7 .2.5, however, that G modulo its center is in a natural way the component of the identity of a real linear algebraic group. Since the finite center is contained in the maximal compact subgroup K, it is easy to see that the KA K decomposition also holds here.)

Lemma 8.3.3 Let :,r be as in the previous theorem. If lim (:,r(a)u, v} = 0

A3a-+oo

for all u, v E H, then the conclusion of the theorem holds.

Proof. By the KA K decomposition each element g of G can be written as g = kak', where k, k' EK and a EA. Let gn = knank~ be a sequence in G such that gn -+ oo in G, that is, gn eventually leaves any compact subset in G. Since K is compact, the condition is equivalent to an -+ oo in A. Let u, v be arbitrary elements of H and suppose for a contradiction that for some subsequence, still denoted gn, there exists E > 0 such that j{:,r(gn)u, v}I :::: E for all n. Passing to yet another subsequence, we may assume that kn and k~ converge to k and k', respectively. Then, from the fact that :,r is continuous in the operator norm, it follows that l{1r(k)1r(an)1r(k')u,v)I :::: E/2 for all sufficiently large n. Therefore, j{:,r(an) :,r(k')u, :,r(k- 1)v)I ::::E/2 for all sufficiently large n, a contradiction. D For any topological group G and a E G, define the stable group of a, G!, as the closure of {g E G I limm-++ooamga-m = e}. Also define the unstable group of a as G~ = G!_, .

Lemma 8.3.4 Let H be a separable Hilbert space and G a locally compact, second countable group. Let :,r: G-+ U(H) be a (strongly continuous) unitary representation of G. Let v be a vector in H that is fixed by an element a e G. Then v is also fixed by all elements in G! and G~.

158

8

Ergodic Theory - Part II

Proof. Suppose that amga-m converges to the identity element and let v be any vector fixed by rr(a). Hence vis also fixed by rr(a)- 1 and, since rr(a) is unitary, we have for all m

Therefore, by continuity, rr(g)v ~~

= v for all g E G!.

A similar argument appli-

~-

D

Lemma 8.3.5 Let G be a connected noncompact semisimple Lie group with finite center, and let A be the abelian subgroup defined in the KA K decomposition. Let H be a separable Hilbert space and ,r: G -+- U(H) a unitary

representation such that no normal subgroup of G not contained in the center leaves invariant a nonzero vector in H. Fix an element a E A, a =I-e, and define W = {v EH

I rr(a)v = v}.

Then W =0. Proof. It suffices to show that W is stable under G since, if this is the case, a would be in the kernel of g E G H- rr(g)lw, whence the kernel would be a noncentral normal subgroup of G fixing W pointwise. Therefore W = 0. We now show that W is stable under G. Notice that the subgroup G of G that stabilizes W is closed, hence a Lie subgroup. If g denotes the Lie algebra of G,then it suffices to show that g = g. The group A is easily seen to be diffeomorphic to its Lie algebra a via the exponential map. (Recall that we can identify a and A with their images under ad and Ad, for the adjoint representations of g and G, respectively, and that ad(a) may be assumed to lie in the subalgebra of diagonal matrices of gl(n, JR), where n is the dimension of G, whereas ad(A) will then lie in the subgroup of diagonal matrices in GL(n, JR)with positive entries.) Therefore, we can write a = exp X for some X E a. Let ( a, g) be the set of roots of ( a, g). The Lie algebra g decomposes as a direct sum

where J is the Lie algebra of the centralizer of a, u- is the subspace of g spanned by the root spaces ga such that a(X) < 0, and u+ is the subspace of g spanned by the root spaces ga such that a(X) > 0. Clearly J C g since any element that centralizes a must stabilize W. On the other hand, u- (resp., u+) must be

159

8.3 Moore's Ergodicity Theorem

contained in the Lie algebra of the stable (resp., unstable) group of a, since for each Y e Da,a e 4>(a,g), and c = exp U e A, we have c(exp Y)c-

1

= exp(Ad(c)Y) = exp(eadY) = exp(ea(U)Y).

It is now a consequence of the previous lemma that u+, u- C

9. Therefore

g=g.

D

We recall the following basic fact, known as Alaoglu's theorem. Denote by B(H) the space of bounded operators on a separable Hilbert space H and let B denote the unit ball in B(H). Then Bis compact in the weak operator topology.

Lemma8.3.6 Denote by ,r(A} the closure of the subset ,r(A)

c

U(H)

c Bin

the weak operator topology of B(H). Then

Proof. For each a e A, it is immediate to verify that the map U

e B(H)

i-+

,r(a)U

e B(H)

is continuous in the weak operator topology. Therefore, the preimage of ,r(A) under this map is closed. That preimage contains ,r (A) so it also contains ,r (A). Since a is arbitrary, we have ,r(A) ,r(A) C ,r(A).

Apply now the same argument to the map U e B(H)

1-+

UT e B(H),

where T is an arbitrary element of ,r(A). We conclude that ,r(A) T c ,r(A) for all T e ,r(A), which is the claim. D

Exercise8.3. 7 Let Tmbe a sequence of unitary operators on a Hilbert space H converging weakly to a bounded operator T. Let Umbe another sequence of unitary operators on H converging strongly to a bounded invertible operator U whose inverse u- 1 is also bounded. Show that the sequences TmUmand UmTmconverge weakly to UT and TU, respectively.

160

8 Ergodic Theory- Part II

If T is a bounded operator on a Hilbert space H, its adjoint operator T* is defined by the identity {T*u, v)

= {u, Tv)

for all u, v e H. It is a basic fact in functional analysis that the identity indeed defines a bounded operator on H and that IIT* II = IITU.Moreover, the map T e B(H) i-+- T* e B(H) is easily seen to be continuous in the weak operator topology. Although not necessary, we make now the simplifying assumption that our semisimple Lie algebra is linear and 0-stable, where 0 is the standard Cartan involution introduced in chapter 7. Recall (proposition 7.7 .8) that if Y is a nonzero root vector in .9a, then Y and 0(Y) generate a three-dimensional subalgebraof g, which is isomorphicto sl(2, JR),and that [Y,0(Y)] e a. We now begin the proof of theorem 8.3.2 (keep in mind lemma 8.3.3). Let am e A be a sequencetending to infinity such that ,r(am) convergesin the weak operator topology to an operator T e B. We want to show that T = 0. We write am = exp(Xm).By choosing a subsequence,we may assume without loss of generality that the line JRXmconverges to JRX C a, where X is a nonzero element of a. Clearly there must exist a e (a,g) such that a(X) :f:.0, since otherwise (having in mind the root space decompositionof g) X would commute with every element of g. But a semisimple Lie algebra has trivial center, which would be a contradiction. Therefore, we may select a root a such that a(Xm) tends to -oo as m tends to +oo. (Recall that -a is a root if a is a root.) Choose a nonzero Y e Da and denote Ua(t) := exp(tY).

Then, for all t e JR,we have (by using the identities that appear at the end of the proof of lemma 8.3.5)

Recall that ,r(am) convergesweakly to T and observe that :,r(ua(eat)) as well as ,r(u-a(eat)) converge strongly to the identity operator as m tends to oo. Therefore, we can apply exercise 8.3.7 to conclude T,r(ua(t))

= T,

for all t e JR.Of course we also have T*T,r(ua(t)) = T*T. These equations show that T*T and T cannot be invertible; if they were, the kernel of ,r in G would contain the infinite group generated by the elements ua(t), whence it

8.3

Moore's Ergodicity Theorem

161

could not be contained in the finite center of G. (Recall that we are assuming that no noncentral nonnal subgroup of G has invariant vectors under the representation 1r.) Denote S = T*T E 1r(A) 1r(A) c 1r(A). Since S is not invertible, it must belong to the boundary 1r(A) - 1r(A). Therefore, Sis a weak limit for a sequence 1r(a~), with a~ E A tending to infinity. We now repeat for S the same argument that was used for T. Namely, write a~ = exp(X~), and assume after passing to a subsequence that the line !RX~ converges to !RX' for some nonzero X' E a. Fix a root a' such that a' (X') < 0. Choose a nonzero Y' E 9a' and denote now ua,(t) := exp(tf')

and

u_a,(t) := exp(t0Y').

Then, for all t E IR,we have as before

Applying 1r and passing to the limit m -+ oo, we conclude, as we did for T, that

Notice that

Therefore, we have

for each t E IR. The closure in G of the group generated by ua,(t) and u_a,(t), t E IR, is a Lie group locally isomorphic to SL(2, IR), and its Lie algebra contains 0 cf; [Y', 0Y'] E a (proposition 7.7.8). By strong continuity of 1r, we have 1r(a))S = S, where a := exp([f', 0Y'] e A. We claim that S = 0. Notice that 1r(a) fixes each element in the image of S in H, so W := {v E H I 1r(a)v = v} contains that image. By lemma 8.3.5, W = 0, so S is indeed 0. On the other hand, S = T* T, so for each v e H 0 = IISvll

= (T*Tv,

v)

= (Tv,

Tv)

= IITvll,

whence T itself must be zero. This concludes the proof of theorem 8.3.2, and with it the proof of Moore's theorem.

162

8

Ergodic Theory - Part 11

Corollary 8.3.8 Let G be a simple noncompact Lie group with finite center, and let r be a lattice in G. Then any closed noncompact subgroup L of G acts ergodically on G / r by left translations. Proof. G clearly acts ergodically on G / r, since the action is transitive. By Moore's theorem, the L-action must also be ergodic. D

8.4 Birkhoff'sErgodicTheorem The next theorem is a fundamental result in ergodic theory. We state the theorem for JR-actions; a similar statement holds for Zr-actions, for which the integrals J{ (·· ·)dt are replaced with summations¾

i

E7.=-J} and {L}2>} will be said to be cohomologous if there exists a tempered sequence {;E GL(n, R) I i e Z} making the following diagram commute:

!~1+1

i~l-1 ...

L 121

~

L 121

Rn~

L 121

L 121

Rn......!.......+ Rn~•••

It is easy to check that if two sequences arecohomologous and one of them is tempered, then the other one also is. We will refer to cohomologous sequences· simply as equivalent. (Later on, our sequences of matrices will be interpreted as the values of a "cocycle over a group action," and in that context the term "cohomologouscocycles" is standard usage.) Lemma 9.2.2 Let L; be a tempered sequence of lower-triangular matrices in G L(n, R), and let A (m) be as defined before. We denote by a;i (m) the entries of A(m). Suppose that, for some x e Rn and each j, 1 ~ j ~ n, we have limm.-.±00¾lnlaii (m) I = x. Then, for each nonzero v e Rn,

lim _!_lnllA(m)vll m.-.±oom

= X·

,

This holds for an arbitrary norm II· IIon Rn. Moreover, if {L}1>} is another sequence equivalent to {L;} and A Cl) (m) is the sequence of m-fold products of the L}1>,then the corresponding limit exists and takes the same value x. Proof. We denote by l;j(m), l;i(m), and a;j(m) the entries of Lm, L-;,1, and A(m), respectively. Notice that, form ~ 1, A(m) is given recursively by A(m) = Lm-1A(m-l)andA(-m) = (L-mr 1A(-m+ 1).Sincethematrices are lower triangular, it follows that

a;;(m)

= lu(m

- l)a;;(m - 1),

a;;(-m)

= lu(-m)au(-m

+ 1),

9.2 Products of Triangular Matrices whereas, for 1 ::::j < i ::::n, we obtain aij(m)

=

and a;j(-m) L~=/;k(-m + l)akj(-m by the previous two, we obtain aij(m) a;;(m) aij(-m) a;;(-m)

= aij(m

- 1) a;;(m - 1)

= L~=j

l;k(m - l)akj(m - 1)

+ 1). Dividing these two equations

i-1

+ ~ l;k(m

- 1) akj(m - 1),

(9.3)

{;;; l;;(m - 1) a;;(m - 1)

i-1 = aij(-m + 1) + ~ ~k(-m)

a;;(-m

181

+ 1)

{;;; l;;(-m)

akj(-m a;;(-m

+ 1).

(9.4)

+ 1)

We claim that the absolute value of aij(m)/a;;(m) increases, as 1ml tends to oo, slower than eclml,for an arbitrary positive constant c. To show the claim, notice first that since lim; .....± 00 lnllLf 11= 0, there must be for each positive E a constant C ::::1 such that

f

for all m e Z. Moreover, since l;;(m)- 1 is the (i, i) entry of L-;.1, we have ll;;(m)- 11 ::: IIL-;.111:5 CeElmland ll;;(m)I ::: IILmII ::: CeElml. Therefore,

for all m e Z. Consequently, it follows from (9.3) and (9.4) that a;j(m) - a;j(m - 1) < C2e2Elml~ ,akj(m - 1) l a··(m) a11.. (m - 1) L,.; - 1) ' 11 k=j a··(m II

I

a;j(-m) l a .. (-m) 11

- a;j(-m aII.. (-m

+ l) + 1)

I

I
0, as m ~ ±oo, since by assumption . hm

1 lajj(m)I -In --

m->±oo m

a;;(m)

. = m-.±oo hm

1 -lnla m

.. (m)I-

11

. hm

I -lnla;;(m)I

m->±oo m

= X - X = 0.

Ll:~

Using inequalities (9.5) and (9.6) and eEi = C' eElml,we obtain the claim by an induction argument, having in mind that E > 0 is arbitrary.

9

182

Oseledec's Theorem

Fix an index i and define the matrix a(m) = (aki(m)/a;;(m)). The next claim is thatlimm-+-±oo¾lnlla(m)II = 0. Recall that, for some constant C ~ 1, we have by exercise 9 .2.1

~ lla(m)II ~ cf:max{lakj(m)l

~ldeta(m)I¾

11

~ j ~ n}.

k=I

Since a(m) is lower triangular, for any positive E there is a constant C~ ~ l such that

Therefore, 0

~ liminf-

~ limsup-

1-lnlla(m)II 00 lml m-+-±

1-· lnlla(m)II m-+-±oo1ml

~ E.

Since E is arbitrary, the second claim follows. In particular,

.

l

lim - lnllA(m)II m-+-±00 m We repeat the preceding argument for A (m )of A(m)- 1• Notice that a;i(m)

= a;j(m

aii(m) aii(-m)

- 1)

+

ajj(m - 1)

t t

iijj(-m

+ 1)

1•

Let aii (m) denote the entries

a;k(m - 1) !kj(m - 1),

k=i+• aii(m - 1) lii(m - 1)

= a;j(-m + 1) +

iijj(-m)

= X·

ii;k(-m

k=i+I iijj(-m

+ 1) lki(-m) + 1) ljj(-m)

replace (9.3) and (9.4) is this case. We proceed along the lines of the foregoing argument, with the obvious modifications. For example, in the induction process the index j starts at i - l and decreases by l at each step. Since the diagonal entries of A(m)- 1 are aii(m)- 1, we obtain lim lnllA(m)m-->±oo

1 11

= -x.

Finally, the inequality 1 s IIAl~:tll IIA(m)- 1 11-

s

IIA(m)II,

1 1 UA(m)vU · l"1es 1·1mm-->±oo v ,..J. 0 , imp ;;, n llvll = X, cone 1ud"mg th e proo f .

D

9.2 Products of TriangularMatrices

183

We now consider a tempered sequence of lower-triangular L; e G L(n, JR) and A(m) = (aij(m)) as before, such that the limit lim _!_lnlajj(m)I m-+±oom

= lnlJ..jl

exists for each 1 ::: j ::: n, but may talce different values for different j. If X;, 1 ::: i ::::k, are the distinct values of lnlJ..jl, we would like to know whether there exists an equivalent lower-triangular sequence that has blockdiagonal form, with k block matrices on the diagonal, such that for all the diagonal entries of the ith block on the diagonal the preceding limit is X;, As an example, notice that the constant sequence L}1> = (~~)is equivalent to the sequence given by L}2> = (~ ~)and the equivalence is realized by

;= (7

I 0

I).

We first consider the two-dimensional case.

Lemma 9.2.3 Let L;

= c:;:~g ~i)) e GL(2, JR)be a tempered sequence such 122

that lim

~ lnl/11(i)I = 1-+±00 .lim ~l lnll22(i)I = 0,

i-+±00 l

and denote by L~the diagonal matrix with the same diagonal entries as L;. Let A(m) (aij(m)) be them-fold product of the L;, as defined before. Suppose that the limits

=

. 1 hm - lnla11(m)I m-+±oom

= Xt,

= x2

lim .!..1n1a22(m)I m-+±oom

exist and that Xt < x2. (If Xt > x2 we consider instead the sequence L 11.) Then L 1 and L 1 are equivalent and there exists a unique tempered sequence

;= ::1:oo . -11- 1tnllLmll- m->::l:oo . 1 hm hm -lnll22(m)I 1ml

. l tn - -(m) hmsupm->::1:oo Im1 122(m)

~

m

=0.

From this and (9.8), it follows that

I

. 1 1a21(m+ 1) a21(m) hmsup- 1 - 1tn ( l) - -(-) ~ ±(x1 - x2), m->::1:oo m a22 m + a22 m

Therefore, a21(m)/a22(m) is a Cauchy sequence form limit by c0 • Notice that

I

. sup -1 ln 1a21(m) lim -(-) - co ~ m->+00m a22 m

x, -

~

(9.9)

+oo. We denote its

X2-

Therefore

I

. sup -1 lnlcmI = hm . sup -l In1a22(m) ( co - a21(m)) hm -(-) -(-) ~ 0. m->+oom m->+oom au m a22 m

(9.10)

In order to prove that ;is tempered, it remains to show that .

I

hm sup -1 - 1 InlcmI ~ 0. m-++oom

(9.11)

Using (9.9) and a telescoping series argument,we obtain

I

. I- In-(-) 1a21(m) hmsupm-+-oo1m 1 a22 m

~

x2-

x,.

(9.12)

9.2 Products of TriangularMatrices

185

From this and (9.7), we obtain . 1 hmsup-lnlcml.:::

. 1 hmsup-ln

m-+-oo 1ml

m-+-oo 1ml

Ico- -a21(m)I a22(m)

lim - 1- lnla22l - lim - 1-lnla11 m-+-oojml m-+-oolml _:::(x2 - x1) + XI - X2 0.

+

I

=

D

Therefore, the sequence ;is tempered, concluding the proof.

Next, we generalize the previous lemma for arbitrary n. The general case will, in fact, be reduced to n = 2, after we make the following definitions. Let P = (P 1, ••• , P 1) denote a partition of {1, 2, ... , n} into l disjoint subsets. We denote by P(i) the element of the partition that contains i and define

G(P, i)

= {(ars) E GL(n, JR)such that ars = Oifr < sorbothO < r -s < i andP(r)-=/; P(s)}.

Exercise 9.2.4 Show that G('P, i) is a closed subgroup of the group oflower-triangular matrices in GL(n, IR) and that G('P, I) :::>G('P, 2) :::>• • • :::>G('P, n).

Notice that G('P, l) is the full group of lower-triangular matrices and G('P, n) is the subgroup of lower-triangular matrices (a,,) such that a,, = 0 if 'P(r) =I='P(s).

Let r, s, I .::: s < r .:::n, be such that r - s define the map Prs : A

= (a;j)

E

G(P, io)

1-+

( ass ars

= io and P(r) O) arr

E

-=/; P(s). We

GL(2, JR).

For the same r and s, we also define the map

that sends(! ~) to the matrix with ls on the diagonal, a in the (r, s) entry, and 0 in all other places. Exercise 9.2.5 Show that p,, is a group homomorphism. Also show that left or right multiplication of any (uij) e G('P, i 0) by t(A) can only affect the entries u,j, j ~ s, or u;,, i ,:: r. In particular, right and left multiplication by t(A) map G('P, i) onto itself for each i < io.

9

186

Oseledec's Theorem

Lemma 9.2.6 Given a tempered sequence Lm of lower-triangular matrices in GL(n, 11), let A(m) = (aij(m)) be them-fold product of the L; as defined earlier. Suppose that liIDm-+±ooOnla;;(m)l)/m exists for each i, 1 ::::i ::::n, and let x, < x2 < · · · < Xk be the different values of these limits. Also suppose that limm-+±ooOnll;;(m)l)/m = 0 for each i. Let 'P = ('P1, ... , A) denote the partition of {1, 2, ... , n} such that

r

E

'P;iff lim (lnla,,(m)l)/m m-+±oo

= Xi·

Then Lm is cohomologous to a tempered sequence in G('P, n) and the tempered sequence m that realizes the equivalence can be taken so as to lie in the group of lower-triangular matrices with ls on the diagonal.

Proof. The proof is by induction, making use of the previous lemma and exercises. Clearly, L; E G('P, 1), since this is the full group of lower-triangular matrices. Suppose that L; E G('P, k) for some k, 1 ::::k:::: n - 1, and fix s, r, 1 ::::s < r ::::n, such that r - s = i and 'P(r) ,f. 'P(s). Applying the previous lemma to the sequence P,s(L;), we obtain a tempered sequence ii,; = (;; ~) that realizes the equivalence between P,s(L;) and the sequence of diagonal matrices with the same diagonal elements as P,s(L;). Set \II; = t,s(il,;). Then \II; realizes the equivalence between L; and a sequence L)0 whose (r, s) entry is 0 and entries other than (r, j), for j ::::s, and (i, s), for i ::: r, are the same as the corresponding entries of L;. By repeating this operation a finite number of times we obtain an equivalence between L; and a tempered sequence in G('P, i + 1), the equivalence being realized by a sequence with ls on the diagonal. The proof ends when i + 1 = n. D Let L; and 'P be as in the previous lemma and denote by L; the equivalent sequence that the lemma yields. Let A(m) be the corresponding m-fold product. By permuting the elements in the standard basis {e1 , ••• , en} of Rn,we may assume that

'P, = {1, ... ' Ii},

A

= {lk-t + t, ...

, n}.

Therefore, we may assume that L;, hence alsoA(m), has block-diagonal form and that within each diagonal block ofA(m), limm-+±ooOnla;;(m)l)/m has the same value for all i. The results obtained so far imply the following proposition.

Proposition 9.2. 7 Let Lm be a tempered sequence of lower-triangular matrices in GL(n, 11), and let A(m) = (aij(m)) be, as before, them-fold product of

9.3

Proof ofOseledec's Theorem

187

the L;. Suppose that limm-.±ooOnla;;(m)l)/m exists for each i and let x1 < x2 < · · · < Xk be the different values of these limits. Also suppose that liffim-.±oo(lnll;;(m)l)/m = 0 for each i. Then we can find for each 1 :::::l :::::k a sequence of subspaces V,(i) c Rn such that the following hold

=

V,(i + I), for each l and each i e Z. 1. L; V,(i) 2. R_n = Vi(i) EB· · · EBVk(i), for each i e Z. 3. For each I :::::I :::::k and each nonzero v e V,(0), we have . 1 bm - lnllA(m)vll m--.±00m

= XI·

4. Given a nontrivial partition of {1, 2, ... , k} into sets I and ZC,let 0; be the V,(i) and V,(i). Then angle between

EB,ez

EB,ez•

lim _!_In sin 0m = 0. m--.±00m

Exercise9.2.8 Show that the previous proposition indeed follows from the lemmas proved earlier.

9.3 Proof of Oseledec's Theorem Let p

:M - M be a fiber bundle over M with compact fibers, and suppose that

J :M - M is an automorphism of M covering a map f : M - M, that is, such that f op = rr o J. We call J a compact extension of f. By exercise 2.1.5, ifµ, is an !-invariant probability measure on M, there exists an ]-invariant probability measure µ on M such that ;;.p, µ,. Ifµ, is ergodic, we can choose µ to be ergodic. If rr : E - M is a vector bundle, we define the pull-back of E to M as follows (see section 6.1):

=

ft: E := p* E = {(i, v)

eM x EI p(i) = rr(v)} -M,

where ft (i, v) = i. A norm or inner product on E induces a norm or inner product on E in a natural way, since the fibers of E are actually fibers of E. Moreover, if F : E - E is an automorphism of E covering f : M - M (so that rr o F = f o rr ), we obtain an automorphism F of E defined by Fx(i, v) := ..is real, set Vi.:= V,.Cn Rn, and if}..and .l are a pair of complexconjugateeigenvaluesof L, set V,i.,;; := (V,.CEBV{) n Rn. If Xi < · · · < X, are the distinct valuesoflnl>..I,we let E.,,1 denote the direct sum of subspaces V,. or V,.,;;such that lnl}..I= Xi· Show that the subspacesEx." 1 ~ i ~ r, are L-invariantand Rn= Ex.,EB··· EBE.,,,. Moreover,for any norm II·IIon Rn and any nonzero v e Ex.,, show that lim ..!.. lnllLmvll= Xi· m-+:1::00 m

Exercise9.3.5 Given A e SL(n, Z), define a diffeomorphismf of then-torus 'Il'n= Rn;zn such that /([x]) = [Ax] for all [x] = x + zn e yn_ Show that the tangent bundle T'Il'ndecomposesas a direct sum of smooth subbundlesL .,,1 , where Xi are the distinct valuesof In l>..Iand}..rangesover the eigenvaluesof A, such that for each x and ~ lnllDfxmvll= Xi· (The norm is induced each nonzero v e Li(x), we have limm-+:l::oo by an arbitraryRiemannianmetric on the torus.) Exercise9.3.6 Let M = G/ r, where G is a connectedsemisimpleLie group and r is a uniformlattice in G. To each X in the Lie algebrag of G, associatethe vectorfield X on M obtainedby projectingto the quotientthe right-invariantvectorfieldon G determined subalgebraof g. Denoteby g,. the by X. Let a be a maximalabelianJR-diagonalizable root space associatedto a root a of (a, g). For each X e a, let 111, denote the flowof X and let E,. be the vector subbundleof TM inducedby g,.. Show that lim ! lnll(Dl{l,)xvll= a(X) t-+:l::oo t for all nonzero v e E,.(x), x e M. Therefore, the Oseledec decompositionfor 1{11 is inducedby the root space decompositionof g. Let 7C : E -+ M be a measurable vector bundle with fiber dimension n over a Borel probability space (M, B, µ,). Given an automorphism F of E and an F-invariant measurable subbundle Eo of E, we may consider the restriction Fo = FIEo•as well as the induced automorphismP on the quotient E = E/ E 0 • The latter is defined as follows: For each x e M, the fiber E(x) is the quotient of vector spaces, E(x)/ Eo(x), and F(v

+ Eo(x)) = Fv + Eo(/(x))

e E(f (x))

for each v + E0 (x) e E(x ). We denote by p : E -+ E the natural bundle map. We also assume that the map f : M -+ M such that 7C o F = f o 7C is a measure-preserving transformation of M and that the integrability condition on

9.3

Proof of Oseledec's Theorem

191

F needed for theorem 9.1.5 is satisfied. It is clear that the same condition will also hold for Fo. It is also immediate that the integrability condition holds for F, if we define onE a norm IIii llxas the infimum of IIv llxover all v e E (x) such that p(v) = ii. (Notice that llp(v)II::Sllvll.) Therefore, Oseledec's theorem applies to Fo and F, and it is natural to ask how the Lyapunov spectra of F, F, and Fo are related. The answer is given by the next proposition [19, 2.3). Proposition 9.3.7 Let (M, B, µ), F,F, Fo, E,E, Eo, and p be as before, and

suppose that F satisfies the integrability condition of theorem 9.1.5. Then the same is true for F and Fo. Let x,(x) < · · · < Xk..,µ}.Define, for each x e A and some S > 0, the sets w;(x) w;(x)

= {y e MI d(f-n(x), = {y E M I d(r(x),

1-n(y)) .:::se-(µ-E)n, n r(y))

e N},

.:::se-(l.-E)n, n EN}.

Theorem 9.4.3 (Local stable manifolds) Let f: M ~ M be a smooth diffeomorphism, and let A = A(>..,µ, E) be a Pesin set. Then there exists Eo > 0

such that for each x e Ak, k e N, and S = Eoe-ik, the following hold:

l. W](x) and W8(x) are smooth submanifolds of M diffeomorphic to disks of dimensions dim E-(x) and dim E+(x), respectively. 2. TxW](x) = E-(x), TxW1(x) = E+(x). As in the remark after theorem 1.5.1, the local stable and unstable manifolds W](x) and W8(x) give rise to (global) stable and unstable manifolds: 00

ws :=

LJ1-nw;cr(x))

00

and

n=O

wu:=

LJrw;u-n(x)). n=O

It is possible to use the stable and unstable manifolds in the way they were used in theorem 1.5.4, to prove the existence of fixed points for J. The precise statement is as follows (see [16]). Theorem 9.4.4 Let J : M ~ M be a smooth diffeomorphism of a compact manifold M, and let v be an !-invariant Borel probability measure. Suppose that v is a hyperbolic measure, that is, at v-a.e. x e M, none of the Lyapunov exponents is zero. Then the support of v is contained in the closure of the set

of periodic points for f. We now return to the more general situation, of measures possibly having zero exponent. The measure-theoretic entropy can also be estimated using the Lyapunov spectrum, as follows. Let J be a smooth diffeomorphism of a compact manifold M. It can be shown, using Oseledec's theorem, that there exists an !-invariant Borel set A C M consisting of regular points, such that for each I-invariant Borel probability measureµ on M, we have µ(A) = 1. Define a function x : A ~ R by x(x) :=

L X;(x)>O

l;(x)x;(x),

9.4 Nonuniform Hyperbolicity and Entropy

195

where the sum ranges over the set of positive Lyapunov exponents of / and x is a measurable /-invariant function.

l;(x) is the multiplicity of the exponent X;(x) (see section 9.1). Then

Theorem 9.4.5 Let f be a smooth diffeomorphism of a compact manifold M, and let A and x be as defined in the previous paragraph. 1. (Ruelle) If vis an /-invariant Borel probability measure on M,

2. (Pesin) If v is absolutely continuous with respect to the natural smooth measure class on M, then

A proof of the theorem can be found in [16] and [23].

10 Rigidity Theorems

Although the existence of the Oseledec decomposition for a Z-action on a vector bundle E is very useful by itself, it is certainly desirable to know the precise values of the Lyapunov exponents. When the Z-action is part of a G-action with an invariant probability measure, where G is a connected simple Lie group of real rank at least 2, it is possible to give a very detailed description of the Lyapunov spectrum in terms of then-dimensional linear representations of G, where n is the fiber dimension of E. It will be shown, in fact, that E admits a measurable trivialization with respect to which the action, in a sense, "reduces" to a homomorphism from the universal covering group of G into a subgroup H C GL(n, JR). (This is actually true only modulo a compact normal subgroup of H, as explained later.) This is the content of theorem 10.7.3. The theorem does not tell which representation arises; that has to be studied separately, in any particular situation to which the theorem is applied; this is illustrated by theorem 10.5.3. Moreover, as theorem 10.5.3 also shows, one can sometimes prove that the trivialization is continuous or even differentiable. The main technical result of the chapter is the er rigidity theorem, given in section 10.4.

10.1 Straightening Sections The main results of the chapter can be cast in terms of finding sections of a principal bundle P that transform in a special way under the action of a group of automorphisms of P. These special sections are defined as follows. Let p : P -+ M be a smooth principal H-bundle over a manifold M, and G a Lie group acting smoothly on P by bundle automorphisms. Let u :M -+ P be a er section of P, where, as 196

10.1

Straightening Sections

197

usual, we allow r = meas. Then u will be called a straightening section, or a p-section, if there exists a smooth homomorphism p : G - H such that for eachg e G andx e M, gu(x)

Notice that, for a general

= u(gx)p(g).

er section u that is not necessarily

straightening,

gu(x) is an element of Pgx, and hence it can be written as a right translation of u(gx) by some element a(g, x) e H. The map a: G x M - His clearly er. It also satisfies the cocycle identity:

for all g1, g2 E G and all XE M. In fact, u(g1g2x)a(g1g2, x)

= (g1g2)u(x) = g1u(g2x)a(g2,

x)

=u(g1g2x)a(g1,g2x)a(g2,x)

and the identity follows since H acts freely on P. We say that a is a er cocycle over the G-action on M. Therefore, a section u of P is straightening exactly when the cocycle a does not depend on x, so a(g, x) p (g) for some homomorphism p : G - H.

=

Exercise10.J.J Let G be a connected Lie group with Lie algebra g, La subgroup of

=

G, and M G/ r, where r is a lattice in G. Consider the action of L on M by left translations. Denote by P the frame bundle of M, and consider the natural action of L on P. Show that P admits a smooth straightening section and that the homomorphism p is (equivalent to) the adjoint representation Ad : G -+ G L(g).

Straightening sections, when they exist at all, are very special, even when we only require that they be measurable, or that they exist only over some open G-invariant subset of M. The following proposition makes this claim precise. We state the proposition for the measurable case only. Proposition 10.1.2 Let G be a connected simple noncompact Lie group, acting on a principal H-bundle p : P - M by bundle automorphisms. Suppose that H is a real linear algebraic group and that the G-action on M leaves invariant an

ergodic probability measure µ. Let u;, i

= 1, 2, be measurable

straightening

198

JO Rigidity Theorems

sections of P. Then there exists ho E H such that u2(x) xeM.

= u1 (x )ho for µ-a.e.

Proof. Suppose that u; is a measurable p;-section of P, where p; : G -+- His a smooth homomorphism, i = l, 2. Define a measurable map A : M -+- H by the property u2(x)

= 0-1(x)A(x).

Our goal is to prove that A is essentially constant. The map A satisfies the following identity: A(gx) = P1(g)A(x)P2(g)- 1

forallg

E G andx EM. In fact, 0-1(gx)A(gx)P2(g)

= u2(gx)p(g) = gu2(x) = gu1(x)A(x) = u1 (gx)p1 (g)A(x).

We know that G contains a group locally isomorphic to S L(2, IR),and in this subgroup we select a diagonalizable one-parameter subgroup, which we denote by C. The group C acts on H by (c, h) ~ T(c, h) := Pt (c)hP2(c)-

1•

This action is algebraic, which can be seen by the explicit description of the linear representations of SL(2, IR)given in section 7.7. By Moore's ergodicity theorem, the action of Con Mis also ergodic. Moreover, the action of C on H leaves invariant the probability measure v := A.µ. In fact, the identity Aoc = TeoA, where Tc= T(c, ·), gives A.c.µ = (Tc).A.µ, and, since g.µ =µfor each g E G, we conclude that v (Tc).v. We can now apply corollary 4.9.5 to conclude that vis supported on the set of fixed points for the action of C on H. This means that for each c and µ-a.e. x, TcA(x) = A(x). Therefore, A(cx) = A(x) for each c and almost every x. Applying proposition 2.3.1 (and 2.3.5), we conclude that A is constant µ-almost everywhere. D

=

Exercise 10.1.3 Let 1r: E -+ M be a vector bundle over a manifold M with fiber dimension n, and let G be a group that acts on E by bundle automorphisms such that the action on M leaves invariant a finite measure µ. Denote by :F(E) the frame bundle of E and suppose that there exist a measurable section a of :F(E), a homomorphism

10.2 H -Pairs

199

p: G -+ GL(n, JR),anda measurablecocyclek : G x M -+ K, whereK is a compact subgroupof GL(n, JR)centralizingp(G'), suchthatthe followingholds: gu(x)

= u(gx)k(g,

x)p(g)

for eachg e G andµ-a.e. x e M. Showthat,for eachg e G, the Lyapunovexponents for the Z-actiongeneratedby g are the numbersIn IAI,wherethe Aare the eigenvalues of p(g). Moreover,by identifyingthe fiberE(x) withRn ateachx viathe isomorphism u(x), concludethat the Oseledecdecomposition correspondsto the decompositionof Rn givenin exercise9.3.4. (Noticethat therewouldbe no difficultyif, with respectto somenorm II· IIon E, the normof the isomorphismu (x) : ]Rn -+ E (x) was bounded uniformlyoverx. Althoughthat maynot be true,by Poincarerecurrencealmostevery orbitwillvisita set wherellu(x)IIis boundedinfinitelymanytimes.) 10.2 H-Pairs

The reader is advised at this point to review the general definitions in the first two sections of chapter 6. The results of this chapter generally will have two versions, one topological and the other measurable. Whenever possible, a common language will be used for both cases. For example, by a er map, r = meas., we mean a Borel measurable map. When we say that a er map is defined over a set U, we implicitly assume that U is open, if r ~ 0, or Borel measurable, if r = meas. In the measurable case we will implicitly assume, and often will make explicit mention to, a measure class represented by some measure µ,. U will usually denote a subset of the base M of some fiber bundle and µ, will usually refer to a measure on M. (Even in the measurable case, M will always be at least a topological manifold, even though we could allow for much greater generality. The reader will have no difficulty formulating the main (measurable) results of the chapter for standard Borel spaces rather than topological manifolds.) If a group B acts by bundle automorphisms on a fiber bundle with base M, such that the action is er,then B naturally acts on M and the action is also er. When r = meas., we assume that the measure class of µ, is preserved by this action. We say that U c M is afull subset if U is open and dense, when r ~ 0, or µ,(M - U) = 0, when r = meas. Let H be a real algebraic group and P a principal H -bundle over a manifold M. The projection map will be denoted p : P --+-M. We suppose that P is a er bundle for r ~ 0 or r = meas. (We refer to the first sections of chapter 6 for the basic definitions. In the measurable case, P is isomorphic to M x H .) Let V be a real algebraic H -space, that is, a real variety equipped with a real algebraic action of H. Recall from section 6.2 that a ergeometric structure of type V may be defined as a er H-equivariant map (g1B, ~) •... , 4>(gkB, ~)). Then, by the lemma, g maps each element of P into the H-orbit of w = (p(gi)w, ... , p(gk)w). But we have chosen the g; so that the isotropy subgroup Ho,is trivial. Therefore, g is a er H-equivariant map into H. Therefore, we obtain a er section ii of P defined byg(ii) = e. It is immediate that ii coincides with as the structure group of the bundle. By Moore's ergodicity theorem, the JR-splitone-parameter subgroups also act ergodically on M and no subgroup of LI contains a nontrivial normal subgroup of L 1, since L I is simple. Therefore, all that we need in order to apply the previous theorem to the present situation is to show that the measurable algebraic hull of some subgroup of G is a proper subgroup of G. We first claim that there exists a proper algebraic subgroup B of L I such that Li/ B is compact. To see this, let R: L 1 ~ GL(d, R) be a faithful representation of L1, where Risa morphism (say, the adjoint representation), and consider the algebraic action of L I on the projective space pd-I (R) induced

222

JO Rigidity Theorems

fromR. Anyorbitof this actionhavingsmallestdimensionwillbe closed(hence compact),due to corollary4.9.2. On the other hand, if B denotes the isotropy group of a point in that closed orbit, it followsby corollary4.9.3 and theorem 1.2.2 that Li/ B is compact. Since the action is real algebraic, B is a real algebraicsubgroup. B is also a proper subgroupsince, otherwise, Li would fix a point in pd-i(JR), which means that the representationR would have a one-dimensionalinvariant subspace. But the kernel of this one-dimensional representationwouldbe a normalsubgroupof Li of codimension1, and this is not possible since Li is semisimple. Let Go ;;.;;Z be the group generatedby an element ?' =I-e contained in an JR-splitCartansubgroup.Definethe associatedbundle Qi xL, Li/Band apply proposition6.7.3to it. (In the proposition,Qi XL, Li/ B takes the role of Pv.) It follows that Qi admits a reductionwith group L2, where L2 is an algebraic subgroupof Li that leaves invarianta Borel probabilitymeasure Voon Li/ B. L 2 is a proper subgroupof Li since, otherwise,Li would fix v0 , which is not possibleby corollary4.9.5. (By this corollary,Li would have fixed points on Li/ B, but no fixed points exist since B is a proper subgroup. Notice that the subgroupof Li generatedby its one-parameteralgebraicsubgroupsis a normal subgroup,so it is equal to Li since Li is simple.) It follows that the algebraichull of Go on Qi is a proper subgroupof the algebraichull of G acting on the same bundle. This is the last conditionthat remainedto be checkedin order to apply theorem 10.3.1. This concludesthe proof. D For the followingresult, recall that if L is a connectedalgebraicgroup,there is a solvablealgebraicsubgroupR (L ), calledthe radicalof L, suchthat L / R (L) is semisimple. (This is the Levi decomposition,theorem 4.10.1.) Denote by 'Ir : L __. Li the projection from L into any one of the noncompactsimple factorsof L / R(L). (Weassumethat 'Ir quotientsout the finitecenterof L / R (L) so that Li has trivialcenter.) We will be referringto 'Ir in the next corollary. Theorem 10.6.2 (Margulis) Let r be a latticeof a connectednoncompactsimple Lie group G of real rank at least 2. Let R: r __. GL(m, JR)be a linear representationof r, and denote by L the connectedcomponentof the Zariski closureof R(r). Let'lr: L __. Li beasdefinedbeforeanddefinethefinite-index subgroupr 0 = r n R-i(L) ofr. Then there is a homomorphismp: G __. Li such that Piro = 'Ir o Rlr0 • Proof. We write M = G/ r. Let µ be the G-invariantBorel probabilitymeasure on M, whichexistsby virtueof r being a lattice. Let P = G x r Li be the

10.6 The TheoremsofZimmerandMargulis

223

principal L 1-bundle over M definedby the suspensionconstructionof section 1.3 (see also exercise6.1.4). By exercise6.5.3, the measurablealgebraichull of the G-action on Pis L 1• Then the previouscorollaryimplies the existenceof a measurablesectionu of P and a smooth homomorphismp' : G ~ L I such that for µ,-a.e. x e M and each g e G, we have gu(x) = u(gx)p'(g). Fix x0 = g0 r in this conull set and write u(xo) = [go, lo]. Then xo is a fixedpoint for the action of gofgo on Mand goyg0 1u(xo) = u(xo)p'(goyg 0 1), for each y e r. Notice that

On the other hand, for each y e

r,

goygo 1[go, lo]= [goygo 1go, lo]= [goy, lo]= [go, (,r o P)(y)lo],

so lop'(goyg 01) = (_,ro P)(y)lo, for each y e the corollaryholds for the homomorphism

r. Therefore,the conclusionof D

Theorem 10.6.3 (Mostow-Margulis) Let G and G' be connectednoncompact simple Lie groups with trivial center, and suppose that the real rank of G is at least 2. Let r c G and r' c G' be lattices and consider an isomorphism ,r : r ~ r'. Then ,r extends to a smooth isomorphism,r : G ~ G'.

Proof. We may identify G' with the connected component of e of a real algebraic group, by means of the adjoint representation. By the Borel density theorem (theorem 4.9.7), the Zariski closure of r' is G'. (Notice that the smallest algebraic subgroup of G' containing all the algebraic one-parameter subgroupsof G' is a normal subgroup,whence coincideswith G'.) Therefore, the present theorem is an immediateconsequenceof the previousone. D Exercise 10.6.4 Let r be a lattice of SL(n, JR),acting on a manifold M of dimension n. Suppose that the action is smooth and leaves invariant a volume form, that is, a nonvanishing alternating n-form. Suppose that the action is ergodic with respect to the probability measure defined by the volume form. Show that there exist measurable vector fields X 1, ••• , Xn on M, almost everywhere linearly independent, and a linear representation p of SL(n, JR),such that

y.X;

= ±1)(y);iXi j

for each i. (The unspecified sign means that the vector fields only define a section of the frame bundle "modulo" the center of SL(n, JR).Notice that this center is trivial if n is odd and is {±/} if n is even.)

224

10 Rigidity Theorems

10.7 The LyapunovSpectrumand Entropy We now show that the Lyapunov spectrum of a smooth measure-preserving actionof a connectedsemisimpleLie group G of real rank at least 2 on a compact manifold M of dimension n is determinedby the linear representationsof (a coveringgroup of) G in dimensionn. It will be convenientto use the following notation. Let p: G -+ GL(n, IR)be a linear representation. For each g e G, let J..;,i = 1, ... , l, be the distinct eigenvaluesof p(g), having multiplicities d;. The set of values ln IJ..;I, with multiplicitiesgiven by the sum of the d; for which the IJ..;1are equal, will be referred to as the Lyapunov spectrum of p(g). We also define the entropy of p(g) as

where the sum runs over the indices i such that IJ..;I > 1. Theorem 10.7.1 Let G be a connectedsemisimpleLie group with finite center and simple factors of real rank at least 2. Suppose that G acts by bundle automorphismson a vector bundle E over a compact manifold M, preserving an ergodic Borel probabilitymeasure µ on M. Let n be the fiber dimensionof E, and denoteby Gthe universalcoveringgroupof G. Then there existsa linear representationp : G -+ GL(n, IR) such that for each g e G, the Lyapunov spectrumassociatedto the Z-actionon E generatedby g is equalto the Lyapunov spectrumof p(g'), where g' is any elementof G that projects onto g. Proof. The firstremarkis that the Lyapunovexponentsare not affectedby passing to a finiteergodicextensionof the action,so we can applyproposition6.5.7. Let E; be as in that proposition. Furthermore,proposition9.3.7 says that it is enough to prove the theorem for the G-action on E;+i/ E;, i = 1, ... , m - 1. The key property of the G-action on these quotients is that the algebraichulls are reductive and Zariski-connected.Therefore, there will be no loss of generality in assuming that the measurablealgebraic hull H for the G-action on E is Zariski-connectedand reductive. Recall theorem 4.10.1, which says that H = (H, H)Z,wherethecommutator(H, H)isaZariski-connectedsemisimple group, Z is a Zariski-connectedsubgroupof the centerof H, and the product is almostdirect,that is, (H, H) and Z centralizeeachotherand their intersection is finite. Let P be a measurableG-invariant H-reduction of the frame bundle of E, and form the associatedbundle P = P /(H, H). This is a principal Z-bundle, where Z = Z/(Z n (H, H)). It is immediatethat Z is the algebraic hull of the G-action on P. Therefore, Z is compactby theorems 8.6.6 and 8.6.4. The conclusionis that H is a reductivegroup with compact center.

10.7

225

The Lyapunov Spectrum and Entropy

Collecting the center of H and the compact simple factors of (H, H) into one single compact group So, and the noncompact simple factors into a semisimple S0 S. This is an almost direct product. Let F be group S, we can write H the finite central subgroup of H such that

=

H/F

= SoX

S1X

•"

X

sk,

=

a direct product, where So So/(So n F) and the S;, i :::: l, are the simple center-free factors of S S/(S n F). Denote by 11:: H -+- S the natural projection. Applying theorem 10.6.1 to the G-action on P = P /So= P x 8 S, we conclude that there exists a surjective homomorphism p : G -+- S and a measurable straightening p-section ii of P. Denote by p the homomorphism from G onto S that lifts p. We claim that there exists a measurable section -+- P. Then C, where m ®o