Causal Symmetric Spaces: Geometry and Harmonic Analysis

Causal Symmetric Spaces Geometry and Harmonic Analysis

P E R S P E C T I V E S IN MATHEMATICS, Vol. 18 S. Helgason, editor

Causal Symmetric Spaces Geometry and Harmonic Analysis Joachim Hilgert Department of Mathematics University of Wisconsin p

Gestur Olafsson Department of Mathematics Louisiana State University

ACADEMIC PRESS San Diego

London

Boston

New York

Sydney

To~o

Toronto

This book is printed on acid-free paper. ( ~ Copyright 9 1997 by Academic Press. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. ACADEMIC PRESS, INC. 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA 1300 Boylston Street, Chestnut Hill, MA 02167, USA http://www.apnet.com ACADEMIC PRESS LIMITED 24-28 Oval Road, London NW1 7DX, UK http://www.hbuk.co.uk/ap/

Library of Congress Cataloging-in-Publication Data ISBN 0-12-525430-X Printed in the United States of America 96 97 98 99 00 BB 9 8 7 6 5 4 3 2 1

Contents Preface

viii

Introduction

x

1 Symmetric Spaces

1

1.1

1.2

Basic Structure Theory

Dual S y m m e t r i c Spaces 1.2.1

1.2.2

2

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

T h e c-Dual Space A:i c . . . . . . . . . . . . . . . . .

1

7 7

1.3

T h e Associated Dual Space A4 a . . . . . . . . . . . 1.2.3 T h e R i e m a n n i a n Dual Space M r . . . . . . . . . . . T h e M o d u l e S t r u c t u r e of T o ( G / H ) . . . . . . . . . . . . . .

9 12

1.4 1.5

A-Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . The Hyperboloids . . . . . . . . . . . . . . . . . . . . . . . .

22 24

Causal Orientations 2.1 C o n v e x Cones a n d T h e i r A u t o m o r p h i s m s

..........

29 29

2.2

Causal Orientations

2.3 2.4

Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . The Order Compactification ..................

43 45

2.5

Examples

50

2.6

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

9

39

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

2.5.1

T h e G r o u p Case

2.5.2

The Hyperboloids

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

50

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

S y m m e t r i c Spaces R e l a t e d to T u b e D o m a i n s

51 ........

52

2.6.1

Boundary Orbits ....................

56

2.6.2

The Functions ~m

58

2.6.3

T h e C a u s a l C o m p a c t i f i c a t i o n of A4 . . . . . . . . . .

63

2.6.4

S V ( n , n) . . . . . . . . . . . . . . . . . . . . . . . . .

65

2.6.5

Sp(n, R ) . . . . . . . . . . . . . . . . . . . . . . . . .

68

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

vi

3

CONTENTS

I r r e d u c i b l e C a u s a l Symmetric Spaces 3.1 Existence of Causal S t r u c t u r e s . . . . . . . . . . . . . . . . 3.2 T h e Classification of Causal S y m m e t r i c Pairs . . . . . . . .

71 71 83

4

Classification of Invariant Cones 4.1 S y m m e t r i c SL(2, R ) R e d u c t i o n . . . . . . . . . . . . . . . . 4.2 T h e Minimal and M a x i m a l Cones . . . . . . . . . . . . . . . 4.3 T h e Linear Convexity T h e o r e m . . . . . . . . . . . . . . . . 4.4 T h e Classification . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Extension of Cones . . . . . . . . . . . . . . . . . . . . . . .

91 91 98 105 110 115

5

The 5.1 5.2 5.3 5.4 5.5 5.6 5.7

120

6

T h e O r d e r Compactiflcation 6.1 Causal Galois Connections . . . . . . . . . . . . . . . . . . . 6.2 A n Alternative Realization of ~.pt . . . . . . . . . . . . . . 6.3 6.4 6.5

7

8

Geometry T h e Bounded Realization of H / H N K ........... T h e Semigroup S ( C ) . . . . . . . . . . . . . . . . . . . . . . T h e Causal Intervals . . . . . . . . . . . . . . . . . . . . . . Compression Semigroups . . . . . . . . . . . . . . . . . . . . The Nonlinear Convexity T h e o r e m . . . . . . . . . . . . . . The B~-Order . . . . . . . . . . . . . . . . . . . . . . . . . . T h e Affine Closure of B ~ . . . . . . . . . . . . . . . . . . . .

T h e Stabilizers for ,AA . , +cpt . . . . . . . . . . . . . . . . . . . . T h e Orbit S t r u c t u r e of A 4 ~pt . . . . . . . . . . . . . . . . . T h e Space S L ( 3 , R ) / S O ( 2 , 1 ) . . . . . . . . . . . . . . . . .

121

126 130 132 143 152 157

172 172 178 180 183 190

Holomorphic Representations

198

7.1 7.2 7.3 7.4 7.5 7.6

199 203 209 214 216 219

Holomorphic R e p r e s e n t a t i o n s of Semigroups . . . . . . . . . Highest-Weight Modules . . . . . . . . . . . . . . . . . . . . T h e Holomorphic Discrete Series . . . . . . . . . . . . . . . Classical H a r d y Spaces . . . . . . . . . . . . . . . . . . . . . H a r d y Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . T h e Cauchy-Szeg5 Kernel . . . . . . . . . . . . . . . . . . .

Spherical Functions 8.1 T h e Classical Laplace T r a n s f o r m . . . . . . . . . . . . . . . 8.2 Spherical Functions . . . . . . . . . . . . . . . . . . . . . . . 8.3 T h e A s y m p t o t i c s . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Expansion Formula . . . . . . . . . . . . . . . . . . . . . . . 8.5 T h e Spherical Laplace T r a n s f o r m . . . . . . . . . . . . . . . 8.6 T h e Abel Transform . . . . . . . . . . . . . . . . . . . . . .

222 222 224 228 230 232 235

CONTENTS 8.7 9

vii

Relation to Representation Theory . . . . . . . . . . . . . .

The Wiener-Hopf

Algebra

A Reductive Lie Groups A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Finite-Dimensional Representations . . . . . . . . . . . . . . A.3 Hermitian Groups . . . . . . . . . . . . . . . . . . . . . . .

237 239 246 246 249 252

B The Vietoris Topology

257

Notation

262

Bibliography

272

Index

284

Preface In the late 1970s several mathematicians independently started to study a possible interplay of order and continuous symmetry. The motivation for doing so came from various sources. K.H. Hofmann and J.D. Lawson [67] tried to incorporate ideas from geometric control theory into a systematic Lie theory for semigroups, S. Paneitz [147, 148] built on concepts from cosmology as propagated by his teacher I.E. Segal [157], E.B. Vinberg's [166] starting point was automorphism groups of cones, and G.I. Ol'shanskii [137, 138, 139] was lead to semigroups and orders by his studies of unitary representations of certain infinite-dimensional Lie groups. It was Ol'shanskii who first considered the subject proper of the present book, causal symmetric spaces, and showed how they could play an important role in harmonic analysis. In particular, he exhibited the role of semigroups in the Gelfand-Gindikin program [34], which is designed to realize families of similar unitary representations simultaneously in a unified geometric way. This line of research attracted other researchers such as R.J. Stanton [159], B. Orsted, and the authors of the present book [159]. This book grew out of the Habilitationschrift of G. Olafsson [129], in which many of the results anounced by Ol'shanskii were proven and a classification of invariant causal structures on symmetric spaces was given. The theory of causal symmetric spaces has seen a rapid development in the last decade, with important contributions in particular by J. Faraut [24, 25, 26, 28] and K.-H. Neeb [114, 115, 116, 117, 120]. Its role in the study of unitarizable highest-weight representations is becoming increasingly clear [16, 60, 61, 88] and harmonic analysis on these spaces turned out to be very rich with interesting applications to the study of integral equations [31, 53, 58] and groupoid C*-algebras [55]. Present research also deals with the relation to Jordan algebras [59, 86] and convexity properties of gradient flows [49, 125]. Thus it is not possible to write a definitive treatment of ordered symmetric spaces at this point. On the other hand, even results considered ~standard" by the specialists in the field so far have either not appeared in print at all or else can be found only in the original viii

PREFACE

ix

literature. This book is meant to introduce researchers and graduate students with a solid background in Lie theory to the theory of causal symmetric spaces, to make the basic results and their proofs available, and to describe some important lines of research in the field. It has gone through various stages and quite a few people helped us through their comments and corrections, encouragement, and criticism. Many thanks to W. Bertram, F. Betten, J. Faraut, S. Helgason, T. Kobayashi, J. Kockmann, Kh. Koufany, B. KrStz, K.-H. Neeb, and B. Orsted. We would also like to thank the Mittag-Leflter Institute in Djursholm, Sweden, for the hospitality during our stay there in spring 1996.

Djursholm Gestur Olafsson Joachim Hilgert

Introduction Symmetric spaces are manifolds with additional structure. In particular they are homogeneous, i.e., they admit a transitive Lie group action. The study of causality in general relativity naturally leads to "orderings" of manifolds [149, 157]. The basic idea is to fix a convex cone (modeled after the light cone in relativity) in each tangent space and to say that a point x in the manifold precedes a point y if x can be connected to y by a curve whose derivative lies in the respective cone wherever it exists (i.e., the derivative is a timelike vector). Various technical problems arise from this concept. First of all, the resulting "order" relation need not be antisymmetric. This describes phenomena such as time traveling and leads to the concept of causal orientation (or quasi-order), in which antisymmetry is not required. Moreover, the relation may not be closed and may depend on the choice of the class of curves admitted. Geometric control theory has developed tools to deal with such questions, but things simplify considerably when one assumes that the field of cones is invariant under the Lie group acting transitively. Then a single cone will completely determine the whole field and it is no longer necessary to consider questions such as continuity or differentiablility of a field of cones. In addition, one now has an algebraic object coming with the relation: If one fixes a base point o, the set of points preceded by o may be viewed as a positive domain in the symmetric space and the set of group elements mapping the positive domain into itself is a semigroup, which is a very effective tool in studying causal orientations. Since we consider only homogeneous manifolds, we restrict our attention to this simplified approach to causality. Not every cone in the tangent space at o of a symmetric space j~4 leads to a causal orientation. It has to be invariant under the action of the stabilizer group H of o. At the moment one is nowhere near a complete classification of the cones satisfying this condition, but in the case of irreducible semisimple symmetric spaces it is possible to single out the ones which admit such cones. These spaces are then simply called causal symmetric spaces, and it turns out that the existence of a causal orientation puts severe restrictions

INTRODUCTION

xi

on the structure of the space. More precisely, associated to each symmetric space .A4 = G / H there is an involution v: G --+ G whose infinitesimal version (also denoted by r) yields an eigenspace decomposition g = ~ + q of the Lie algebra g of G, where [], the Lie algebra of H, is the eigenspace for the eigenvalue 1 and q is the eigenspace for the eigenvalue - 1 . Then the tangent space of jr4 at o can be identified with q and causal orientations are in one-to-one correspondence with H-invariant cones in q. A heavy use of the available structural information makes it possible to classify all regular (i.e., containing no lines but interior points) H-invariant cones in q. This classification is done in terms of the intersection with a Cartan subspace a of q. The resulting cones in a can then be described explicitly via the machinery of root systems and Weyl groups. It turns out that causal symmetric spaces come in two families. If g - E+ p is a Cartan decomposition compatible with g = []+q, i.e., r and the Cartan involution commute, then a regular H-invariant cone in q either has interior points contained in E or in p. Accordingly the resulting causal orientation is called compactly causal or noncompactly causal. Compactly causal and noncompactly causal symmetric spaces show a radically different behavior. So, for instance, noncompactly causal orientations are partial orders with compact order intervals, whereas compactly causal orientations need neither be antisymmetric nor, if they are, have compact order intervals. On the other hand, there is a duality between compactly causal and noncompactly causal symmetric spaces which on the infinitesimal level can be described by the correspondence [1 + q ," " [] + iq. There are symmetric spaces which admit compactly as well as noncompactly causal orientations. They are called spaces of Cayley type and are in certain respects the spaces most accessible to explicit analysis. The geometry of causal symmetric spaces is closely related to the geometry of Hermitian symmetric domains. In fact, for compactly causal symmetric spaces the associated Riemannian symmetric space G / K , where K is the analytic subgroup of G corresponding to E, is a Hermitian symmetric domain and the space H / ( H n K) can be realized as a bounded real domain by a real analog of the Harish-Chandra embedding theorem. This indicates that concepts such as strongly orthogonal roots that can be applied successfully in the context of Hermitian symmetric domains are also important for causal symmetric spaces. Similar things could be said about Euclidian Jordan algebras. Harmonic analysis on causal symmetric spaces differs from harmonic analysis on Riemannian symmetric spaces in various respects. So, for instance, the stabilizer group of the basepoint is noncompact, which accounts for considerable difficulties in the definition and analysis of spherical functions. Moreover, useful decompositions such as the Iwasawa decomposition

x]i

INTRODUCTION

do not have global analogs. On the other hand, the specific structural information one has for causal symmetric space makes it possible to create tools that are not available in the context of Riemannian or general semisimple symmetric spaces. Examples of such tools are the order compactification of noncompactly causal symmetric spaces and the various semigroups associated to a causal orientation. The applications of causal symmetric spaces in analysis, most notably spherical functions, highest-weight representations, and Wiener-Hopf operators, have not yet found a definitive form, so we decided to give a mere outline of the analysis explaining in which way the geometry of causal orientations enters. In addition we provide a guide to the original literature as we know it. We describe the contents of the book in a little more detail. In Chapter 1 we review some basic structure theory for symmetric spaces. In particular, we introduce the duality constructions which will play an important role in the theory of causal symmetric spaces. The core of the chapter consists of a detailed study of the H- and b-module structures on q. The resulting information plays a decisive role in determining symmetric spaces admitting G-invariant causal structures. Two classes of examples are treated in some detail to illustrate the theory: hyperboloids and symmetric spaces obtained via duality from tube domains. In Chapter 2 we review some basic facts about convex cones and give precise definitions of the objects necessary to study causal orientations. These definitions are illustrated by a series of examples that will be important later on in the text. The central results are a series of theorems due to Kostant, Paneitz, and Vinberg, giving conditions for finite-dimensional representations to contain convex cones invariant under the group action. We also introduce the causal compactification of an ordered homogeneous space, a construction that plays a role in the analysis on such spaces. Chapter 3 is devoted to the determination of all irreducible symmetric spaces which admit causal structures. The strategy is to characterize the existence of causal structures in terms of the module structure of q and then use the results of Chapter 1 to narrow down the scope of the theory to a point where a classification is possible. We give a list of symmetric pairs (g, T) which come from causal symmetric spaces and give necessary and sufficient conditions for the various covering spaces to be causal. In Chapter 4 we determine all the cones that lead to causal structures on the symmetric spaces described in Chapter 3. Using duality it suffices to do that for the noncompactly causal symmetric spaces. It turns out that up to sign one always has a minimal and a maximal cone giving rise to a causal structure. The cones are determined by their intersection with a certain (small) abelian subspace a of q and it is possible to characterize

INTRODUCTION

xiii

the cones that occur as intersections with a. Since the cone in q can be recovered from the cone in a, this gives an effective classification of causal structures. A technical result, needed to carry out this program but of independent interest, is the linear convexity theorem, which describes the image of certain coadjoint orbits under the orthogonal proction from q to ft.

Chapter 5 is a collection of global geometric results on noncompactly causal symmetric spaces which are frequently used in the harmonic analysis of such spaces. In particular, it is shown that the order on such spaces has compact intervals. Moreover, the causal semigroup associated naturally to the maximal causal structure is characterized as the subsemigroup of G which leaves a certain open domain in a flag manifold of G invariant. The detailed information on the ordering obtained by this characterization allows to prove a nonlinear analog of the convexity theorem which plays an important role in the study of spherical functions. Moreover, one has an Iwasawa-like decomposition for the causal semigroup which makes it possible to describe the positive cone of the symmetric space purely in terms of the "solvable part" of the semigroup, which can be embedded in a semigroup of affine selfmaps. This point of view allows us to compactify the solvable semigroup and in this way makes a better understanding of the causal compactification possible. In Chapter 6 we pursue the study of the order compactification of noncompactly causal symmetric spaces. The results on the solvable part of the causal semigroup together with realization of the causal semigroup as a compression semigroup acting on a flag manifold yield a new description of the compactification of the positive cone which then can be used to give a very explicit picture of the G-orbit structure of the order compactification. The last three chapters are devoted to applications of the theory in harmonic analysis. In Chapter ? we sketch a few of the connections the theory of causal symmetric spaces has with unitary representation theory. It turns out that unitary highest-weight representations are characterized by the fact that they admit analytic extensions to semigroups of the type considered here. This opens the way to construct Hardy spaces and gives a conceptual interpretation of the holomorphic discrete series for compactly causal symmetric spaces along the lines of the Gelfand-Gindikin program. Chapter 8 contains a brief description of the spherical Laplace transform for noncompactly causal symmetric spaces. We introduce the corresponding spherical functions, describe their asymptotic behavior, and give an inversion formula. In Chapter 9 we briefly explain how the causal compactification from Chapter 2 is used in the study of Wiener-Hopf operators on noncompactly

xiv

INTRODUCTION

causal symmetric spaces. In particular, we show how the results of Chapter 6 yield structural information on the C*-algebra generated by the WienerHopf operators. Appendix A consists of background material on reductive Lie groups and their finite-dimensional representations. In particular, there is a collection of our version of the standard semisimple notation in Section A.1. Moreover, in Section A.3 we assemble material on Hermitian Lie groups and Hermitian symmetric spaces which is used throughout the text. In Appendix B we describe some topological properties of the set of closed subsets of a locally compact space. This material is needed to study compactifications of homogeneous ordered spaces.

Chapter 1

Symmetric Spaces In this chapter we present those parts of the theory of symmetric spaces which are essential for the study of causal structures on these spaces. Since in this book we are mainly interested in the group theoretical aspects, we use here the definition in [81] which is given in group theoretical rather than differential geometric terms. Apart from various well known standard facts we describe several duality constructions which will play an important role in our treatment of causal structures. The central part of this chapter is a detailed discussion of the module structure of the tangent spaces of semisimple non-Riemannian symmetric spaces. The results of this discussion will play a crucial role in our study of causal structures on symmetric spaces. Finally we review some technical decomposition results due to Oshima and Matsuki which will be needed in later chapters. In order to illustrate the theory with examples that are relevant in the context of causal symmetric spaces we treat the hyperboloids in some detail.

1.1

Basic Structure Theory

D e f i n i t i o n 1.1.1 A symmetric space is a triple (G, H, r), where 1) G is a Lie group, 2) r is a nontrivial involution on G, i.e.,r: G -r G is an automorphism with r 2 = IdG, and

CHAPTER1.

S Y M M E T R I C SPACES

3) H is a closed subgroup of G such that Gor C H C G ~. Here the subscript o means the connected component containing the identity and G r denotes the group of v-fixed points in G. By abuse of notation we also say that (G, H) as well as G / H is a symmetric space. The infinitesimal version of Definition 1.1.1 is D e f i n i t i o n 1.1.2 A pair (g, r) is called a symmetric pair if 1) g is a Lie algebra, 2) r is a nontrivial involution of g, i.e., r: It --+ g is an automorphism with r 2 = Id a. [:1 Let (g, r) be a symmetric pair and let G be a connected Lie group with Lie algebra {I. If H is a closed subgroup of G, then (G, H) is called associated to (9, r) if v integrates to an involution on G, again denoted by r, such that (G, H, r) is a symmetric space. In this case the Lie algebra of H is denoted by Ij and it is given by: I} = g(1, v ) : = {X e g I r ( X ) = X}.

(1.1)

Note that g = I~ @ q, where q = g ( - 1 , r ) : = {X E g l r ( X ) = - X } .

(1.2)

We have the following relations: [Ij, I}] C I~,

[t}, q] C q,

and

[q, q] C Ij.

(1.3)

From (1.3) it follows that adq: 11 --r End(q), X ~ ad(X)Jq, is a representation of Ij. In particular, (g, I}) is a reductive pair in the sense that there exists an [j-stable complement of I~ in g. On the other hand, if (g, I}) is a reductive pair and the commutator relations (1.3) hold, we can define an involution v of g by rl~ = id and rlq = - i d . Then (g, r) is a symmetric pair. E x a m p l e 1.1.3 ( T h e G r o u p C a s e I) Let G be a connected Lie group. Define 7- by T(a,b) := (b,a). Then (G x G) r = A(G) := {(a,a) l a E G} is the diagonal in G x G and :~/1 = (G x G ) / A ( G ) is a symmetric space. The map G • G 9 (a,b) ~ ab -z E G induces a diffeomorphism M "~ G intertwining the left action of G • G on M and the operation (a, b). c = acb -z of G x G on G. The decomposition of g • g into r-eigenspaces is given by

1.1.

BASIC STRUCTURE

THEORY

3

and q=

{(X,--X) lX e g}

g.

E x a m p l e 1.1.4 ( T h e G r o u p C a s e II) Let g be a real Lie algebra, gc its complexification, and a: gc -~ gc the complex conjugation with respect the real form g of gc, i.e., a(X + iY) = X-

iY,

X , Y e g.

Further, let Gc be a connected complex Lie group with Lie algebra gc such that a integrates to a real involution on Gc, again denoted by a. Then GR, the analytic subgroup of Gc with Lie algebra g, is the connected component of (Gc) ~. In particular, it is closed in Gc and G c / G R is a symmetric space. The decomposition of gc into a-eigenspaces is given by gc = [~+ q = g + ig. o E x a m p l e 1.1.5 ( B o u n d e d S y m m e t r i c D o m a i n s ) Let g be a Hermitian Lie algebra and let G be a connected Lie group with Lie algebra g. Let K be a maximal compact subgroup of G. Then D - G / K is a bounded symmetric domain. Let H ( D ) be the group of holomorphic isometries f : D -~ D. Then H ( D ) o , the connected component containing the identy map, is localy isomorphic to G. Let us assume that G - H ( D ) o . Let a D " D ~ D be a complex conjugation, i.e., a D is an antiholomoprhic involution. We assume that a(o) - o, where o - {K}. Define a : G --4 G

by o'(f) -- O'D

o f o O'D.

Then a is a nontrivial involution on G In this case G ~ = {f e G If(DR)= DR}

where DR is the real form of D given by DR = {z E

D I an(z)

=

z}.

c~

E x a m p l e 1.1.6 ( S y m m e t r i c P a i r s a n d T u b e D o m a i n s ) We have the following construction in the case where G / K is a tube domain and G is contained in the simply connected Lie group Gc. Consider the elements Z ~ - Zo E ~(t), Xo E p and Yo E p, as well as the space a C p, from p. 253 and p. 254. We have spec(adZ ~ = { O , i , - i } and spec(adXo) = spec(adYo) = {0, 1 , - 1 } . Let b := g(0, Yo), q+ : - g(1,Yo), q- : - g(-1,Yo) and q - q+ @ q-. Then {j is a 0-stable subalgebra of g. As [q+, q+] C g(2, Yo) - {0} and [ q - , q - ] C g(-2,Yo) - {0}, it follows that q+ and qare both abelian subalgebras of g.

CHAPTER1.

S Y M M E T R I C SPACES

Define

r = Ad(exp~iYo) : gc ~ gc. Then r]br = id and r[, e = - i d . Hence r is a complex linear involution on 9c. We notice that r = C 2 in the notation of L e m m a A.3.2. As usual, we denote the involution on Gc determined via v(exp X) = exp(v(X)) for X E 9c by the same letter. We set H := G r a n d . M = G / H . Then (G, H, r) is a symmetric space. As r = C 2, we get from L e m m a A.3.2 and Theorem A.3.5 that r(Zo) = - Z o . Hence the induced involution on the tube domain G / K is antiholomorphic, o Let (G, H, v) be a symmetric space. We set M := G / H and o := 1/H. Then the map q B X ~ )to E To(.M), given by d X o ( / ) := ~ / ( e x p ( t X ) .

o)l~=0, I e C ~ ( M ) ,

(1.4)

is a linear isomorphism of q onto the tangent space T o ( M ) of r at o. Two symmetric pairs (9, r) and ([, ~) are called isomorphic, (9, v) ~_ ([, ~), if there exists an isomorphism of Lie algebras )~ : 9 ~ [ such that )~ o r = ~ o )~. If )~ is only assumed to be injective, then we call (9, T) a subsymmetric pair of ([, to). In the same way we can define isomorphisms and homomorphisms for symmetric pairs (G, H) and also for symmetric spaces ,At[ = G / H . The symmetric pair (9, ~) is said to be irreducible ff the only v-stable ideals in 9 containing the Lie algebra 9 lix := ~ g c a Ad(g)tj are 9 Iix and 9. Note t h a t it plays no role here which group G with Lie algebra 9 we use in the definition of 9 lix. Let M = G / H be a symmetric space. For a E G we denote the diffeomorphism m ~-r a . m by ea : ,h/[ --~ ~ . L e m m a 1.1.7 Consider the pointwise stabilizer v

= {. e G I T

e M

:

= m} = ker(a ~ ta) c H

of M in G. Then the Lie algebra of G ~ is 9 lix. Proo]!. X E 9 is contained in the Lie algebra of G ~ if and only if (exp t X ) g H C gH for all g E G and all t E R. But this is equivalent to A d ( g ) X E tj for all g E G and hence to X E ~ g ~ a Ad(g)lj = 9 fix. I-1 Denote by Af(H) the set of normal subgroups in G contained in H. Then A/'(H) is ordered by inclusion.

1.1. B A S I C S T R U C T U R E T H E O R Y Proposition

1.1.8

1) The group G "~ is a normal subgroup of G and

contained in H. 2) If N is a normal subgroup of G contained in H, then N C G "~. This means that G ~ is the unique maximal element of Af(H). 3) Let Adq: H ~ GL(q) be defined by Adq(h) = Ad(h)l q. Then G ~ = ker Adq. Proo~ The first part is obvious. If N is a normal subgroup of G contained in H, then for all a E G and n E N we have n. (ao) = a(a-Zna)

9o = a . o

as a-Zna E N. Hence N C G ~ and 2) follows. In order to prove 3) we first observe t h a t G ~ C k e r i d q since h e x p ( X ) h - Z H = e x p ( X ) H for all h E G ~ and X E q. To show the converse it suffices to prove that ker Adq is normal in G. But exp q and H are clearly in the normalizer of ker Adq and these two sets generate G. [=! R e m a r k 1.1.9 Proposition 1.1.8 shows the assumption ~l f i z "-- {0} means t h a t the pair (g, [j) is effective, i.e., that the representation adq of Ij, X ~-+ adXIq , is faithful. If (g, I~) is effective, we have [q, q] = 1~ since [q,q] + q clearly is an ideal in g. [] R e m a r k 1.1.10 Let (it, r) be an irreducible and effective symmetric pair. If r is the radical of g, then r is r-stable. Therefore g is either semisimple or solvable. If g is solvable, then [g, g] is a r-stable ideal of g. Hence g is abelian. As every r-stable subspace of an abelian algebra is an r-stable ideal, it follows that g is one dimensional. • The following theorem is proved in [97], p.171. T h e o r e m 1.1.11 Let G be a connected simply connected Lie group and T: G --+ G an involution. Then the group G r of r-fixed points in G is connected and the quotient space G I G r is simply connected. • Let G be a connected simply connected Lie group with Lie algebra g. Then any involution r 9g -+ g integrates to an involution on G which we also denote by v. Thus Theorem 1.1.11 yields a canonical way to construct a simply connected symmetric space from a symmetric pair. D e f i n i t i o n 1.1.12 Let (8, v) be a symmetric pair and (~ the simply connected Lie group with Lie algebra g. Then the symmetric space A/[ := G I G ~ described by Theorem 1.1.11 is called the universal symmetric space associated to (g, T). []

CHAPTER 1. SYMMETRIC SPACES Note that, given a symmetric space .M = G/H, the canonical projection (~ -+ G induces a covering map f14 ~ f14, so that fist is indeed the universal covering space of f14. Another canonical way to associate a symmetric space to a symmetric pair is to complexify g and then use a simply connected complex Lie group. D e f i n i t i o n 1.1.13 Let It be a real Lie algebra and {to the complexification of ~t. Consider the following involutions of the real Lie algebra {tc: 1) a = a~ is the complex conjugation of gc with respect to g,

a ( X + iY) = X -

iY,

X , Y 6g,

2) r: gc --+ gc, the complex linear extension of r to {tc. 3) r / ' = r o a = a o r, the antiUnear extension of r to "Yc. The corresponding Lie algebras of fixed points are g~=g,

g~=I}c,

g~={te:=lJ+iq.

Note that r/depends on r as well as on the real form g. If necessary we will therefore write r/(r, {t). Given a symmetric pair ({t, r), let Gc be a simply connected complex Lie group with Lie algebra gc. Then, according to Theorem 1.1.11, r, a, and r/can be integrated to involutions on Gc with connected fixed point groups G~, G~, and G~, respectively. We set t~ "= G~., t~ e := G~ and /z/:= (G~z Cl G~z) = (G~z n G~:). Then 351 := a~/(a6 N a~) = r

(1.5)

:Qe := G~/(G~ fl G~.) = Ce/[-I are symmetric spaces associated to (g, v) and (ge, r), respectively. L e m m a 1.1.14 Let (g, r) be a symmetric pair with It semisimple and de-

note the Killing form of g by B. Then the following holds:

0 q = { x e g I B(X, b)= 0}. 2) The ideal gl [q,q] ~ q T-stable ideal a n d [ : = I X 6 g l V Y 6 gx 9 B ( X , Y) = 0} /s an ideal of g contained in b. In particular, if gyiz = {0), then 01 = {t. --'~

1.2. DUAL S Y M M E T R I C SPACES Proof. 1) As B is nondegenerate and B ( X , Y) = B(r

Y) = B(X, r

=

-B(X, Y)

for all X E I} and Y E q, the claim follows from g = I} ~ q. 3) Since B([X, Y], Z) = - B ( Y , [X, Z]) for all X, Y, Z e g, this follows from (1). El R e m a r k 1.1.15 Let (G, H, r) be a symmetric space with G a noncompact semisimple Lie group and ~" not a Cartan involution. In this case we call .A/I := G / H a nonRiemannian semisimple symmetric space. For such spaces it is possible to find a Cartan involution 0 that commutes with r (cf. [99], p. 337). Consequently we have g = Z ~ p = I j ~ q = ~k ~q/,,

9

~qp,

(I.7)

where ~ = g ( + l , 0 ) , p = g ( - 1 , 0 ) and the subscripts k and p, respectively denote intersection with {~ and p, respectively. The above decompositions of g are orthogonal w.r.t, the inner product (. ]')0 on g defined in (A.9). Any subalgebra of g invariant under 0 is reductive (cf. [168]). In particular, the Lie algebra f) is reductive. Moreover, the group H is 0-invariant and 0 induces a Cartan decomposition H = ( H N K) exp lip.

(1.8)

In fact, let x = k e x p X E HwithkE KandX Ep. Thenx =r(x) = r(k) e x p r ( X ) . Thus k = r(k), X = r ( X ) . In particular, exp(X) e Ho C H whence k = x e x p ( - X ) E H. []

1.2

Dual Symmetric Spaces

In this section we fix a symmetric pair (g, r) with semisimple g and a Cartan decomposition 0 of g commuting with r. Denote the complex linear extension of 0 to gc also by 0 and recall the involutions a and r] from Definition 1.1.13. The decomposition (1.7) of g allows us to construct further symmetric pairs. 1.2.1

The

c-Dual

Space

A4 ~

The fixed point algebra of r/is gc = t} + iq. Then we have two natural symmetric spaces ~ associated to the symmetric pair (go, r), the universal symmetric space M c (cf. Def. 1.1.12) and the space ~d c (cf. (1.6)). We call M e the c-dual of .M.

CHAPTER1.

SYMMETRIC SPACES

The equality ge = (IJk @ iqp) @(Ijp @ iqk)

(1.9)

yields a Caftan decomposition of fl~ corresponding to the Cartan involution Or. We will write ~ := IJk ~ iqp and p~ := Ijp ~9 iqk . Analogously, we write q~ for i q . If we want to stress which involution we are using, we will write (i~, r)" or (g, r, O)~. Comparing the decompositions (1.7) and (1.9), we see that the c-duality "interchanges" the compact and the noncompact parts of the respective ( - 1)-eigenspaces. There are cases for which c-dual symmetric pairs are isomorphic. The following lemma describes a way to obtain such isomorphisms. L e m m a 1.2.1 Let g be a semisimple Lie algebra with Caftan involution 0 and corresponding Cartan decomposition g = [g+ p. 1) Let X e gc be such that gc = gc(O,X) ~9 gc(i, X) @ g c ( - i , X ) . Further, let r x := Ad(exp r X ) = e ~ad X . Then r x is an involution on gc such that

g c ( + l , r x ) = gc(O,X)

and

g c ( - 1 , r x ) = gc(i,X) ~ g c ( - i , X ) .

If X E ~ tO ip, then g is rx-stable and rxlg commutes with O. 2) If X E ~ tO ip, define

qox = Ad (exp(~X)) = e (~r/2) ad X and

(g,~) := (g,g(1,rx)). Then T2x = r x , TxI~ = id, Txlg(+i,x) = rl:i id, and in the case that X E ip, ~ox defines an isomorphism (g, 11) - (ge, 11). Proof:. All statements except the last part of 1) follow from the fact that

exp (z ad(X))lac(),,x) = e'x id. Let X E t~Uip. Then a ( X ) = X if X E t~ and a ( X ) = - X if X E ip. As r x is an involution, we have r x = r - x . Hence orx =r~(x) o a = r x o

and r x commutes with a. Thus O is rx-stable. That r x commutes with O follows in the same way. El

1.2. D U A L S Y M M E T R I C

1.2.2

The

SPACES

Associated

Dual

Space

A4 ~

The involution T~ := ~-0 is called the associated or a-dual involution of v. We have l)':' = g'" = l)k $ %, qa = g ( - 1 , "ra) = qk 9 and call (g,r,0) a = (g, ra,0)

(1.10)

the a-dual or associated triple. Since all ideals in ]fg are 0-invariant we see that v-invariant ideals are automatically va-invariant and conversely. In particular, we see that (9, v) is irreducible and effective if and only if (0, Ta) is irreducible and effective. To define a canonical symmetric space associated to (g, ra), let G~oa C H a := (K 91 H) exp(qp) C G r~

(1.11)

H a is a group since K 91 H is a group normalizing qv" Thus (G a, H a, ~.a), where G a = G, is a symmetric space. We denote G / H a by a~4a.

1.2.3

The

Riemannian

S p a c e ~d"

Dual

The dual (Riemannian) triple (g', T', 0") is defined by (g', r ' , 0") := (Q', Olg~, O~)~

(1.12)

As al~ - r / , it follows that (go, o) =

=

o

=

(g,o o

Thus

g'=

( g ~ 1 7 6= ( g ~

(:.:3)

Furthermore,

O" = (lJk ~ il}p) @ (iqk 9 qp) = {~"E9p"

(1.14)

is a Cartan decomposition of g" corresponding to the Cartan involution 0" = TIg.. The symmetric pair (g~, r ~) is Riemannian and it is called the Riemannian dual of 9 or the Riemannian pair associated to (g, %9). Let G r be a connected Lie group with Lie algebra gr and K r the analytic subgroup of G ~ with Lie algebra tr. Then M r = G r / K r is said to be the (dual) Riemannian form of .M. We note that M r is automatically simply connected and does not depend on the choice of G r.

10

CHAPTER1.

SYMMETRICSPACES

Example 1.2.2 ( T h e G r o u p C a s e III) Let L be a connected semisimpie Lie group and A4 = (L • L ) / A ( L ) as in Example 1.1.3. The associated symmetric pair is ([ • [, r), where r(X, Y) = (Y, X ) for X, Y E L We define {] := [ • [, G := L • L and H := A(L). Then we have gc = {(X, X) + i ( Y , - Y ) IX, Y e [}, and the projection onto the first component

( X , X ) + i ( Y , - Y ) = ( X + iY, X - iY) ,-.+X + i Y is a linear isomorphism ge _+ [c which transforms r into the complex conjugation a'fr -~ [c with respect to [. Thus the symmetric pair (it ~, r) is isomorphic to (~c, a) (cf. Example 1.1.4). If now Lc is a simply connected complex Lie group with Lie algebra Ir and L~, is the analytic subgroup of Lc with Lie algebra I, then a integrates to an involution on Lc, again denoted by a and LR = L~ (cf. Theorem 1.1.11). Thus Lc/LR is simply connected and hence isomorphic to r Let 0L be a Cartan involution on L. Let [ = Ik ~ ~p be the corresponding Cartan decomposition. Define a Cartan involution on G by O(a,b) = (Oz,(a),Oz.(b)). Then

r~

b) = (O(b), O(.)).

In particular, H

CI

0a

qa

= = =

{ ( ~ , 0 ( ~ ) ) I ~ e L}_~ L, { ( X , 0 ( X ) ) I X e ~} _~ f, {(Y,-0(Y))IY ~. [} _~ I,

and A4 a _~ L, where the isomorphism is now given by (a,b)H a ~-~ aO(b)-1. For the Riemannian dual we notice that

qk = q~ =

{ ( x , - x ) I x e t,} ~_ ~, { ( x , - x ) I x e ~} ~_ ~.

The projection onto the first factor, ( X + i Y + iS + T, X + i Y - i S - T) ~ X + i Y + iS + T, defines an isomorphism {t" -~ Ic mapping Iff into the compact real form u = [k ~iIp. Hence A4 r "~ Lc/U, where U is the maximal compact subgroup corresponding to u. rn

1.2.

DUAL

SYMMETRIC

11

SPACES

E x a m p l e 1.2.3 Let M = G / H be as in Example 1.1.6. We want to determine the dual symmetric spaces of A4. To do this we have to invoke Lemma 1.2.1. We recall that r = riy. .

Hence ~oiy~ = Ad

7ri ) ge exp-~'Yo " g "+

is an isomorphism commuting with r. Furthermore, ~,w. o o = o o ~,,-~o = (o o r) o ~ o w . .

Hence ~iy. 9(g, r, 0) ~ (g', r, 0r) is an isomorphism. We note that ~iy~ = Ch in the notation in Lemma A.3.2, page 255. An e[2-reduction proves the following lemma.

(

L e m m a 1.2.4 X o = Ad e x p - ~ Z o

Yo = [ - Z o , Yo].

Let ~ = T - Z o . As O = rzo we get T o r = r o ~ - 1 = r a o T a n d r a = ViXo. In particular, ~ defines an isomorphism (g, I~a) ,.. (g, Ij). Finally, we have ~OiXo o 0 = 0 o ~o-ix~ = r o ~oix~ 9

As ~OiXolllk~qp = id and ~iXo is multiplication by 4-i on Ijp ~ qk, we get that ~iXo " (g, E) -+ (gr, Er) is an isomorphism interchanging the role of r and 0. We collect this in a lemma.

L e m m a 1.2.5 L e t r = riYo a n d O = rzo as above.

T h e n the f o l l o w i n g

holds: 1) ~OiYo " (g, r, O) -+ ( g ' , r, Or) is an i s o m o r p h i s m . 2,) L e t X o = [-Zo, Yo]. T h e n r " = r i x . is an i s o m o r p h i s m .

a n d ~ o - z , "(g, r,O) --+ (g, r ~, 0)

3) ~OiXo "(g, r , O ) - + (g~, r~,O ~) is an i s o m o r p h i s m .

12

1.3

CHAPTER1.

The Module Structure of

S Y M M E T R I C SPACES

To(G/H)

Let M = G / H be a non-Riemannian semisimple symmetric space as defined in Remark 1.1.15. Let r be the corresponding involution and 8 a Cartan involution of G commuting with r. Further assume that ({~,[j) is irreducible and effective. We will use the notation introduced in Remark 1.1.15. In this section we will study the H- and [}-module structures of q under the assumptions just spelled out. The spaces

qHnK := { X e q lVk e H n K

9 Ad(k)X = X}

qHoNK __ { X ~. q ] Vk ~. Ho ["1K

9 Ad(k)X

(1.15)

and --

X}

will play a crucial role in the study of causal structures on irreducible non-Riemannian semisimple symmetric spaces. We will explore the special properties of symmetric spaces for which qHnK or qHong are nontrivial. L e m m a 1.3.1 Let [ be a O-stable subalgebra of g. Let [• "= { X E glVY E [" B ( Y , X ) - 0}. Then 0 - [(9 [• and the Lie algebra generated by [• gl "- [[• [• + [• is a O-stable ideal in g.

Proof: That g - [• [• follows from the fact that - B ( . , 0 ( . ) ) is positive definite inner product on g. Let X, Y E [ and let Z E [• Then B ( X , [Y, Z]) - - B ( [ Y , X], Z) - O, as [ is an algebra. By the Jacob• identity it follows now that gl is an ideal. As [ is 0-stable, the same holds true for [• and hence for gl. El L e m m a 1.3.2 The algebra [} is a maximal 8-stable subalgebra of 9.

Proof: Let [ be a 0-stable subalgebra of g containing D and assume that [ # 9. Then [• # {0}. Furthermore, [• C D-L = q. Now the above lemma shows that [[• [• ~ [• is an r-stable ideal in g. As (0, [}) is assumed to be irreducible, we get [[• [• @ [• = g. Thus [• = q and the lemma follows. El R e m a r k 1.3.3 Lemma 1.3.2 becomes false if we replace 0-stable by Tstable as is shown by s[(2,R) with the involution r:

(~ C

(o --"

--C

--a

1.3. THE MODULE S T R U C T U R E OF T o ( G / H ) In this case ~ = R

0

-

0)

13

. Furthermore,

D C Pmin =

{(~ 0

--a

a,x E R

}

,

Pmin is T-stable and a proper subalgebra of 5[(2, R). L e m m a 1.3.4 Let (g, 7") be an irreducible effective semi.simple symmetric pair. If q is not irreducible as an D-module, then the following holds.

1) q splits into two irreducible components, q+ := I](1, yO)

and

q- := 11(-1, yO),

(1.17)

such that O(q+) = q- for any Caftan involution 0 commuting with T. 2,) The submodules q+ are isotropic for the Killing form and abel• subalgebras of g. 3} The subalgebras 19+ q+ of g are maximal parabolic. 4) The ~-modules q+ and q- are not isomorphic. In particular, q+ and q- are the only nontrivial O-submodules of q. Proof:. Let V C q be an I~-submodule and V • C q its orthogonal complement w.r.t, the Killing form B, which is nondegenerate on el. But b := V + IV, V] and note that this is an Ij-invariant subalgebra of g. Moreover, we have B([V • V], Ij) = B ( V • [V, Ij]) C B ( V i , V) = {0}. Since the restriction of B to I~ is also nondegenerate, we conclude that

[v

v] = {0}.

If V C q is a nontrivial irreducible Ij-submodule which is not isotropic, then the restriction of B to V is nondegenerate and so q = V @ V • Since [b, V • = {0} and [b, V] C V, we see that b is a r-stable ideal of g, i.e., equal to g. But this is impossible unless V = el. Thus every proper submodule of q must be isotropic. Moreover, if q+ is such a submodule, the subspace q+ + 0(q +) is also an Ij-submodule which is not isotropic, hence coincides with q. To complete the proof of (1) and (2) we only have to note that (q+)• = q+, since q+ is isotropic and of half the dimension of q. To show (3), note first that we now know

[q+,q-] c

[q+,

c

q+, [q+,q+] c {o}.

CHAPTER1.

14

S Y M M E T R I C SPACES

Therefore q+ acts on g by nilpotent linear maps and the subalgebra [~+ q+ is not reductive in g. On the other hand, it is maximal since any subalgebra of g strictly containing [}+ q+ contains an [}-submodule of q strictly containing q+, i.e., all of q. Now [10], Chap. 8, w Cor. 1 of Th. 2, shows that [j+q+, and hence also [~ + q- is parabolic. Part (3) shows that there is an 0 i~ X E [}f'l p that is central in ~) and for which ad(X)l,+ has only positive eigenvalues. Since O(X) = - X , it follows that ad(X)l q- has only negative eigenvalues. Thus q+ and q- cannot be isomorphic as D-modules. D Let [ be a Lie algebra and a, b C [. Then we denote the centralizer of a inbby a~(,) = { x e b IVY e , : [x, Y] = 0 }. (1.18) The center a,([) of[ will simply be denoted

by a(0.

L e m m a 1.3.5 Let X G q. Then the following holds.

1) Assume X e qk. Then [X, Ijk] = 0 if and only if [X, qk] = O, i.e., ~,~ ( ~ ) = ~,~ (qk) = 3(~) n q. 2) Assume X e qp. Then [X, Ijk] = 0 if and only if [X, qpl = o, i.e., 3,,(~k) = 3,, (%) = ~(~") n q.

3) qHOK and qHoOg are O-stable, i.e., qHOK ___ qHnK ~

qp _zing

and qUong = qHong (~ qpHoOK

~) qHong C 3([~) and qp ..ltong C ~(Da)

Proo.fi. It is obvious that qHng is 0-stable. Let B(., .) be the Killing form of g. Then B(., .) is negative definite on [~. Let X E qk be such that [X, t)k] = 0. If Y e qk, then IX, Y] e Dk. Thus 0 >_ B([X, Y], [X, Y]) = - B ( Y , [X, [X, Y]I) = 0. Thus IX, Y] = 0. The other claims are proved similarly,

n

From Lemma 1.3.5.4) we immediately obtain the following corollary. C o r o l l a r y 1.3.6 If qffonK # 0, then ~(~) ~t O.

0

L e m m a 1.3.7 Suppose that qttong ~s {0}. Then H ~t K and one of the following cases occurs:

1) 9 is noncompact, simple with no complex structure, and r is not a Cartan involution.

1.3. THE M O D U L E S T R U C T U R E OF T o ( G / H )

15

2) g "-- ~ 1 X gl, ~where 01 is noncompact, simple with no complex structure, and 7" is the involution (X, Y ) ~ (IF, X). 3) g is simple with a complex structure and I} is a noncompact real form

oSg. Proof. If H = K, then A4 = G / K is Riemannian. Then [t is simple since (g, D) is irreducible and effective.. Therefore qHnK C p and qHnK commutes with t~. But this is impossible, as t~ is a maximal subalgebra of [t (apply Lemma 1.3.2 to 8). In order to prove the lemma, according to [33], p. 6, we have to exclude two further possibilities: a) Suppose that [t is complex and r is complex linear. Then it follows from Lemma 1.3.5 t h a t qHonKis a complex abelian algebra that commutes with D since I) - Dk 9 iI)k. But this contradicts Lemma 1.3.2. b) Suppose that [t = [tl x [11, where {tx is noncompact, simple with complex structure, and r is the involution (X, Y) ~-~ (IF, X). This case is c-dual to Case a), as can be seen from Example 1.2.2 (we have to use the complex structure on {tc given by the identification with ({tx x {t~) x ([1~x Its )), so it also cannot occur if qHonL ~s {0}. r-1 For information and notation concerning bounded symmetric domains, refer to Appendix A.3. As 7" commutes with 0, we can define an involution on G / K , also denoted by 7", via r ( a g ) = r ( a ) g . T h e o r e m 1.3.8 Suppose that .M = G / H is an irreducible effective nonRiemannian semisimple symmetric space such that qH.nK ~ {0}. Then we have:

1) G / K is a bounded symmetric domain, and the complex structure can be chosen such that 7": G / K -r G / K is antiholomorphic. 2) g is either simple Hermitian or of the form gl • gl with Itl simple Hermitian and T(X, Y) = (Y, X), X, Y G {tl.

s)

{0}

Proof. Lemma 1.3.5 shows that {0} ~ qHonK C ~(I~). Therefore, according to Lemma 1.3.7, It is either simple, of the form It = l}c with Ij simple, or of the form g = {tl • gl with {tl simple. Suppose that Case 3) of Lemma 1.3.7 holds, i.e., r is the complex conjugation of g with respect to I}. Then iqk =Ijp so that Ijp has a (HonK)-fixed point which contradicts the assumption that tl is simple. Suppose that Case 1) of Lemma 1.3.7 holds. Then qff~ C ~(t~) so that [t is Hermitian. In fact, we even see that qHonK = ~(~) = R Z 0 in the

CHAPTER1.

16

S Y M M E T R I C SPACES

notation of Appendix A.3, and obviously Z ~ E qHnK. Note that ad(Z ~ induces a complex structure on p and then, by K-invariance, also on G / K . As r(Z ~ = - Z ~ it follows that r : G / K -r G / K is antiholomorphic w.r.t. this complex structure. Finally we suppose that Case 2) of Lemma 1.3.7 holds. Then, similarly as for Case 1), we find

qHonK k

= {(x,-x)

lx e

3(~,)}.

Thus $(1~1) # {0} and {Ix is Hermitian. Let Z define a complex structure on G x / g x . Then Z ~ = ( Z , - Z ) e q~nK defines a complex structure on G / K anticommuting with r. This implies the claim, cl R e m a r k 1.3.9 There is a converse of Theorem 1.3.8, as Example 1.1.5 shows: Every antiholomorphic involution of a bounded symmetric domain D = G / K fixing the origin o - { g } gives rise to an involution 7- with qHong r {0}. This follows from the fact that an involution on D is antiholomorphic if and only if it anticommutes with ad(Z~ But then [r(Z ~ + Z ~ g] C t~, which implies 7-(Z~ = - Z ~ Therefore v and/9 commute. Consider the special case G = G1 • G1. Then we have

rCz,,l,) = (~,z) for z, w ~. Gx/K1, where the complex structure on the second factor is the opposite of the complex structure on the first factor. 1:3 L e m m a 1.3.10 Suppose that .A4 = G / H is an irreducible non-Riemannian semisimple symmetric space with q/-t.nK ~ {0}. Then we have:

1) Z(H) n K is discrete. In particular, the center of b is contained in p. Furthermore, dime(b) < 1.

z) Ho n K is connected with Lie algebra bt. s) b~

c

[b, b].

~) Let H~o be the semisimple analytic subgroup of G with Lie algebra [I3, I3]. Then H o O K = H~o n K. Proo~ If ~(b) -~ {0}, then g is simple without complex structure (Lemma 1.3.7). Then [44], p. 443, implies that Itc is simple. Thus gr is also simple. Since ~(b) is 0-invariant, we have

~(~") = ~(b) n t + i(3(b) n p) # {o}.

1.3.

T H E M O D U L E S T R U C T U R E OF T o ( G / H )

17

Therefore gr is Hermitian and ~(t r) is one-dimensional. Thus ~([j) C [}k or ~([}) C [Jp. Moreover, ad Zlpr is nonsingular for every nonzero Z e 3(tr). 1) Let Z e ~(I~) be nonzero. If Z e tJk, then ad(Z)]q,onK - 0, which contradicts the regularity of ad(Z)lpr. Thus ~([}) C [}p, and 1) follows. 2) Since Ho is t?-invariant, the Cartaninvolution of G restricts to a Cartan involution on Ho as was noted in (1.8). Therefore K N Ho, being the group of 0-fixed points in the connected group Ho is connected. 3) This follows from the 0-invariance of [tj, [}] and ~([~) C p. 4) In view of 1) and 2), this is an immediate consequence of 3). o T h e o r e m 1.3.11 Let .M - G / H be an irreducible non-Riemannian semisimple symmetric space with qHonK ~t {0}. Then the following statements are equivalent: 1) dim3([})= 1. 2) q is reducible as an [}-module. 3} dimqH~

> l.

~) dimqH~

= 2.

5) G / K is a tube domain and up to conjugation by an element of K , we have [} = g(0, }To) (cf. Appendix A.3 for the notation). If these conditions are satisfied, then in addition the following properties hold: a) There exists an, up to sign unique, element yo e }([J) N p such that

b = g(O, rO),

and

q = {](+1, yO) (9 g ( - 1 , yO)

(1.19)

is the decomposition of q into irreducible l)-modules. b} The spaces g(-l-1, y o ) are not equivalent as lj-modutes and

0(9(+1, yO)) = g(-1, yO).

g(+1. yO).o K # {o} # g(-z. yo).~

..d

qHonK= g(+l, yO)HonK EBg(-1, yO)HonK. _HonK = dimqp _Honk = I d) dimqk

e) qk _Honk

_

q~nK

(1.20)

18

CHAPTER 1. SYMMETRIC SPACES

Proof: (1)~(2): Suppose that ~(~) # {0}. Then up to sign there is a unique Z ~ E ~(tr) such that ad(Zr~ has the eigenvalues • on p~. since g~ is Hermitian (cf. L e m m a 1.3.10). Then yo := _iZ o e ~(0)n p and adqc(Y ~ has eigenvalues 1 and -1. Let ~ 9gc -4 gc be the conjugation with respect to g. Then a(yo) = yo. Hence the eigenspaces g(j,yo), j = -1, 0, 1, are ~r-stable. It follows that go(j, yo) = g(j,YO)c" Therefore 0(yo) = _yo implies g(• yo) # {0}, q = g(+l, yo) ~ g(-1, yo) and, 0(9(+1, yo)) = 9(-1, yo). As yo is central in 0, we get that g(+l, yo) and 9(-1, yo) are [j-invariant.This shows that q is reducible as 0-module, i.e., (2). (2)=~(1): Conversely, suppose that q is reducible. Then q is the direct sum of two irreducible t-modules q• with 8(q +) = q- (Lemma 1.3.4). We get {0 } # qHonK = (q+) Honk q~ (q-) Honk

and both of the spaces on the right-hand side are nonzero. Let 0 # X E (q+) Honk and define

z := IX, 0(x)]

p n [q, q] c

Then Z commutes with 0k. Apply Lemma 1.3.5 to the involution r o 0 to see that Z is central in 0. Thus we have to show that Z is nonzero. To do that define Y := X + O(X) E qHonK VI it. Then Y E ~(~) by Lemma 1.3.5 and Y # 0. From Lemma 1.3.7 we see that g is either simple Hermitian or the direct sum of two copies of a simple Hermitian algebra. Calculating in each simple factor separately, we obtain o # IX + o ( x ) , x - 0(X)l = - 2 z .

(1),(2)=~ a),b),c): Assertion a) follows from Lemma 1.3.4 and the fact that y0 has different eigenvalues on the two irreducible pieces. To prove b) and c), consider 0 # X E qHonK. Let X+ and X_ be the projections of X onto g ( + l , y0) and g ( - 1 , y0), respectively. Then X• E g ( + l , yO)HonK, and either X+ or X_ is nonzero. Obviously, 0 (0(+1, yO)HonK) = 9(--1, y0)HonK, and 0 o Ad(k) = Ad(k) o 0 for all k e H f3 K. Thus (1.20) follows. (3)=~(1): Suppose that dim qUonK > 1. As the commutator algebra [0, [J] is semisimple, it follows from Lemma A.2.5 that q is reducible as an [0, [}]-module. If 3(0) were zero we would have (2) and a contradiction to the equivalence of (1) and (2). (1)=~(4): In view of Theorem 1.3.11.4), we see that Lemma A.2.5, applied to H'o, proves dim(q HonK) = 2, since q contains precisely two irreducible

[b, b]-submodules by a).

1.3.

19

T H E M O D U L E S T R U C T U R E OF T o ( G / H )

(4)=~(3): This is obvious. (1)=~ d)" Since the spaces g(4-,Y~ H~ get interchanged by O, we see that qUonK contains nonzero elements in ~ and in p. This proves dim(q+ ) HoNK dim(q- ) HoNK 1. .

_

. - -

( z ) ~ ~): Apply Theorem Z.3.S to o b t ~ q ~

= e H - ~ # {0} ~om d).

(1)=~(5): Since qHong r {0}, we can apply Theorem 1.3.8, which shows that g is Hermitian since I) is not semisimple. Choose a maximal abelian subspace a ~ of p containing y0. Since {j = 9(0, y0), we have a ~ C f). The restricted roots vanishing on y0 are precisely the restricted roots of the pair (I), a~ In particular, there is at least one restricted root not vanishing on y0. But the spectrum of ad(Y ~ is { - 1 , 0, 1} so that Moore's Theorem A.3.4 shows that G / K is of tube type (cf. also Theorem A.3.5). There exists a k E K such that Ad(k)a ~ = a is the maximal abelian subspace of p described in Appendix A.3. Thus we may assume that y0 E a. Conjugating with a suitable Weyl group element, we may even assume that y0 in the closure of the positive Weyl chamber. But then all positive restricted roots which do not belong to the pair (I), a) take the value 1 on y0, which proves that y o _ Yo. (5)=~(1): Since f) is the centralizer of an element in g it clearly has a nontrivial center. [::] In the following we will choose an element y0 E 3({))N p in the way Theorem 1.3.11 describes it, whenever it is possible. E x a m p l e 1.3.12 Consider the conjugation a

o 0)( c

b

0) ( b a )/

(1.21)

on G - SL(2, R). It commutes with the Cartan involution 0(g) - tg-Z and satisfies G~=+

h(t):=

sinht

cosht

(

(coss sins

-sins) coss

teR

.

Moreover, we have K=G ~

k(s):=

} seR

.

Note that G is contained in its simply connected complexification Gc SL(2, C). The above formula defines an involution v on Gc which on 9c is the complex linear extension of v on 9. The involution rI on Gc is given by r/

c

d

-

b

CHAPTER 1. SYMMETRIC SPACES

20

which implies (~e = SUo(1,1) and [I = S0(1,1). In particular, we see that /-/is not connected. The corresponding space ~ / = G/[-I can be realized as the Ad(G)-orbit of the element

(10) 0

-

E g, and this orbit is

{(o~_ ~ ~1+ ~ )}, o,, +_~ -o ~ , i.e., a one-sheeted hyperboloid. On the Lie algebra level, r is given by the same conjugation and we find

1)

1 0 qk

=

R(01

qp

---~

R(O

Cp,

1

O)'

-

o)

9

Moreover, we see that gc = au(1,1) and t e = iqp. Set

1(o 1)

xO

-

1(1

01)

~%'

-

2

0 -

--

2

1

Y+

=

(O 1 ) = Y ~ 1 7 6 0 0

y_

=

(00)=yO+zO-O(Y+)=T(Y+) 1 0 -

1(o :)

Eqk,

X+ -

1(1 - 1 ) = x O + zO, ~ 1 _1

x_

~

=

1( 1

-x-t

1 )=X~176

=

'

-O(X+).

Then we have

g(+l, yO) = R X•

g(0, yO) = R yO = [j.

The spaces g(d:l, yO) are the irreducible components of q as G r- and hmodules. More precisely, we have

Adh(t)(rX+ + sX_) = e2trX+ + e-2tsX_.

1.3.

T H E M O D U L E S T R U C T U R E OF T o ( G / H )

21

Further, we have qG r AK = R Z O,

qp -G~AK = RX 0

spec(ad X ~) = spec(ad yO ) = { _ 1, O, 1}, and spec(ad Z ~ = {--i, O, i}.

El

C o r o l l a r y 1.3.13 Let A4 be an irreducible non-Riemannian semisimple symmetric space with qHonK ~ {0}. Then q is reducible as an [~-module if and only if (ge, [j) is isomorphic to (g, [j). In that case there exists an y o E 3 ( IJ) n p such that

r = exp(i~ ad yO)

(1.22)

1to = H'o x e x p R Y ~ .

(1.23)

and Proof. Suppose that q is reducible as an h-module. Then Theorem 1.3.11 shows that we can find an element y0 E 3([~)f3 p such that q = g ( + l , y0) ~Bg ( - 1 , yO).

Now apply L e m m a 1.2.1 to i Y ~ to see that r = Tiyo and that ~Oixo : (g, [7) -} (go, [9) is an isomorphism. Conversely, suppose that (g, Ij) and (go, Ij) are isomorphic. Choose an isomorphism A: gc __+ g. We may assume that qffonK # {0}. In fact, otherwise we replace A by A-l"g -+ go. As iqk = q~, we get (qC)~onK # {0}. But qk n A (q~) = {0}, so dim qHonK = 2. Thus Theorem 1.3.11 shows that q is reducible as an [} module and [~ = [t}, [~]+ R Y ~ Considering the Cartan decomposition of Ho, we see that we also have the global version of this fact, which is (1.23). E! Lemma

1.3.14 Let .A4 be an irreducible non-Riemannian semisimple symmetric space. Suppose that q is reducible as an Ij-module but irreducible as an H-module. Fix y o as in Theorem 1.3.11 and set Hx = {h e H I A d ( h ) ( g ( + l , yO)) = g ( + l , y 0 ) } . Then H1 is a O-stable normal subgroup of H . element k E K t'l H such that

Moreover, there exists an

Ad(k)g(+l, yO) = g(-1, yO) and

H = H~ O kHa.

CHAPTER1. SYMMETRIC SPACES

22

Proo~ We write q• := 0(4-1, y 0 ) for the ~-irreducible submodules of q. Since q is irreducible as an H-module, and H = ( H N K)expflp by (1.8), we can find a k E K f3 H such that Ad(k)q + r q+. As Ad(k)q + f3 q+ is Ho-stable and q+ is irreducible as an Ho-module, we get Ad(k)q + N q+ = {0}. Thus q = q+ ,~ Ad(k)q + = q+ ~ q-. The Ho-representations q+ and q- are inequivalent by Theorem 1.3.11. Thus Ad(k)q + = q - . Fix an h E H \/'/1. Then it follows as above that Ad(h)q + = q- = Ad(k)q +. Hence A d ( k - l h ) q + = q+ and/-/1 is subgroup of index 2 in H. Therefore/-/1 is normal in H. To prove the 0-stability let h E/-/1 and recall that exp p C Ho C/-/1. Then (1.8) shows that there is a k E H f3 g and an X e ~p with h = k exp(X). Thus k = h e x p ( - X ) E/-/1 and H1 is 0-stable. [::! The following example shows t h a t the situation of L e m m a 1.3.14 actually occurs. E x a m p l e 1.3.15 Let jVI = G/H be as in Example 1.1.6 and consider the space A d ( G ) / A d ( G ) r, where Ad(G) r = {~ E Ad(G) [ ~ r = r~}. Then A d ( G ) / A d ( G ) ' . is a non-Riemannian semisimple symmetric space which is locally isomorphic to ,~4. Moreover, 0 E Ad(G)'., so that q is irreducible as an Ad(G)"-module. D

1.4

A-Subspaces

Let ,Ad = G/H be a symmetric space with G connected semisimple. In this section we recall some results from [143] on the orbit decomposition of M with respect to H. It can be skipped at first reading because there are no proofs in this section and the results will be used only much later. A maximal abelian subspace aq of q is called an A-subspace if it consists of semisimple elements of g. For X E q consider the polynomial n

det (t - a d ( X ) ) = ~

dj(X)t j.

j--1

Let k be the least integer such that dk does not vanish. Then the elements

q' := { x e q I d (X) # O}

23

1.4. A-SUBSPACES

are called q-regular elements . The set of q-regular elements is open and dense in q. Let r G --4 G be the map defined by r

----gT-(g-1).

This map factors to G / G r and defines a homeomorphism between G / G r and the closed submanifold r of G ([143], p.402). We have the fonowing maps:

G

~

G/H GIG ~

For x E r

~ r

consider the polynomial n

det (1 - t - Ad(x)) - ~

D j ( x ) t j.

j-----1

Then the elements of

r

:=

er

I Dk( ) # O}

are called the r G)- regular elements . The centralizer

Aq := ZCj(G) ( aq ) of an A-subspace aq is called an A-subset. We set A~ := Aq n r Oshima and Matsuki, in [143], prove the following results. Theorem

1.4.1 Let the notation be as above. Then the .following holds:

1) Each A-subspace is conjugate under (Gr)o to a O-invariant one. 2) The number of H-conjugacy classes of A-subspaces is finite. number can be described in terms of root systems.

This

3) The decomposition of an A-subset into connected components has the

/o,~n Aq = U k1 expaq, jEJ

where k i e r and J is finite if the center of G is finite (which is the case if G C Gc).

C H A P T E R 1. S Y M M E T R I C S P A C E S

24

4) If aq is an A-subspace, then Z4~(Gc) ((aq)C) = exp(aq)C.

0

is open dense in r

5) r

T h e o r e m 1.4.2 Let (aq,x , . . . , aq,r) be a set of O-invariant A-subspaces representing the H-conjugacy classes. Then the set r

U ~-1 (a'~,j)

j=l

is open dense in G.

1.5

The

0

Hyperboloids

Let p, q E Z +, n = p + q, and V = IR". We write elements of V as v = (~) with x E IRP and y E ]Rq. Note t h a t for p = 1, x is a real number. We write pr 1( v ) " = x and pr2(v ) = y. Define the bilinear form Qp,q on JR" by

Qp,q(V,W)

=

(prx(v)lprx(w))-(pr2(v)lpr2(w))

=

VlW 1 "~-...

"~" V p W p

-- l)p+lWp+

1 --...

(1.24) -- VnW n.

(1.25)

Here (.1.) is the usual inner product on R n. F o r r E I R +, p, q E N , n = p + q _ > l d e f i n e Q~q = Q_~ := {v e R "+x

and

I Qp,q+~ (v, v)

= - r 2}

QP'q + ~ = Q+~ := {x E R n+l I Q~+I,~(~,~) = +~2}.

T h e n Q•

(1.26)

(1.27)

has dimension n and n Tm(Q+~) "~ {v e ll~n+l " p + l [ Qp+x,q(v,m) = 0} ~- Rp.

T h e linear isomorphism Ln"

R" ~ t(Xl,...,~,)~ t(X,,...,Xl) E Rn

satisfies Qq,p o (Ln, Ln) = -Qp,q. Hence Ln+I maps ~+rg'lP'qbijectively onto Qq_'P~. Furthermore,

s~ • R. ~ (~, ~ ) ~ F(~, ~ ) . = ' (~, V'~~ + II~ll2 ~) ~ R "§

1.5. THE HYPERBOLOIDS

25

satisfies

Qv,q+l(F(v, w),F(v, w))

=

II~ll 2 -- ( ~ + IIwll ~) = - - ~

and induces a diffeomorphism

Q_,. ,,,, S q • R v,

(1.28)

where the inverse is given by

(:) (,,+,, Using that on Q_., we have Jlwll2 = r 2 + Ilvl[2. In the same way, or by using the map Ln+l, it follows that S v x R n-v "., Q+r. In particular, Q - r , respectively Q+r, is connected except for p = n (respectively p = 0), where it has two components. Let O(p, q) be the group

o(v, q):= {~ e GL(,,R)I W, w e R". Q~.~(~, ~ ) = Q~.~(~,~)}. Consider lv,q := ( Iv 0

O ) , where Im is the identity matrix of size m x m. -Iq

With this notation we have

O(p,q) = {a e GL(n,R) l taIv,qa = Ip,q}. In block form O(p, q) can thus be written as the group of block matrices (cA ~9) e GL(n, R) satisfying the following relations: AEM(p,R),

B e M(p • q,R),

DeM(q,R),

C e M(q • p,R),

and t A A _ tCC tAB-tCD t B B - tDD

= = =

Iv, O, -Iq.

(1.29)

Here M(l • m, K) denotes the m • matrices with entries in IK and M(m, K) "= M ( m x m, IK). The group O(p, q), pq ~ 0 has four connected components, described in the following way, writing (~ D B) E O(p, q). 1) d e t A > 1 and d e t D > 1 (the identity component), representative In,

26

CHAPTER1.

SYMMETRIC

2) det A > 1 and det D < - 1 representative ( I P --"

---

k 0

o) o) Ip--1,1 01 q--l,1

'

det D > 1 representative ( Ip_ 1,1

3) det A < - 1 and m

4)

'

SPACES

"="

'

k

0

det A < - 1 and det D < - 1 representative( ~

'

k

Iq--l,1

0

"

Let SO(p, q) := 0(/9, q) N SL(n, R). Then

O(p, q)o = SOoO,, q) := SO(p, q)o. For the Lie algebras we have

..(p. q) = so(p. q) = { x e gt(.. R) I z,..~x + 'xz,..~ = o) or, in block form, so(p, q) =

C

tA = -A,'D = -D, tB

D

= C}.

(1.30)

It is clear that O(p + 1, q) acts on Q+r. Let { e l , . . . , en } be the standard basis for R".

L e m m a 1.5.1 For p , q > 0 the group SOo(p + 1, q) acts transitively on Q+r. The isotropy subgroup at re1 is isomorphic to SOo(p, q). Whence, as a manifold, Q+r " SOo(p + 1, q ) / S O o ( p , q). Proofi. We may and will assume that r = 1. Let v = (~) 6 Q1. Then, using Witt's theorem, we can find A 6 SOo(p + 1) and D e SOo(q) such that

A~--II~lle,

As(A O) 0

D

and

Dy--IlYlle~+~.

E SOo(p + 1, q), we may assume x = Ael and y = pep+2 with

A, p > 0 and A2 - # 2 = 1. But then a(el) = v with

a

--

A 0 0 Ip p 0

p 0 A

0

o I~_~

0

0 / 0 0 eSOo(p+l,q),

where the block structure of a is according to the partition (1, p, 1, q - 1) of n + 1. The last statement is a direct calculation and is left to the reader.[:]

1.5. THE HYPERBOLOIDS

27

Now Ln+l(r, 0 , . . . , 0) - ( 0 , . . . , 0, r), and on the group level the conjugation Ad(Ln+l)" a ~-~ Ln+laL'~. 1 sets up an isomorphism of groups O(q + 1,p) ~_ 0(/9, q + 1) mapping the stabilizer of (r, 0 , . . . , 0) onto the stabilizer O(p, q) of ( 0 , . . . , 0, r). Thus Q - r -~ 0(/9, q + 1)/0(/9, q) = SOo(p, q + 1)/SOo(p, q). Next we show that the hyperboloids Q:t:r are symmetric spaces. We will only treat the case O(p + 1, q)/O(p, q), as the other case follows by conjugation with Ln+I. The involution 7 on 0(/9 + 1, q) is conjugation by II,n. Then 0(/9 + 1, q)r = SOo(p, q) C O(p + 1, q)re, C. O(p + 1, q)r. The involution ~" on the Lie algebra g = $o(p + 1, q) is also conjugation by Ii,n. Then [~ = so(p, q) = it r and defining

(0 .(v,..))

q(,,) :=

v

0

(1.al)

for v E R n we find a linear isomorphism R n B v ~

q(v)

e q

(1.32)

which satisfies

q(av)

--

aq(v)a -1, a e SOo(p,q),

Qp,q(V, W)

=

--~1 Tr q(v)q(w).

The c-dual of a hyperboloid is again a hyperboloid or at least a covering of a hyperboloid. More precisely, (so(2, n -

1),so(1,n-

1))c= (so(1, n),so(1, n-- i))

which, by abuse of notation, can be written Q~.i "" Q-1. In fact, let q and ql denote the map from (1.32) for the case of (so(2, n - 1), so(l, n - 1)) and (so(1, n ) , s o ( 1 , n - 1)), respectively. For X e so(l, n - 1) C 5 o ( 2 , n - 1) we define A(X) e so(l, n - 1) C so(l, n) by

x)) (x o) Then

s . . ( 2 . . - 1) ~ 9 x + iq(v) ~ a ( x ) + qx (v) e 5o(1..)

CHAPTER1.

28

S Y M M E T R I C SPACES

is a Lie a l g e b r a i s o m o r p h i s m .

Notes for Chapter 1 The material of the first section in this chapter is mostly standard and can be found in [81, 97]. The classification of semisimple symmetric spaces is due to M. Berger [5]. The dual constructions presented here and the relations between them can be found in [146]. The importance of the c-dual for causal spaces was pointed out in [129, 130], where one can also find most of the material on the D-module structure of q. The version of Lemma 1.3.4 presented here was communicated to us by K.-H. Neeb.

Chapter 2

Causal Orientations

In this chapter we recall some basic facts about convex cones, their duality, and linear a u t o m o r p h i s m groups which will be used t h r o u g h o u t the book. T h e n we define causal orientations for homogeneous manifolds and show how they are determined by a single closed convex cone in the tangent space of a base point invariant under the stabilizer group of this point. Finally, we describe various causal orientations for the examples t r e a t e d in C h a p t e r 1.

2.1

Convex

Cones

and Their Automorphisms

Let V be a finite-dimensional real Euclidean vector space with inner product ('l'). Let ]R+ := {)~ E R I)~ > 0} and R + = IR+ N {0}. A subset C C V is a cone if ]R+C C C and a c o n v e x cone if C in addition is a convex subset of V , i.e., u, v e C and A E [0,1] imply )~u + (1 - )0v E C. Thus C is a c o n v e x cone if and only if for all u, v E C and )~, # E IR+, )~u + #v E C. The cone C is called n o n t r i v i a l if C ~ - C . Note t h a t C ~ {0}, and C i~ V if C is nontrivial. We set 1) V c := C M - C ,

2) < c > : = c - c = 3) C* :=

-

{=eVlV, e c : (,1=)_>0}.

T h e n V c and < C > are vector spaces called the edge and the s p a n of C. The set C* is a closed convex cone called the d u a l cone of C. Note t h a t 29

CHAPTER2.

30

CAUSALORIENTATIONS

this definition agrees with the usual one under the identification of the dual space V* with V by use of the inner p r o d u c t (-[.). If C is a closed convex cone we have C** = C and (C* n - C * )

=< C >~,

where for a subset U in V we set U x = {v e V lVu e U

(2.1) 9 (u Iv) = 0}.

D e f i n i t i o n 2.1.1 Let C be a convex cone in V. T h e n C is called generating if < C > = V and pointed if there exists a v E V such t h a t for all u E C \ {0} we have (u [ v) > 0. If C is closed, it is called proper if V c = {0}, regular if it is generating and proper, and and self-dual if C* = C. T h e set of interior points of C is denoted by C ~ or int(C). T h e interior of C in its linear span < C > is called the algebraic interior of C and denoted algint(C). Let S C V. T h e n the closed convex cone generated by S is denoted by cone(S)"

cone(S) "= { n~itr.s.

seS,r.>_0}.

(2.2)

T h e set of closed regular convex cones in V is denoted by Cone(V).

El

If C is a closed convex cone, t h e n its interior C ~ is an open convex cone. If f~ is an open convex cone, then its closure ~ "= d ( f l ) is a closed convex cone. For an open convex cone we define the dual cone by f~* "= {u E V IVy E ~ \

{ 0 } ' ( u Iv) > 0} = int(~*).

If fZ is proper we have 12"* = f~. W i t h this modified version of duality for open convex cones it also makes sense to talk a b o u t open self-dual cones. E x a m p l e 2.1.2 ( T h e F o r w a r d L i g h t C o n e ) For n _> 2, q = n p = 1 we define the (semialgebraic) cone C in R n by

1 and

c :- {~ e R" I Q,,~(~, ~) >_ 0, x >_ 0} and set

n :=

c ~

= {~ e R" I Q,,~(~, v) > 0, z > 0}.

C is called the forward light cone in R n. We have v = (~) E C if and only if x _> Ilyll. u ~ e c n - c , then 0 < z < 0 and thus x - O. We get Ilyll- o and hence y = O. Thus v = 0 and C is proper. For v, v' E C we calculate

(,/I v) - ~'= + (v' I v) >_ IIv'll Ilvll + (v' I v) >__o

2.1.

CONVEX CONES AND THEIR A UTOMORPHISMS

31

so t h a t C C C*. For the converse let v = (~) e C*. Then it follows by testing against el E ~ t h a t x _> 0. We may assume, t h a t y ~t 0. Define w by pr l(w) = Styliand pr2(w ) = -y. Then w e C and

0 _<

=

II II- llli= = ( -ll ll)llyll.

If follows that x >_ llyll.Therefore y E C so that C* C C and hence C is self-dual. In the same way one can show t h a t also ~ is self-dual, n R e m a r k 2.1.3 Let C be a closed convex cone in V. Then the following are equivalent:

1) C ~ is nonempty. 2) C contains a basis of V.

3) < C > = V .

o

P r o p o s i t i o n 2.1.4 Let C be a nonempty closed convex cone in V . the following properties are equivalent:

Then

1) C is pointed, 2) C is proper. 3) int(C*) r 0. Prooj!. The implications "(1)=~(2)" and "(3)=~(1)" are immediate. Assume now t h a t C is proper. Then by (2.1) we have < c* >= (vC) • = v,

so C* is generating. Now R e m a r k 2.1.3 shows int(C*) # 0 and (3) follows. ff] C o r o l l a r y 2.1.5 Let C be a closed convex cone. Then C is proper if and only if C* is generating. Q C o r o l l a r y 2.1.6 Let C be a convex cone in V . Then C E Cone(V) if and only if C* E Cone(V). B

A ]ace F of a closed convex cone C C V is subset of C such t h a t v, v ~ E C\Fwithv, v~ E C i m p l i e s v E C \ F o r v ~EC\F. We denote the set of faces of a cone C by Fa(C). Note t h a t Fa(C) is a lattice with respect to the inclusion order. A cone C in a finite-dimensional vector space V is said to be polyhedral if it is an intersection of finitely m a n y half-spaces. For the following result see [55].

CHAPTER2.

32

CAUSALORIENTATIONS

R e m a r k 2.1.7 Let V be a finite-dimensional vector space and W C V. Then 1 ) - 3 ) are equivalent and imply 4), and 5). 1) There exists a finite subset E C V such that W = cone(E) =

~eER+v.

2) W is a polyhedral cone. 3) The dual wedge W * C V* is polyhedral. 4) For every face F 6 Fa(W) there exists a finite subset D C E such t h a t the following assertions hold: a) f = cone(D). b) W - F = cone(E O - D ) is a cone. c) V W - F = f -

f = cone(D U - D ) .

d) < F > A W = F . 5) The mapping op" Fa(W) ~ Fa(W*),

F ~ f•

W*

(2.3)

defines an antiisomorphism of finite lattices. Moreover, < op(F) > = F •

VF e Fa(W).

[::]

We introduce some notation that will be used throughout the book. Let W be a Euclidean vector space and let V be a subspace. Denote by p r y or simply pr the orthogonal projection pry: W -+ V. If C is a cone in W , then we define PYw(C), IYw(C) C V by pW(c) = pry(C)

and

I W(C) = C n V .

(2.4)

If the role of W and V is clear from the context, we simply write P(C)

I(c). L e m m a 2.1.8 Let W be Euclidean vector space and let V be a subspace of W . Denote by pr: W --+ V the orthogonal projection onto V. Given C 6 C o n e ( W ) , we hare I(C*) = P(C)*.

Proo~ Let Z e C and let X = pr(Z) E P(C). If Y e I(C*), then (YIX) = (YIZ) > O. Thus Y e P(C)*. Conversely, assume t h a t X e P(C)* and Y e C. Then Y = pr Y + Y• with Y• 2. V. In particular, pr(Y) e P(C). Therefore we have

(YlX) and hence X E I(C*).

(pr(Y)[X) > 0

o

2.1. C O N V E X C O N E S A N D T H E I R A U T O M O R P H I S M S

33

L e m m a 2.1.9 Suppose that r : W --+ W /s an involutive isometry with fixed point set V . Let C E Cone(W) with - v ( C ) = C. Then I W ( C ) = p w ( ~ ) and I W ( C* ) = I W ( C)* . Proof:. Let X = X+ + X _ E C, where the subscripts denote the projections onto the ( + l ) eigenspaces of v. As -~'(C) = (~, it follows that - r ( X ) = - X + + X _ E C. Hence X _ = 2X-(X- v(X)) e C. Thus P ( C ) C I(C). But we always have I ( C ) C P(C). Hence P~C) = I(C). Let Y = Y+ +Y_ e (7". Since -7"((~) = C implies - v ( C * ) = C*, we find Y_ E C*. Lemma 2.1.8 shows that P ( C * ) = I(C)* and thus I ( C * ) = I((~)*. 0 Next we turn to linear automorphism groups of convex cones. For a convex cone C we denote the automorphism group of C by Aut(C) "= {a E G L ( V ) I a(C ) = C}.

(2.5)

If C is open or closed, then Aut(C) is closed in GL(V). In particular, Aut(C) is a linear Lie group. If we denote the transpose of a linear operator a by t a, we obtain Aut(C*) = t A u t ( C ) (2.6) whenever C is an open or closed convex cone. R e m a r k 2.1.10 Equation (2.6) shows that if C is an open or closed selfdual cone in V, then Aut(C) is a reductive subgroup of GL(V) invariant under the involution a ~-+ 0(a) : - ta-1. The restriction of 0 to the commutator subgroup of the connected component Aut(C)o of Aut(C)o is a Cartan involution. D D e f i n i t i o n 2.1.11 Let G be a group acting (linearly) on V. Then a cone C C V is called G-invariant if G . C = C. We denote the set of invariant regular cones in V by ConeG(V). A convex cone C is called homogeneous if Aut(C) acts transitively on C. D For C E Cone(V) we have Aut(C) - Aut(C ~ and C - OC U C ~ (C \ C ~ U C ~ is a decomposition of C into Aut(C)-invariant subsets. In particular, a nontrivial closed regular cone can never be homogeneous. R e m a r k 2.1.12 Let V C W be Euclidian vector spaces with orthogonal projection pry: W ~ V. Suppose that L is a group acting on W and N is a subgroup of L leaving V invariant. Let C C W be an L-invariant convex cone. Then I ( C ) is N-stable. If furthermore t N - N, then P ( C ) is N-invariant, too. In fact, the first claim is trivial. For the second, note that t N = N implies that V • is N.invariant. Hence p r ( n . w) - n . pr(w) for all w E W and P ( C ) is N-invariant.

CHAPTER2.

34

CAUSALORIENTATIONS

Now let c be an N-invariant cone in V. We define the extension ~V,N ~W,L (c) of c to W by (2.7) By,W,L N (C) = conv (L 9C). If the roles of L, N, V, and W are clear, we will write E(C), E L ( C ) ~W,L E W ( c ) instead of ~V,N (C). If N1 is a subgroup of N acting trivially V, then the group N/N1 acts on V. By abuse of notation we replace N N/N1 in that case.

or on by El

T h e o r e m 2.1.13 Let G be a Lie group acting linearly on the Euclidean vector space V and C E C o n e a ( V ) . Then the stabilizer in G of a point in

C ~ is compact. Proo~ Let ft := C ~ First we note that for every v E f~ the set U = f~fl ( v ft) is open, nonempty ( 89 G U), and bounded. Thus we can find closed balls B r ( l v ) C V C BR( 89 Let a e Aut(f~) v = {b e Aut(f~) I b. v = v}. From a.fl C ~ and a.o = v we obtain a(U) C U. Thereforea(Br( 89 C B R ( l v ) and a ( 8 9 89 implies Ila[[ < R/r. Thus Aut(f~) v is closed and bounded, i.e., compact.

El

E x a m p l e 2.1.14 (H+(m, IK)) For K = IR or C we let V be the real vector space H(m, IK) of Hermitian matrices over K, V = H(m, K) := {X G M ( m , K) IX* = X}, where M ( m , K) denotes the set of m • m matrices with entries in lK and X* := t ~ . Then n = dimR V is given by

1) . = m ( ~ + 1)/2 for K = R 2) n = m2 for IK = C~ Define an inner product on V by

(X I Y) := R e T r X Y * ,

X , Y q. V.

Then the set

n = H +(m, K) := { X e H(m, ~) I X > 0},

(2.8)

where > means positive definite, is an open convex cone in V. The closure C of f~ is the closed convex cone of all positive semidefinite matrices in H(m, K). Let Y = Im denote the m • m identity matrix. Then for X G C, X i~ 0, (X I Y) > 0 as all the eigenvalues of X are non-negative and X r 0. Thus C is proper.

2.1. C O N V E X CONES A N D T H E I R A U T O M O R P H I S M S

35

We claim that C and fl are self-dual. To prove that, let e , , . . . , em denote the standard basis for K 'n, ej = t ( 6 I , j , . . . , 6re,j) and define Eij e M ( m , K) by Eijek = 6j,kei. Then E X l , . . . , Emm is a basis for M ( m , K). Now suppose that X, Y 6 f~. Then Y = an* for some matrix a and hence T r ( X Y ) = Tr(a*Xa) > 0 so that f~ C fl*. Conversely, let Y 6 ~*. As Y is Hermitian, we may assume that Y is diagonal: Y = ~ i =m1 )~iEii. Let X - Ejj 6 ft \ {0}. Then

o < ( x IV) = ,x~. Thus Y is positive definite and hence in f~. This proves that f~ is self-dual and hence also the self-duality of C. As a consequence, we see

{0} = C n - C

= C* n - C *

= < C >.L

and C 6 Cone(V). The group G = GL(m, K) acts on M ( m , K) by

a. X := aXa*, a 6 GL(m, K), X 6 M ( m , K) and f~ and C are GL(m, K)-invariant. As every positive definite matrix can be written in the form an* = a. I for some a 6 GL(m, K), it follows that f~ is homogeneous. The stabilizer of I in G is just K = U(m, K). We have 1) K fl SL(m, K) = SO(m) for K = lilt, and 2) K N SL(m, K) = SU(m) for K = C_. We can see that Theorem 2.1.13 does not hold in general for an element in the boundary of a convex cone. In fact, let X - (/k0~ Let g - (A~)) 6 GL(m, K) x . Here A 6 M ( k , K), B 6 M ( k x (m - k), K), C 6 M ( ( m - k) x k, K) and D e M ( m - k, K), where M ( r x s, K) denotes the r x s matrices with entries in K Then

X = gXg* = Thus

GL~(K)x = { ( A

AA* CA*

) CC*

"

DB 1 A E U(k) D 6 GL~-~(K)'B E Mk•

and thisgroupisnoncompact.

13

36

CHAPTER2.

CAUSALORIENTATIONS

Consider a subset L C V. The convex hull of L is the smallest convex subset of V containing L. We denote this set by cony(L). Then conv(L) = { E j )UvJ

J finite, Aj _> 0, Ej Aj = 1, vj ~. L } .

Caratheodory's theorem ([153], p. 73) says that one can always choose Aj

and vj such that the cardinality of J is less than or equal to dim V + 1. If L is a cone, then the convex hull of L can also be described as conv(t) = { Ej )~jvj J finite, Aj E 0, vj ~. L } . L e m m a 2.1.15 Let G be a Lie group acting linearly on V and let K C G

be a compact subgroup. If C C V is a nontrivial G-invariant proper cone in V, then there exists a K-fixed vector u E C \ {0}. If C is also generating, then u may be chosen in C ~ ProoJl. Choose v e C* such that (ulv) > 0 for all u E C, u r 0. Fix a u e C \ {0}. Then (k. ulv ) > 0 for all k e g . It follows that UK "= fK(k" u)dk ~. conv(K, u) C C is K-fixed and

(UKI v) = f K ( k . u]v) dk > O. Thus UK ~ O. As K is compact, it follows that K . u is also compact and thus c o n v ( g , u) - c o n v ( g , u) is compact, too. If u E C ~ then Ug E cony K . u = cony K . u C C ~ since C is convex. [:! E x a m p l e 2.1.16 ( T h e F o r w a r d Light C o n e C o n t i n u e d ) Recall the notation from Example 2.1.2 and the groups SOo(p, q) described in Section 1.5. Obviously, the forward light cone C is invariant under the usual operation of SOo(1,q) and under all dilations AIn, A > 0. We claim that the group SOo(1, q)R+Iq+l acts transitively on f~ - C ~ if q _> 2. In particular, this says that f~ is homogeneous. To prove the claim, assume that q _> 2. We will show that

SOo(1

2.1.

37

CONVEX CONES AND THEIR AUTOMORPHISMS

In fact, using at :=

cosh(t) sinh(t) 0

sinh(t) cosh(t) 0

0 ) 0 e SOo(1, q)

I.-2

we get at. ~ 0 ) = , ~ t (cosh(t),sinh(t),O,... ,0) E f~

for all t E R. As SO(q) acts transitively on S q-1 and

0

A

E SOo(1, q)

for all A E SO(q), the claim now follows in view of the fact that coth(t) runs through ]1, er as t varies in ]0, ~[. We remark that the homomorphism SO(q) ~ SOo(1, q) realizes SO(q) as a maximal compact subgroup of SOo(1, q), leaving the nonzero vector el E invariant, and a straightforward calculation shows that SOo(1,q) el = SO(q). According to Lemma 2.1.15, any SOo(1, q)-invariant regular cone in JR" contains an SO(q)-fixed vector, i.e., a multiple of el. Therefore homogeneity of Q implies that Conesoo(~,q) (IR') = { C , - C } .

(2.9)

for q > 1. Note that for q = 1 the equality (2.9) no longer holds. In fact, the four connected components of Ix[ r lYl in R 2 are all SOo(1,1)-invariant cones. [::] We now prove two fundamental theorems in the theory of invariant cones. The first result is due to Kostant [157], whereas the second theorem is due to Vinberg [166]. For the notation, refer to Appendix A.2. T h e o r e m 2.1.17 ( K o s t a n t ) Suppose that V is a finite-dimensional real vector space. Let L be a connected reductive subgroup of GL(V) acting irreducibly, and G = L ~ the commutator subgroup of L. Further, let K be a maximal compact subgroup of G. Then the following properties are equivalent: 1) There exists a regular L-invariant closed convex cone in V . 2) The G-module V is spherical. Proo]!. Note first that G is closed by Remark A.2.7, so K is a compact subgroup of GL(V). We may assume that V is Euclidian and that K C SO(V). Then the implication (1)=~(2) follows from Lemma 2.1.15. Lemma A.2.8 shows that the connected component of Z ( L ) acts on V by positive real numbers. Thus it only remains to show the existence of

38

CHAPTER2.

CAUSALORIENTATIONS

a G-invariant proper cone. To do t h a t we note that V c is irreducible by L e m m a A.2.5 and consider a highest-weight vector u of Vc. Let v g G W K be a nonzero K-fixed vector. We can choose VK such that (U[Vg) > 0. In fact, if u and v g were orthogonal, then G -- K A N would imply t h a t all of V is orthogonal to V K . Now (G.UlVK) -- (A.u[vK) = R +. This shows t h a t cone(G, u)) C_ (R+u) * is a nontrivial G-invariant convex cone in V. It has to be regular by irreducibility. O R e m a r k 2.1.18 1) In the situation of Theorem 2.1.17 we see t h a t K . u is a compact subset of C bounded away from zero. This shows t h a t for v G conv(G, u ) \ {0} there exists a compact interval J C R + such that v e conv 7r(g)Ju. In particular, (conv G . u) U {0} is closed. 2) The assumption in the next theorem that V is irreducible is not needed for the implication (1) =~ (2). [:3 T h e o r e m 2.1.19 ( V i n b e r g ) Let L be a connected reductive Lie group and V a finite-dimensional irreducible real L-module. Then the following properties are equivalent:

I) Con (v) # O. 2) The G-module V is spherical, where G = L' is the commutator subgroup of L. 3) There exists a ray in V through 0 which is invariant with respect to some minimal parabolic subgroup P of G. If these conditions hold, every invariant pointed cone in V is regular. Proof!. Note first that any nontrivial L-invariant cone C is automatically generating, as < C > is an L-invariant subspace of V and V is assumed irreducible. Denote the representation of L on V by ~r. Then 7r(G) is closed in G L ( V ) and 7r(K) is compact, as linear semisimple groups always have compact fixed groups for the Cartan involution. In fact, this shows that It(K) is maximal compact in ~r(G). Thus the equivalence of (1) and (2) follows from Kostant's Theorem 2.1.17 applied to lr(L). Now suppose that V is K-spherical. Then L e m m a A.2.6 yields the desired ray. Finally, assume that the ray R +. u is MAN-invariant. Since N is nilpotent and 7r(M) compact, we see that n and m act trivially on u. Thus u is a highest-weight vector and the corresponding highest weight satisfies the conditions of Theorem A.2.2, i.e., V is K-spherical. [::!

2.2.

CAUSAL ORIENTATIONS

39

R e m a r k 2.1.20 In the situation of Vinberg's theorem with a spherical Gmodule V, Theorem A.2.2 shows that for any highest-weight vector v E V c also ~ is a highest-weight vector. This implies that V contains a highest weight vector u. In other words, V is a real highest-weight module, n Another well-known fact about invariant cones that we will often use is the following description of the minimal and maximal invariant cones due to Vinberg and Paneitz. T h e o r e m 2.1.21 ( P a n e i t z , V i n b e r g ) Assume that L is a connected reductive Lie group. Let V be a finite.dimensional real irreducible L-module with ConeL (V) # ~. Equip V with an inner product as in L e m m a A.2.3. Then there exists a-unique up to multiplication by ( - 1 ) - m i n i m a l invariant cone Cmin E ConeL(V) given by

Cmin --- conv(G, u) U {0} -- cony G (R +"

VK),

where u is a highest-weight vector, V K 18 a nonzero K-fixed vector unique up to scalar multiple, and (ulvg) > O. The unique (up to multiplication by - 1 ) maximal cone is then given by Cmax = Cmin. Pro@ We know by now that there is a pointed invariant cone in V if and only if dim V g = 1 and that every pointed invariant cone contains either v g or --Vg. Furthermore, we know that every pointed invariant cone is regular. Thus Remark 2.1.18 implies that C := conv(G, u) U {0} is a regular cone and Vg E C ~ In particular, we get that the closed G-invariant cone C1 generated by Vg is contained in C. But Lemma A.2.6 shows that u E C1 and thus Cmin = C = C1 is in fact minimal. The equality Cmax = Cmin follows from t~r(g) = r(0(g)) -1. Now the claim follows from L e m m a A.2.8. Q R e m a r k 2.1.22 We have seen in Remark 2.1.20 that in the situation of Vinberg's theorem 2.1.19 the G-module V is a real highest-weight module with highest-weight vector u E V. It follows from L e m m a II.4.18 in [46] that the stabilizer G u of u in G contains the group M N . This implies that u E OC. In fact, according to L e m m a 2.1.10, the stabilizer group of a point in the interior of an invariant cone acts as a compact group. El

2.2

Causal Orientations

Let ~/t be a C~176 For m E M we denote the tangent space of j~4 at m by TIn(M) or Tm~/t and the tangent bundle of jr4 by T(j~/t).

CHAPTER2.

40

CAUSALORIENTATIONS

The derivative of a differentiable m a p f: .)k4 -~ A/" at m will be denoted by

d l: A smooth causal structure on j~4 is a m a p which assigns to each point m in ~ a nontrivial closed convex cone C(m) in Tin.It4 and which is smooth in the following sense: One can find an open covering {Ui}ie1 of ,s smooth maps : x R" T(M) with ~i(m, v) 6 Tm(2r

and a cone C in lR" such t h a t

C(m) = ~i(m, C). The causal structure is called generating (proper, regular) if C(m) is generating (proper, regular) for all m. A m a p f: ~ --+ .M is called causal if dmf (C(m)) C C ( f ( m ) ) for all m 6 j~4. If a Lie group G acts smoothly on .)k4 via (g, m) ~-r g . m , we denote the diffeomorphism m ~-+ g . m by tg. D e f i n i t i o n 2.2.1 Let ./~ be a manifold with a causal structure and G a Lie group acting on .M. Then the causal structure is called G-invariant if all tg, g 6 G, are causal. El If A4 = G / H is homogeneous, then a G-invariant causal structure is determined completely by the cone C := C(o) C T o ( M ) , where o := {H} E G/H. Furthermore, C is proper, generating, etc., if and only if this holds for the causal structure. We also note t h a t C is invariant under the action of H on T o ( M ) given by h ~ cloth. On the other hand, if C 6 ConeH(To(M)), then we can define a field of cones by

M 9 aH ~ C(aH):= dora(C) C Ta.o(./~), and this cone field is clearly G-invariant, regular, and satisfies C(o) = C. W h a t is not so immediate is the fact t h a t m ~ C(m) is smooth in the sense described above. This can be seen using a smooth local section of the quotient map .M = G / H --r G and then one obtains the following theorem, which we can also use as a definition since we will exclusively deal with G-homogeneous regular causal structures. Theorem

2.2.2 Let M = G / H be homogeneous. Then

C ~ (all ~ doto(C)) defines a bijection between ConeH(To(J~r ular causal structures on .s

(2.11)

and the set of G-invariant, rego

2.2.

CAUSAL ORIENTATIONS

41

Let 3/[ - G / H and C E C o n e d ( T o M ) . An absolutely continuous curve 7: [a, b] --+ f14 is called C-causal (also called conal, of. [52]) if "~'(t) e C ('y(t)) whenever the derivative exists. Here a continuous mapping "y" [a, b] ~ 3/[ is called absolutely continuous if for any coordinate chart r U --+ I~" the curve 77 = 0o-~: 7 - I ( U ) ~ R" has absolutely continuous coordinate functions and the derivatives of these functions are locally bounded. We define a relation ~ , (s for strict) on jr4 by saying t h a t m ~ , n if there exists a C-causal curve ~ connecting m with n. The relation ~ clearly is reflexive and transitive. We call such relations causal orientations. Elsewhcre they are also called quasiorders . E x a m p l e 2.2.3 ( V e c t o r S p a c e s ) Let V be a finite-dimensional vector space and C C V a closed convex cone in V. Then we define a causal Aut(C)-invariant orientation on V by u-~v~v-uEC. Then ~ is antisymmtric if and only if C is proper. In particular, H + (n, K) defines a GL(n,K)-invariant global ordering in H ( n , K ) . Also, the light cone C C N "+1 defines a SOo(1, n)-invariant ordering in I~ "+1. The space IR"+1 together with this global ordering is called the (n + 1)-dimensional Minkowski space, rn In general, the graph .M~_. " - {(re, n) E f14 x f l 4 1 m -~8 n} of ~ will be closed in .M x 3/t. This makes ~ difficult to work with and we will mostly use its closure ___, defined via not

m ~_ n: r

(re, n) E M~_.,

instead. It turns out t h a t -z,_is again a causal orientation. The only point that is not evident is transitivity. So suppose t h a t m ~ n ~ p and let ] ,rtk, nk, n k , p k be sequences with mk ~_s nk,,/k ~_spk,

mk ~ m, nk ~ n,n'k ~ n,pk ~ p.

Then wc can find a sequence gk in G converging to the identity such t h a t 7~ - g k ' n k . Thus gkmk ~ m and gkmk -~8 Pk implies m ~ p. Given any causal orientation < on 3,I, we write for A C 3d SA'-{yeYI3aeA"

a<_y}

(2.12)

and SA:-{yeYI3aeA

9y < a } .

(2.13)

42

CHAPTER2.

CAUSALORIENTATIONS

We also write simply 1"x := t { z } and $ x := ${x}. Then the intervals with respect to this causal orientation are [m,n]< "= (z e ~

Im < z <_ . } = 1,m n

and

[m, c~[< := $ m,

l - c~,ml< "= Sin.

The following proposition shows that replacing -~, by _ does not change intervals too much. P r o p o s i t i o n 2.2.4 Let f14 = G / H be a homogeneous space with a causal structure determined by C G Conea(To.M) and -~,,-~ the associated causal orientations. Then Ira, oo[_~= tin, oo[~_. Proot. Let m _ n. Then there exists a sequence (mk, nk) G fl4Z. converging to (m, n). Let U be a neighborhood of m in A4 such t h a t there exists a continuous section a of the quotient map 7r" G -+ .A4 = G / H . Then a ( m k ) --~ tr(m) and hence gk = a ( m ) a ( m k ) -1 converges to 1 G G. Therefore m = a ( m ) . o = gk .ink ~_, gk . n k and g k ' n k --4 n so that n E Ira, oo[___, proving the first claim. For the second, suppose that m __ n ~ 1. Then, according to the first part we can find sequences (nk) and (Ik) converging to n and 1, respectively, such t h a t m ___8nk and n "~8 lk. As above, we find gk G G converging to 1 such that gk 9nk = n. Thus gk 9m ~8 gk "nk = n ~8 lk implies (gk " m, lk) G .M~_. and hence (m, 1) e A4 ~. []

For easy reference we introduce some more definitions. D e f i n i t i o n 2.2.5 Let M be manifold. 1) A causal orientation < on f14 is called topological if its graph A4 < in f14 x f14 is closed. 2) A space (f14, <_) with a topological causal orientation is called a causal space. If < is in addition antisymmetric, i.e., a partial order, then (f14, <) is called globally ordered or simply ordered. )Afbe 3) Let (.M, _<) and (Af, <_) be two causal spaces and let f " .M continuous. Then f is called order preserving or monotone if

2.3. S E M I G R O U P S

43

4) Let G be a group acting on u~4. Then a causal orientation < is called G-invariant if

m < n - - - ~ V a E G" a . m < a . n . 5) A triple (j~r <, G) is called a causal G-manifold or simply causal if < is a topological G-invariant causal orientation. El

2.3

Semigroups

Invariant causal orientations on homogeneous spaces are closely related to semigroups. We assume t h a t .M = G / H carries a causal orientation _< such t h a t (2k4, <, G) is causal. T h e n we define the semigroup S< by

S< := {a e

G Io <_a.o},

called the causal semigroup of (A4, _<, G). If m is another point in A/I, then we can find an a E G such t h a t m = a . o . Thus for the corresponding semigroup S~ := {a E G l m <_ a . m } we have S' = aSa -1. Lemma 2.3.1 closed.

1) For all m, n e .M the intervals Ira, n] and [m, c~[ are

2) The semigroup S<_ is closed. 3) Gs< := S< N S< x is the closed subgroup of G given by

Gs< = {a e G[o_< a.o_< o}. Gs< contains the stabilizer H of o and normalizes S<. 4) Gs< = H if and only if < is a partial order.

Proo~ Let {zj } be a sequence in Ira, n] converging to z E M . T h e n A4 < 9 (re, z j) --+ (re, z). As j~r is closed, it follows t h a t (re, z) E .It4<_, i.e., m < z so t h a t [m, c~] is closed. Similarly, we find z < n whence [m, n[ is also closed. Let {aj} be a sequence in S, l i m a j = a E G. T h e n again M < 9 (o, a j . o) --r (o, a . o) E ~ < as jk4< is closed. It follows t h a t S< is closed -x- = Gs~, and so is Gs<_. Now Gs< _ and the G-invariance of < and the transitivity of < imply (3). Using (3) and the G-invariance of <, we see t h a t < is a partial order if and only if o < g. o < o is equivalent to g. o = o, i.e., if and only if G s< = H. 0

44

CHAPTER2.

CAUSALORIENTATIONS

R e m a r k 2.3.2 If (j~4 = G / H , <_, G) is causal and H = Gs<, then < can be recovered from S< via a . o < b . o ~ a-Xb 6 S<. Conversely, given a closed subsemigroup S of G, one obtains a causal G-manifold ( G / H , <_s, G) m

via H = S n S -1 and a . o < s b . o : .-t--.*. a - l b E S.

D

Let .M = G / H be a homogeneous space with a causal structure determined by a cone C 6 ConeG(Tojgl) and ~ the associated topological causal orientation. Then S__. - {g E G ] o ~ g . o} is a closed subsemigroup of G. But there is another semigroup canonically associated to C: Let W be the preimage of C under T1G = {t ~ T o M = g/I) and S w the closed subsemigroup of G generated by exp W" For the following theorems, refer to [52], Proposition 4.16, and Theorem 4.21. Theorem

2.3.3 S-~ = S w H .

Theorem

2.3.4 The following s t a t e m e n t s are equivalent:

[3

1) ~_ is a partial order.

2) S 5 rqSi<X = H. s)

{x e g I

xpR+x c

= w.

Theorem 2.3.4 shows in particular that one can recover the causal structure from -4 provided -4 is a partial order. To do that one has to calculate the tangent cone L(S___) of the semigroup S-~. We can also build causal orientations starting with a closed subsemigroup S of G. Write H := S n S -x for the group of units of S. If M = G / H is the associated homogeneous space, let 7r 9G --r ./Vl, g ~-r g H be the canonical projection, and o := 7r(1) the base point. We define a left invariant causal orientation on G by the prescription m

g<sg'

if g'6gS.

(2.14)

Then g _<s g' _<s g is equivalent to g H = g~H and the prescription r ( g ) _< lr(g')

if

g _< s g'

(2.15)

defines a partial order on j~l such that ~r" (G, _<s) -'+ (.h4, <) is monotone. We say t h a t a function f " G --+ R is S - m o n o t o n e if

I" (G, <_s) "+ (R, _<) is a monotone mapping. W e write Mon(S) for the set of all S-monotone

continuous functions on G. The construction of a causal orientation from a semigroup is of particular interest if the semigroup S can be recovered from its tangent cone L(S) =

{x e g I exp(R+x) c s}.

2.4. THE ORDER COMPACTIFICATION

45

D e f i n i t i o n 2.3.5 1) S is called a Lie semigroup if (exp L(S)) = S, i.e., if the subsemigroup of G generated by exp L(S) is dense in S. 2) .9 is called an extended Lie semigroup if Gs(exp L ( S ) ) = S. 3) S is called generating if O is the smallest subalgebra of 0 containing

T,(s).

o

R e m a r k 2.3.6 For every generating extended Lie semigroup S C G we know that S ~ is a dense semigroup ideal in S and S ~ C (exp L ( S ) ) H (cf.[52], L e m m a 3.7). Moreover, we have t g = gS and .~g = gS -1 for g E G and these sets have dense interior. Similarly, t x = ~r(gS) and S x = ~r(gS -1) for x = gH E G / H and these sets also have dense interior, o

2.4

The Order Compactification of an Ordered H o m o g e n e o u s Space

Compactifications are an indispensible tool whenever one wants to describe the behavior of mathematical objects at infinity in a quantitative manner. Which type of compactification is suitable depends very much on the specific situation given. For an ordered homogeneous space A~ = G / H one can define a compactification that takes the order into account and therefore turns out to be particularly useful. The basic idea is to identify an element gH E .Ad with the set of elements g~H smaller than gH. One has the Vietoris topology (cf. Appendix B) on the set ~'(A~) of closed subsets of A~ which makes ~'(A4) a compact space. Then one can close up A4 to obtain a compactification. In this section we assume that G is a connected Lie group and S C G an extended Lie semigroup with unit group H; cf. page 45. We describe a compactification of A~ := G / H which is particularly suited for analytic questions taking into account the order structure < on A~ induced from < s . Recall the notation from Appendix B and note that both G and G / H are metrizable and a-compact. Therefore the results of Appendix B apply in particular to ~ ' ( G ) a n d :F(G/H). L e m m a 2.4.1

1) The set ~4(G) "= {F e

~(G)I $F

= F } c :F(G) H

is closed. 2) The set :Fr

:= { f e Y : ( G / g ) l S f = F} is closed.

Proof:. 1) The condition S F = F is equivalent to Fs C F for all s E S -1. For every s E S -1 the set

7 . := {F e

F , r F}

CHAPTER2.

46

CAUSALORIENTATIONS

is closed in view of Lemma B.1.2 because ~'(G) is a pospace. Therefore

sES

is closed. 2) This follows from Proposition B.1.4 and 1). L e m m a 2.4.2 The mapping O" G --+ 3r~(G),g ~ S g factors to a continu-

ous order-preserving injective mapping "~" G / H --4 5r~(G),

gH ~ $(gH)

of locally compact G-spaces. Pro@. The continuity of the mapping r/follows from Lemma B.1.2 and the fact that q(g) = g" 0(1) Vg e G. This mapping is constant on the cosets gH of H in G. Therefore it factors to a continuous mapping ~. To see that ~ is injective, let a, b E G with o(a) = o(b). Then $ a = ~ b and therefore a <_s b _<s a. Hence a H = bH. This proves the injectivity of ~. Finally, suppose g < s g~. Then g~ E gS and therefore $ g - gS-1 C g~S -1 = $ g~. This shows that ~ preserves the order. [3 We write A/I+ := [o, oo) = S . o for the positive cone in jt4 = G / H and set A4 cp' := ~(A4) = r/(G) C 5r(G)

and

~ , - c, +p t

(2.16)

Then .A4ept is called the order compactification of the ordered space (A4, <). We refer to ~ as to the causal compactification map. L e m m a 2.4.3 Let F E .A4ept. Then the following assertions hold:

1) F 6 ,A.#ePt . , + is equivalent to 1 6 F.

z) f = { g e G [ g - i f

cpt

}.

IrA ept , i.e. ~ 3} If F ~t @, then there exists g E G with gF E ,-,+ G . ,.,+ AAcpt u

. ~ t opt

C

Proof:. 1) For s E S we clearly have that 1 e O(s) = $ s. Therefore 1 e F for all F E ,~-e, +p t . If, conversely, 1 E F , and F = limr/(g,), g, E S, then

2.4.

THE ORDER COMPACTIFICATION

47

there exists a sequence a , E rl(gn) "- $ gn such t h a t an ~ 1 ( L e m m a B.I.1). We conclude with L e m m a B.1.2 t h a t 1 6. lim $ (a ~ 1g n ) : lim a ~ 1 ,~ g n --'~ f . 2) In view of 1), this follows from the equivalence of 1 E g - 1F and g 6- F . El 3) If F ~ 0, then there exists g 6- F , and therefore g - i F 6- ,AA . , +opt .

c.

Note that L e m m a

2.4.3 implies that either A4 cpt = G.

MT'u

Proposition

1)

2.4.4

.~,,,q.]IAcpt=

~""I'AAcpt or /~cpt

__

{A E A4 cpt [ 1 E A}.

2) (.hdTt) ~ = {A 6- Mcpt [ 1 6- A~ cpt (M+)o c

s) s = {g e Gig.

#)

s ~

= {g e c Ig

. A,4cP t

,-,+

c

Pro@. 1) This was proved in L e m m a 2.4.3. 2) Let A 6. (.M~t) ~ T h e n 1 6. A. Moreover, there exists a s y m m e t r i c neighborhood U of 1 in G such t h a t U . A C ,AzcPt - , + . Hence U C A, which proves t h a t 1 6. A ~ Conversely, if 1 6. A ~ then there exists an s 6. A ~ gl S ~ and therefore also a neighborhood V of s contained in A N S. T h e n Y := { F 6. Y ( G ) [ F N

Y # 0}

is a neighborhood of A in ~'(G). Since each element A 6- M cpt satisfies

A=$A=

{gEG]gSNA#O},

F 6. V~ n M cpt entails 1 6. $ F = F , hence F E ,/iA - , +cpt . 3) Let s 6- S and A 6- (Ad~t) ~ According to 2), we have

1 <_s s E (sA) ~ whence 1 e (sA) ~ and therefore s. A E ( A 4 ~ ' ) ~ If g E G with g. (A4~P') ~ C

(M~+Pt)~ then g 9M~.Pt C ,AAcPt - , + and gS -1 E r + r because 2~4~pt = ( M +cpt~o j . This implies t h a t g E (S - 1 ) - 1 = S 4) If s E S ~ and A E A4~ 't then 1 E A and we find a neighborhood U of 1 opt . Hence 1 E gs. A for every g E U. This leads in G such t h a t Us. A C A/I+ to U -1 c s . A and hence to 1 E (s. A) ~ i.e., s . A E (.Mc+Pt)~ according to cpt 1). To show the converse, we assume t h a t g E G and g . , .AA , +opt C (A4+)o. opt Then, as in 2), we obtain t h a t gS -1 e ( A / I + ) o and therefore 1 E ($g)O or g E S ~ again by 1). El

C H A P T E R 2. CAUSAL O R I E N T A T I O N S

48

1) Let Y : ~ ( G / H ) denote the set of all closed subsets F C G / H with $ F = F such that for every a E F the connected component of a in t a N F is noncompact. Then Y : ~ ( G / H ) is closed in Y:~(G/H).

L e m m a 2.4.5

2) Let F E ~ ( G / H ) \ ' ~ ( G / H ) and a E F. Then the connected component of a in ~ a I"1F is noncompact. Proof:. 1 ) L e t F = limn-,er Fn, Fn E Y : ~ ( G / H ) , and a E F. Let Un denote the nl ball around a. Then there exists rim ~. N such that Fn f'lUm ~ 0 for all n > n,n. We clearly may assume that the sequence (nm)mer~ is increasing and that nm > m. Let am ~. Urn I"1Fn... Then our assumption implies that the connected component Cm of am in 1"am Iq Fn.~ is noncompact. Passing to a subsequence, we even may assume that Cm ~ C in YZ(G/H). Then Cm C ~ am and the closedness of <_ show that C = lim Cm C l"(lim am) = ~ a. In addition, we know that C =

lim Cm C lim F n .

= F.

m---~oo

Hence C C 1"a f'l F. Next, Lemma B.I.1 entails that the connected component of a in C is noncompact. Therefore A E ~ ' ~ ( G / H ) . 2) Let a , E ($a) ~ C F with an -4 a and F m = Sxm ---r F with Xm E G / H . For ri E N there exists mn >_ ri such that Fm CI (~ an) ~ r 0 for all m >_ m , . This clearly implies that an E Fro, i.e., an < Xm. Then we find monotone curves 7,~ " R + --r G / H and Tn ~. R + such that 7,(0) = a , and 7,(T,) xm. ([114], 1.19, 1.31). Passing to a subsequence, we may assume that the sequence Cn := 7n([O, Tn]) converges to C in ~'(.MTt). Then a = lira an E lim Cn = C, the sets Cn are connected chains, i.e., totally ordered subsets, in the partially ordered set (G/H, ~_), and Un>no Cn is not relatively compact for any no E 1~1because Xm. E Cn and Xm. ~ w since lim~(xm.) r ~ ( G / H ) . Now, Lemma B.I.1 entails that the connected component of a in C is noncompact. Moreover, C=limC,

Cl"lima,=ta

and

C=UmC,

ClimFm.=F.

Finally, this proves that the connected component of a in $ a N F is noncompact. El T h e o r e m 2.4.6 The image ~(,M) of ,It4 in :7:$(G) is open in its closure.

2.4. THE, O R D E R COMPA CTIFICATION

49

Pro@ Let F e fi(G/H). We note first t h a t the following two conditions are equivalent: 1) For every a E F , the connected component of 1"a n F is noncompact. 2) F r ~ ( G / H ) . In fact, 2) implies 1) by L e m m a 2.4.5. If F = W(z) = S x with x E G / H , then x e F and T x n S x = {x} is compact. Therefore 1) implies 2). Now the theorem is a consequence of L e m m a 2.4.5. El P r o p o s i t i o n 2.4.7 Let A C G be a closed subset with $ A = A. Then

p(OA) = O

,,d

A = ~.

Pro@. Suppose t h a t this is false. Then we can find x E O A , the boundary of A, and a compact neighborhood V of x with I~(OA n V) > 0. Choose a sequence s, E S ~ with S,+l E ($sn) ~ and lim,~oo s , = 1. Then As~ 1 C A ~ and therefore A = A ~ L e t / 5 denote a right Haar measure on G. It suffices to prove t h a t ~(OA n V) = 0. If not, we have 15 ((OA n V)sn) = f~(OA n v ) > 0 for every n E l~l and

(OA n v ) , . n (OA n v ) , . because (OA)sn C A~

= ~

for

m <,

This shows t h a t

~n~176 f~(OA n V) = ~,~176 p ((OA n V ) s , )

= # (U.~.(oA n v),.) < # (U.~. v,.) <

Whence f~(OA n V)

=

O.

=.

n

We will see later on t h a t for specific .hd the order compactification can be described in much more concrete terms than has been done in this section. In particular, it will turn out t h a t the space A4 cpt is in some sense the smallest compact G-space X such t h a t there exists an open subset (9 C X with the property t h a t

S-{geGIg.

OcO}

and

S~

(2.~)

This will be used to get information about the structure of 4vL~= AAcpt and the G-space .A4cpt because in special cases there are very natural compact Gspaces with the above property.

C H A P T E R 2. CAUSAL O R I E N T A T I O N S

50

2.5 2.5.1

Examples The

Group

Case

Recall the way of viewing a group G as a symmetric space from Example 1.1.3. A cone D C q belongs to Conea• if and only if it is of the form

n = {(X,-X) lX e C}, where C E Conea(g). There is extensive literature on the classificationof (7 E Conec(g) for arbitrary connected Lie groups G (cf. [50, 52, 118]). Here we only recall some basic facts for the case of simple Lie groups. It turns out that only Hermitian simple Lie algebras (cf. Appendix A.3) admit regular invariant cones. L e m m a 2.5.1 Let G be a simple real Lie group with Lie algebra g. Then Conea(g) is nonempty if and only i.f g is Hermitian.

Proof:. Note first that any nontHvial G-invariant cone in g is automatically regular since it spans an ideal. Suppose that G = K exp(p) is a Cartan decomposition and t C g is the Lie algebra of K. The group A d ( K ) is compact, so Kostant's Theorem 2.1.17 shows that there is a nontrivial Ginvariant cone in g if and only if there exists Z E g, Z ~ 0, such that A d ( k ) Z = Z for all k E K. As g is irreducible as a G-module, it follows from L e m m a A.2.5 that dim gK = 1 and ~tc is simple. Let/9 be the Cartan involution on g corresponding to the Caftan decomposition g = t @p. Then the K-invariance of O(Z) shows that O(Z) = -l-Z. If O(Z) = - Z , then R Z is a t-submodule of p, which is impossible by Lemma 1.3.4 since the Killing form is positive definite on p. El Now suppose that It is Hermitian and Z E ~([), Z y~ 0. minimal cone (cf. Theorem 2.1.21) is given by

Then the

Cmin = conv G . R + Z and the corresponding maximal cone is Cmax --" {X E ~ l ~ / " E Cmin :

(XIY)o > 0}.

Later on we will describe these cones in more detail. L e m m a 2.5.1 shows that each Hermitian Lie group is a causal manifold with causal structures in bijective correspondence with the G-invariant cones in g. Moreover, the lemma shows that every nontrivial G-invariant cone contains a nonzero element Z E l(t). Therefore the causal structure cannot be antisymmetric if K is compact. This follows from the fact that the curve t ~-4 exp tZ is periodic and causal.

2.5. EXAMPLES

51

Multiplication by i maps ConeH(q) into ConeH(iq). In this way causal structures of c-dual spaces are in a canonical bijective correspondence. We make this explicit in the group case: Assume that G is contained in a complex group Gc with Lie algebra gc. Let M := Gc/G. Then the tangent space at the origin o -- 1G is ig. If C is a G-invariant cone in g, then iC C ig is also G-invariant. Therefore L e m m a 2.5.1 shows t h a t ~ carries a causal structure if and only if G is Hermitian. We will show later t h a t j~4 is ordered for each of the topological orientations associated to elements of ConeG(ig) and t h a t the intervals Ira, n] are compact. Furthermore, it will turn out t h a t the semigroup S_3 is given by G e x p i C and L(S) = g ~ iC, where C E ConeG(ig) is the cone inducing _. 2.5.2

The

Hyperboloids

(9x,n Recall the hyperboloid Qr = -,r ~o(2, n), defined by

C._ R "+1 and the map q: R n + l

q(v) := ( O -t(vlx,n) ) v

--~ q C

(2.18)

0

from Section 1.5. Let Cx C R n+l be the forward light cone and

c := {q(o)e q lv e c , } . Then C is a regular cone in q invariant under the group SOo(1,n) and Conesoo (1,n) (q) = { C , - C } according to Example 2.1.16. In particular, Qr ~- SOo(2, n)/SOo(1, n) carries a causal structure. Let ~(t) :=

exp(tq(el)).o = costel + sinte2.

Then &(t) = d~(0)ea(t)(&(0)) =

d~(o)e~(t)(q(el)) ~- C(c~(t)).

Thus c~ is a closed causal curve. Therefore the causal orientation is not a partial order. The case Q_ 1 can now be treated in the same way by using the map q~

~w~

(0 o) t(wIl,n)

0

e q C $o(1,n + l)

(2.19)

and the SOo(1, n)-invariant form (qe(v), ql (w)) := Tr

(qe(v)qe(w)).

Let C :=

{qC(v) 6 q lv 6 C,}

(2.20)

52

C H A P T E R 2. CAUSAL ORIENTATIONS

Then the cone C is SOo(1, n)-invariant and regular. Furthermore, Conesoo(1,n) (q) ---- (C,--C} (cf. Example 2.1.16 again). In particular, Q _ r ~_ SOo(1, n + 1)/SOo(1, n) is causal. The causal curve a(t) = exptqC(el).e,+l is given by a(t) = sinh(t)el + cosh(t)e,+l, and this curve is an embedding of R. We will show later t h a t this space is actually globally hyperbolic, i.e., the causal orientation is ant• and all the order intervals Ira, n] are compact. If n is odd, we consider a = I1,,+1 e SOo(1,n + 1). Then 7(a) = a, a commutes with SOo(1, q) but a r SOo(1, n). On the other hand, a 2 = 1 so t h a t HI := SOo(1, n) U a SOo(1, n) is a group and the group { 1, a} normalizes SOo(1, n) so it acts on Q - r . Then Q - r is a double covering of the quotient space Q _ , / { 1 , a } ~- SOo(1,n + 1)/H1. In particular, locally Q-r~{ 1, a} admits a causal structure. But aq(el)a -1 = -q(el). Thus the light cone is not Hl-invariant and does not define a causal structure on Q - r I{ 1, a} globally. Thus the existence of a causal structure may depend on the fundamental group of the space in question.

2.6

S y m m e t r i c Spaces Related to Tube D o m a i n s

Let (g, b) be a symmetric pair associated to a tube domain. In the notation of Example 1.1.6 this means in particular that we have an element Z ~ E ~(t) N q and elements Xo, Yo e p such that b = 0(0, Yo), q = q+ + q- with q• = 0(4"1, Yo) and Xo = [Yo, Z ~ e qp. For the last relation see L e m m a A.3.2. The same lemma shows that the involution r: g --+ g coincides with C 2. So Z ~ E q is a K-fixed point, whereas Yo E b is an Ho-fixed point. Therefore Xo E qHonK and we are in the situation of Theorem 1.3.11. Since conjugation by an element of K does not move Z ~ we may assume t h a t Iio = yo in the notation of that theorem. Decompose Xo = 89 + X _ ) with X • E (q-t-)HonK and O(X+) = - X _ . As Z ~ e qk and ad Yo : q -4 q is a linear involution, we obtain Z ~ = ad(Y~ = 89( X + - X _ ) . Let {'Y1, 99 9 %} be a maximal system of strongly orthogonal roots with suitable root vectors Ej = ET~ and co-roots H i = HT~ as in Appendix A.3. Then r

1 X • = Xo • Z ~ = ~ E ( X j j--1

.4-iHj),

(2.21)

where Xj = - i ( E j - E_i). By Theorem 2.1.21 there are minimal Hoinvariant cones C• C q• such that X • E C• and C_ = -O(C+).

2.6. S Y M M E T R I C SPACES R E L A T E D TO TUBE DOMAINS

53

L e m m a 2.6.1 The cone C+ is H-invariant if Xo E qHnK.

Proo~ Let X = A d ( h ) S + for some h E 1to and fix some k E H I"1K. Then A d ( k ) X = Ad(kh)X+ = Ad(khk -1) A d ( k ) X + = Ad(khk-1)S+ E C+

because of khk -x E Ho and the lemma follows by C+ = c o n v Ad(Ho)R+X+ and H = (H n K)Ho (cf. (1.S)). D R e m a r k 2.6.2 Consider Ho-stable cones

Ck := {X + o(Y) I X, Y e C+) = C + - C_

(2.22)

Cp := { X -

(2.23)

and

O(Y) I X, Y e C+} = C+ + C _ .

They are pointed and generating in q. The (Ho Mg)-fixed points in the Ho-invariant cones Ck and Cp, guaranteed by Lemma 2.1.15, can be determined explicitly:

z o = x + + o ( x + ) e q~o~K n cZ; 2

go

=

X t "- •(Xt) 2

E -HONK n qP

c~.

Moreover, we have

c~ n p = {0},

cp n ~ = (0}.

[]

P r o p o s i t i o n 2.6.3 Suppose Xo E qHnK. Then the space M = G / H has a regular invariant causal structure.

Proof:. In view of Lemma 2.6.1, the assumption on Xo ensures that Ck and Cp are even H-invariant. I::] E x a m p l e 2.6.4 (cf. Example 1.3.15) Suppose that Xo E qHnK. Then the cone Cp defines an invariant causal structure on G / H but not on the symmetric space A d ( G ) / A d ( G ) ~, where Ad(G) ~ = {~o e Ad(G) ] ~ov = rqo}. In fact, note that by definition C~ fl t ~t 0 and C~ I"1 p ~t 0. Then 0 e Ad(G) ~, which implies the claim since Cp is not 8-invariant. [::1 We will now show that ConeHo (q) -- {=l=Ck,=l=Cp}. L e m m a 2.6.5 X j + iHj E C+ for j = 1,... ,r.

CHAPTER2.

54

CAUSALORIENTATIONS

r

Proofi Recall from (2.21) X• = 89Ej=I (Xj 4- iHj). For s l , . . . , s~ e R, let h = exp s92YI"'" exp s,.Y,. E H. Then 2 Ad(h)X+ = e2~ (X1 + iHi) + ... + e2"r(Xr + iHr). Let sj = 0 and let the other sk tend to - o o . follows that Xj + iHj 6 C+.

Then, as C+ is closed, it

[:3

L e m m a 2.6.6 Let C E Coneu.(q +) such that X+ E C ~

Then C is self

dual, C = C+, and C ~ = Ad(Ho)X+. Pro@. By assumption, we have C+ C C C C~.. Fix an X E C. Then X - O ( X ) E qNp. Applying C, it follows from Lemma A.3.2 and Proposition P A.3.3 that ~ j = l R X j is a maximal abelian subspace of p N q. Since Ho N K is the group of 0-fixed points in the analytic subgroup of G with Lie algebra b a = b I"1t + q N p, there exists a k E K N Ho such that r

A d ( k ) ( X - O(X)) = E t j X j e aq. j=l As X j = 89 + iHj) + 89 - iHj) and Xj 4-iHj 6 q+, it follows that r Ad(k)X = ~1 ]~j=l tj(Xj + igj). Now C C C ; , and by Lemma 2.6.5,

0 <_ (Xj + iHj

I Ad(k)X)a

1 iX j + iHjl 2 = ~tj

Hence tj > O. Again by Lemma 2.6.5 it follows that C C C+. Hence C = C+. If we take C = C~., this implies that C~ = C+ and C+ is self dual. Now assume that X E C ~ Then tj > 0 for j = 1 , . . . , r. Define h=Hex j=l

p

-

logtj

I~

eHo.

Then A d ( h k ) X = X+. Hence C~. = Ad(Ho)X+.

Q

C o r o l l a r y 2.6.7 ConeHo(q +) = { C + , - C + } . If, in addition, Xo 6 qHng, then ConeH (q+) = { C+, - C + }. 17 T h e o r e m 2.6.8 Let G / K be a tube domain, G C Gc with Gc simply connected, and 7 the involution of Gc which on gc is given by 7" = ead inYo

Further, let H = G ~ and C• the closed convex Ho-invariant cones in qqgenerated by X+ = Xo 4-Z ~ Then ConeH. (q) = {+(C+ - C_),•

+ C_)}.

2.6. S Y M M E T R I C SPACES RELATED TO TUBE DOMAINS If, in addition, Xo

E

55

qHnK, then

ConcH(q) = 1-4-(C+ - C_),=l=(C+ + C_)}.

Proof:. We prove this in four steps. Fix C E ConeHo (q). Step 1: Let pr• 9q --~ q• be the H-equivariant projection acording to the decomposition q = q+ @ q-. Fix an X = pr+ (X) + p r _ ( X ) e C and consider h(t) = exp tYo. Then

Ad(h(t))X = e t pr+ (X) + e - ' p r _ ( X ) . Hence lira e -t Ad(h(t))X = pr+(X) fi C . t--~oo

Similarly, we get pr_ (X) E C. Step 2: By the first step it follows that C ( + ) := pr• C C C . Then C C C (+) + C ( - ) . We claim that C ( + ) and C ( - ) are Ho-invariant proper cones in q+ and q-, respectively. As C is generating, it follows that C ( + ) ~ {0}. If X 6 C ( + ) N - C ( + ) , Step 1 implies that X 6 C N - C . Hence X = 0 and C ( + ) is pointed. Let X E q+. Then, as C is generating, there are V, W 6 C such that X = V - W. But then x

=

-

e c ( + ) - c(+).

This shows that C ( + ) is generating. As C ( + ) C C and C, as well as q+, are Ho-invariant, it follows that Ad(Ho)C(+) C C Nq + C pr+(C) = C(+). Thus C ( + ) is an Ho-invariant regular cone in q+. Step 3: By Corollary 2.6.7 we get C ( + ) = d:C+. Similarly, we find c(-) = +o(c(+)). Step 4: As C ( + ) C C, we have C ( + ) + C ( - ) C C. The other inclusion is obvious, so we have

c=c(+)+c(-). Step 3 now implies that either C = =t=Ck= C + - C _ or C = =l=Cp = C+ + C _ . The last claim is an immediate consequence of Lemma 2.6.1. o R e m a r k 2.6.9 The cone C+ C q+ also has a geometric meaning, as G / K is biholomorphically equivalent to the tube domain q+ + ifl, where fl = C~_. The idea of the proof is as follows. Let Q~ = exp(q+)c, Qc = exp(q-)c. Similarily, we let Kc c Gc be the complexification of K. Let c = exp ((i~r/2)Xo). Then c - l K c c = Hc and G C Q~cHcQc. Furthermore, G McHcQc. Thus G / K is diffeomorphic to the G-orbit through cHcQc in G c / H c Q c . This defines a complex structure on G / K . Now if Z e q+ + ifl, then exp Z ~. GcHc(G-)c and this defines a biholomorphic map onto G / K .

CHAPTER 2. CAUSAL ORIENTATIONS

56

2.6.1

Boundary Orbits

We keep the asumptions from the last section. In particular, we assume that G is contained in the simply connected complex Lie group Gc with Lie algebra go- By Theorem 1.1.11, Kc = G~ and Hc = G~. Recall the Cayley transform C = Ad(c) with c = exp((ri/2)Xo), cf. p.255. Then C = tOiXo (cf. Lemma 1.2.1). By Lemma 1.2.5 we have r0 = riXo. Now Lemma 1.2.5 and Remark 2.6.9 yield

i) C - i = C s = r O o C = C o r S . 2) r o C = C o O . 3) c -1HCC = KC. L e m m a 2.6.10 q- = C(p +) f'l g and q+ = C ( p - ) N g.

Proof: We will only prove the first statement, as the second follows in exactly the same way. By Lemma A.3.2, C(Hj) = Xj. Therefore - i C ( Z ~ = -Yo. It follows that for Z E p+:

[Zo,c(z)] = c([c-' (Zo),Zl) =

c([z ~ z]) = -c(z).

Thus Z E q~-. The same calculation shows that if X E q-, then C-I(X) E p+. From this the lemma follows. [] We recall that the realization of G / K as a bounded symmetric domain in p+, cf. (A.23), p. 253; see also Lemma 5.1.4, Remark 5.1.9 and Example 5.1.10. The map

P+ • Kc • P - ~ (p, k, q) ~-~pkq E G c is a diffeomorphism onto an open dense submanifold of Gc and G C P + K c P - . If g e G, then g = p+(g)kc(g)p-(g) uniquely with p+(g) e P+, kc(g) e Kc, and p-(g) e P - . Let log := (explp+)-I . p + ~ p+. The bounded realization of G / K is given by

fl+ = {log(p(g)) l g e a } . For g E Gc and Z E p+ such that gexp Z E P + K c P - , define g . Z E p+ and j(g, Z) e Kc by g exp Z e exp(g 9Z)j(g, Z ) P - .

(2.24)

Thus exp(g. Z) -" p+(gexp Z) and j(g, Z) = kc(gexp Z). We denote the map P + K c P - / K c P - -~ P+, p K c P - ~ logp by p ~ ~(p). Define E e p+ by E = ~(c). The Shilov boundary of f/+ is ,9 = G . E.

2.6. S Y M M E T R I C SPACES R E L A T E D TO T U B E D O M A I N S L e m m a 2.6.11 The stabilizer of E in G is H Q +. G / H Q + = K / K N H is compact.

57

In particular, S =

Proof: The stabilizer of c K c P - G G c / K c P - in Gc is c K c P - c -1 = H c Q ~ . But by construction e x p ( E ) K c P - = c K c P - . 0. View G and Gc as subgroups of G1 "= G x G, respectively (G1)c "= Gc x Gc, by the diagonal embedding g ~ (g, g). This induces a G-action on 81 := S x S and a Gc-action on G c / K c P - • G c / K c P - as well as other homogeneous spaces of G1 and (G1)c, respectively. L e m m a 2.6.12 Suppose that g e x p ( - Z ) E P + K c P - , p+. Then O(g)exp(Z) e P + K c P - and

g E Gc, and Z G

o(g). z = - [ g . ( - z ) ] . Proof" Let g e x p ( - Z ) - exp(g. ( - Z ) ) k p , with k G Kc and p G P - . Then

o(g)

z

=

o(g

---

exp(O(9. ( - Z ) ) ) k p -1 exp (--[g. ( - Z ) ] ) kp -1

xp-Z)

which proves the claim,

o

C o r o l l a r y 2.6.13 g e P + K c P - . we have (:(c -s) = - E .

T h e n - ~ ( g ) = ~(8(g)). In particular,

Proof: Take Z - 0 in Lemma 2.6.12 and note that 8(c) - c -1. L e m m a 2.6.14 Let the notation be as above. Then the following hold:

1) The stabilizer o f - E

is H Q - .

2) 81 := 8 • 8 ~_ G / H Q - x G / H Q +. 3) Let ~o - ( E , - E ) G $1. Then .hJ ~_ G . ~o. Proof" Statement 1) follows from Lemma 2.6.12 and Corollary 2.6.13. The second claim is a consequence of 1) and Lemma 2.6.11, whereas the last claim follows immediately from 1) and 2). o We list here the symmetric spaces, the corresponding Shilov boundary together with the real rank r, and the common dimension d of the restricted root spaces for short roots as provided by Moore's Theorem A.3.4. Here k > 3, T is the one-dimensional torus, Qn is the real quadric in the real projective space RIPn defined by the quadratic form of signature (1, n), and the subscript -4- means positive determinant.

CHAPTER 2. CAUSAL ORIENTATIONS

58 Symmetric

Spaces Related

domains

M = G/H

,5 = K / K CIH

r

Sp(n,R)/GL(n,R)+ SU(n, n)/ GL(n, C)+ SO*(4n)/SU*(2n)R + S0(2, k)/SO(l, k 1)R +

U(n)/O(n) U(n) U(2n)/Sp(2n)

n n

-

Er(-2s) /Es(-28)R + 2.6.2

to tube

The Functions

Q.

EeT/F4

2n 2 3

d 1

2 4 k-2 8

~

Our aim now is to construct an analytic function ~ on S1 such that G / H = {~ 6 81 [ ~(~) # 0}. This implies in particular that G / H is open and dense in $1. For this we need a few facts from representation theory. Let p, be the half-sum of positive noncompact roots. Then

p . = ~ I [1-1- d ( r 2- 1 ) ] (7, + " " + %)

(2.25)

Hence

2(pn 17./) = 1 +d(,-~ X) and

I~r ~

2

2(p. 1 89 + ~i)) = 2+d(r-1). 1 89 + ~rj)l~

(2.26)

From the table we see that

2(p. I~r

6 Z + r g # ap(2n, lR)

or g # so(2, 2 ~ - 1).

T h e o r e m 2.6.15 Fix the positive system A+(pc, tc)U--A+(tC, tO) on A = A(gc, tc). Then the following hold:

1) There exists an irreducible finite-dimensional representation of Gc with lowest weight-2p,. 2) Assume that g # sp(2n, R), so(2,2k+l), n,k > 1. Then there exists a finite-dimensional irreducible representation of Gc with lowest weight Dpn

9

Pro@ We have to check the integrality of p, and 2p,. Let {~/1, O~2,... ,O~k} be the set of simple roots for the positive system A+ (pc, tC)U--A+(~C, tc). Then cq = "/1 is the only simple noncompact root. Furthermore, (pnlC~) = 0

2.6. S Y M M E T R I C SPACES RELATED TO TUBE DOMAINS

59

for a E A(tc, tc). If 7 is an arbitrary noncompact positive root, then 7 = 71 + ~ j > l n~jaj. By (2.25),

2(kp.lT)

2(kp=lT1)[Tll2

1~12

=

i~112

[

1~12 = k

1+

d(r-1)]I~112 2

I~12"

From [46], p. 537, it follows that ")'1 is always a long noncompact root. Thus [')'112/]')'[ 2 e 7/~+. We have (1 + [d(r - 1)]/2) e Z if and only if d is even or r is odd. But in all cases 2 (1 + [d(r - 1)]/2) E 7/.. The claim now follows from the table. O Let m E Z + be such that there exists a finite-dimensional irreducible representation (lrm, Vm) of Gc with lowest weight -mpn. Choose an inner product on Vm as in Lemma A.2.3, e.g.,

~(g)" = ~(~o(g) -I) where a is the conjugation with respect to G. Let u0 be a lowest-weight vector of norm 1. Define ~m " P+ -4 C by

~m(Z) "= (Irm(c -2 exp Z)uo l uo) and set

9~(z, w):= ,~(z-

w).

E x a m p l e 2.6.16 ( T h e case SU(1,1)) Let us work out the special case G = SU(1,1) before we describe the general case. Now Gc = SL(2,C). As an involution on SU(1, 1) take r ( X ) = X. Then r is conjugation by I n t ( O1 01) and the holomorphic extension of r is given by

(o -a

b

Thus

H = + ( h(t) "= ( c~

,(1

Let Z ~ = ~

0

,, ((00

0 )

--1

"

tER 1 9

Then

z) z c} 0

t sinh t ) sinh t

~

a

and

, {(0 00) W

wEC 1 9

CHAPTER2.

60

CAUSALORIENTATIONS

The Cartan involution on SU(1,1) is given by conjugating by Z ~ and the holomorphic extension of O to SL(2, C) is given by

(o ~) (o -~) c

Thus TO(X) = t X. ac=tc=tc=C

d

~

-c

d

Identify p+ with C b y z ~ z ( 00

( 10

- 01 ) " ~ -C a n d K c _ ' ~ C *

01 ) " Similarily,

Asimplecalculationnow

shows that

o~ 1 ~;~)(1~ o)(~ 1 0 ) (~ ~) (0 0 1 if d # O. Thus a

b

~0} o~ ~((~ ~))~ a

b

b

Thus f~+ - (z e C ll~l < 1} and S -- {z E C ll~l = 1}. Furthermore, kc//{{ac \\

db~=l/d'\\ Thus we recover the foUowing well-known facts: //

(a b) . z = az+b c cz+d and

~((o ~) ) (o 1) c

We have

,z

=(cz+d)

El

=

0

0

'

E-1

=

1

0

'

-1.

(oo) ,(0 ~1) Xo - -~ 1 1(o 1) Yo _,_(1 01) =

Zo q+

2

1 0

2

0

-

'

'

- {'( ~ -~-~/ r E R } ,

2.6. S Y M M E T R I C SPACES R E L A T E D TO TUBE DOMAINS

61

and -

ThusexpitXo=

mr

rER

mr

.

( c o s 8 9 sin ~ ) . Hence _ s i n 8 9 cos

c - - - ~1 Wefindthatc-2expZ=

(

_11

1i

)

and

c 2 --

- 01

0

"

( 01 - E 1 )P + K c P - i f a n d ~

thermore, we have

~(c) = 1,

r -1) = - 1

and

j(c-2, Z ) = 1/z.

The functions Cm and ~,~ are given by ~m(Z) = z m and @re(z, w) = o

Define the homomorphisms ~oj 95[(2, C) --+ gc by E• ~+ E• as in Appendix A.3 and denote the corresponding homomorphisms SL(2, C) -+ Gc by the same letters. Then ~oi(SU(1 , 1)) C G and since ra(gc~j) = 8c~j, we can choose the root vectors Ej in the construction such that a(Ej) = r(Ej) = E_j. Thus ~oI o )~ = v o ~oi(X ). Let ~ be a character of Kc and assume it is unitary on K. Let r = R Z ~ Then d~ e ic*. We write k d~ := ~(k), k e Kc. L e m m a 2.6.17 Let r be a finite-dimensional irreduciblerepresentation of Gc with lowest weight p. Let Uo be a nonzero vector of weight p. Then the Kc-module generated by uo is irreducible. I.f tJ E/r then Kc acts on C uo by the character k ~-, k ~.

Proofi Let V be the representation space of Ir and W be the Kc-submodule generated by Uo. Then, as Kc normalizes p - , we find W C V p , where V p is the space annihilated by p - . For m

aeA+(ec,tc) we obtain W ("k)e C V ( " k ) e ~ p -

9

But V ("k)c~p- = Cuo. Thus W ("k)c is one-dimensional, which shows that W is irreducible. Now the last claim is a consequence of the lowest-weight description of the irreducible holomorphie representations of Kc. The claim follows. O

CHAPTER2.

62

CAUSALORIENTATIONS

L e m m a 2.6.18 Let a : Gc -r Gc be the conjugation with respect to G. Let X: K c --+ C* be a character. If XIK is unitary, then X ( a r ( k ) ) = x(k).

Proof" Both sides are holomorphic in k and agree on K as r I, = - 1 . The lemma now follows as K is a real form of Kc. t:3 L e m m a 2.6.19 Let Z E p+. Then the following hold:

1) If exp Z E c2P + K c P - then ~m(Z) - (Trm(j(c -2, Z))uo Itto) --- j(c -2, Z) -rap'.

z) r

Z) = k2~"-Va(Z) lo~ e~e,~ k e Kc.

3) Let Z l , . . . , z ~ ~_ C. Then r

n

(I'~(Ad(k) ~ zjEj) - k 2m~ H z ? ('+d(r-l)/2) j=l .i=x

Proof" 1) Write c -2 exp Z - pkq with p E P+, k - j(c -2, Z) and q E P - . As a(p +) - p - , we get rm(p)*uo - 7rm(aO(p)-l)uo = uo. Thus

9~ ( z )

= = =

( ~ ( ~ - 2 ~xpZ)~o I ~o) (rm(k)Trm(q)uo [~m(p)*UO) (~(k)~ol~o) k--rnPn

where the last equality follows from Lemma 2.6.17. 2) We have k . Z = Ad(k)Z. By Lemma 2.6.17,

6 m ( k . Z)

=

(Trm(c-2kexpZk-1)uo I=o)

= --

(rm(7(k))lrm(c -2expZ)lrm(k-1)uo I uo) kmp"(~rm(C -2 exp Z)uO [ Trm(~rT(k)-l)uo)

= =

k~-(~(k))~.C~(Z) k2'-,'- +,,,(z)

3) This follows from 2) and 512-reduction via ~oj. T h e o r e m 2.6.20 Let k E K c , g E Gc and Z, W E

p+ be such

and g. W are defined. Then the following hold: 1) ~m(g" Z , g . W) = j(g,Z)mP"j(g, W)mP"~m(Z, W).

that g . Z

2.6. S Y M M E T R I C SPACES RELATED TO TUBE DOMAINS

63

2) Let zj,wj E C, j = 1 , . . . , r . Then ~m(k"

zjSj, k.

wjSj) : k 2mpn H ( z j -- Wj) m(l"l'd(r-1)/2)

j--1

j--1

j=l

Proof: 1) We have, by definition, expg. Z - g exp Zj(g, Z) -lq,

q e P-,

and similarily for W. Thus

r

z-g.

w)

~xp(-g. w)~xp(g, z)

--

(gexp(W)j(g, w)-lp)-l gexp(Z)j(g,Z)-lq

=

p-lj(g, W)

exp(-W)exp(Z)j(g,

Z) -1

N o w A d ( c -2) = C 2 = 8r. Thus C 2 ( p - ) = P+.

for s o m e p e P - . above, we get

@,,(g. Z,g. W)

=

-

As

(Tr,,(c-2p-lj(g, W)exp(-W)exp(Z)j(g,Z)-~)uo I uo)

= j(g,z)"..j(g,w)"".~,.(z,w). 2) We have k. E 1 zJEi - k. E 1 wjEj - k ( E j ( z j - w j ) E j ) , as Kc acts by linear transformations. The claim now follows from Lemma 2.6.19. El 2.6.3

The

Causal

Compactification

o f A4

A causal compactification of a causal manifold is an open dense embedding into a compact causal manifold preserving all structures. More precisely, we set the following definition. Definition 2.6.21 Let A4 be a causal G-manifold. A causal compactification of A4 is a pair (A/', ~) such that 1) Af is a compact causal G-manifold. 2) The map ~" A4 ~ Af is causal. 3) 9 is G-equivariant, i.e., r mEA4. 4) r

is open and dense in jV'.

= g. r

for every g e G and every

CHAPTER2.

64

CAUSALORIENTATIONS

In this section we will show that the map A4 9 gH ~ g. ( E , - E ) G S x S is a causal compactification of .A4 and that the image of this map is given

by e& I

# 0}.

Identify the tangent space of 81 at ~0 with (g • g)/((D + q - ) x (D +q+)) ~_ q+ • q-. Let D be the image of C+ • - C _ in Tr 81 under this identification. Then D is an (H • g)-invariant regular cone in Tr (81). L e m m a 2.6.22 D is an ( H Q - • HQ+)-invariant cone in Tr (81).

Proof" Let q = exp X e Q+, p = exp Y e Q - , and (R, T) e q+ • q-. Then Ad(p, q)(R, T)

=

(e ~d Y R, end XT)

=

(R + [Y, R] + [Y, [Y, R]], T + [X, T] + [X, [X, T]]).

But [IF,R] e I) and [Y, [Y, R]] E q-. Thus R+[Y, R]+[Y, [Y, R]] -- R mod(i)+ q-). Similarily, T + [X, T] + IX, [X, T]] = T mod(i) + q+). It follows that Ad(p,q)[Tr ) = id. This implies the claim, o It follows that D defines an invariant causal structure on 81. Recall the maximal abelian subalgebra ~ j =P l R X j of p from Proposition A.3.3. Write Xj = X + + X~- with X ~ G q+. P r o p o s i t i o n 2.6.23

2) Q - - E

1) q- -- Ad(H N K) E~=I R X ; zjEj G S [ [zj[ - 1, zj ~ --1}.

= Ad(K f3 H ) { ~ = I

3) ~ : ( K N U ) { E ; - 1

zjEjE ~11zjI-" 1).

Proof:. 1) Let X E q-, then X k G K N H and xj G IR such that

O(X) E q N p. Therefore we can find

r

A d ( k ) ( X - O(X)) = Z

r

zjXj = Z

j=l

j--1

xjX~ +

x j X + (5 %. j=l

As O ( X ) , X + G q+ it follows that Ad(k)X = E~=I x j X ~ 2) Assume first that G = SU(1 1) Then X 1 = i ( 1 '

( xp tX1)

"

. 1 = ( 1 + it \ -it

k-1

1 )

-I

"

Thus

it ~ 1 + 2it 1 - i t ) = 1 - 2it

1 z-1 The general case now follows So if ]zl = 1, z r - 1 , we choose r = 2i z+X" from 1) and s[2-reduction.

2.6. S Y M M E T R I C SPACES RELATED TO TUBE DOMAINS

65

3) Q + - E is dense in ,~ as Q+ H Q - is dense in G. The claim follows from r that, as ( g N H){~.j= 1 zjEj e 8 [[z/[ - 1} is closed and contained in 8. []

T h e o r e m 2.6.24 Define ~ : ~

~ 81 by O(gH) :-- g. ~o. Then (S1, O) is a causal compactification of M with O ( M ) -- {~ e 81 I ~rn(~) # 0}.

Proof: The G-equivariance of the function ~ is clear. Let us show that ~ is causal. As both the causal structures on A4 and that on 81 are G-invariant, and because 9 is G-equivariant, we only have to show that (dO)o(C) C Dr But this is obvious from the definition of D. To show that the image of ~ is dense, it suffices to show that it is given as stated. It follows from Theorem 2.6.20 that the left-hand side is contained in the right-hand side. Assume now that ~m (Z, W) # 0, ~ - (Z, W) E 81. Let g E G be such that g . W - - E and then choose k E K gl H such that k . (g. Z) = ~, zjEj. Then r

(kg) . ~ = ( E z j E j , - E ) . j'-I

By Theorem 2.6.20 we have @m((kg). ~) # 0. By the second part of that theorem we have zj r - 1 for j = 1 , . . . , r. By using Proposition 2.6.23 we now find q e Q+ such that q. (kg. ~) = ~0. Hence ~ = (qkg) -1. ~o. [] R e m a r k 2.6.25 The compactification in Theorem 2.6.24 is also causal with respect to the causal structure on G / H coming from the cone field

c+ +c_. 2.6.4

S U ( n , n)

Let n -- p + q. Then SU(p, q)

=

{a e SL(n, C) [a*Ip,qa = Ip,q}

A C

=

B D

D*D-B*B=Iq B*A-D*C=O

.

The conjugation in SL(n, C) with respect to SU(p, q) is given by =

where ~ is the Cartan involution a ~ (a*) -1. If a e SU(p, q), then a -1 = Ip,qa*Ip,q. Hence C

D

=

-B*

D*

"

(2.27)

CHAPTER2.

66

CAUSALORIENTATIONS

The Lie algebra of SU(p, q) is given by

{(x ~)

zu(p'q) =

Y*

Z

Y ~. M ( p • q, C), X E u(p),

ZEu(q),TrX+TrZ=0

(2.28)

]"

The maximal compact subgroup K is given by S(U(p) x U(p)). Furthermore,

, {(~o o ~)l {(0 ~)

A E u(p), B E u(q), Tr(A)+ Tr(B) = 0}

and

P=

We choose Z ~ =

Y*

,,0Ip

- ~ 0I q

ad(Z o)

Y e M(p x q , C ) ) .

0

(~) yA

e t . Then

B

=

(0 ,Y) -iY*

0

'

which implies that P~ =

Suppose that

0

C

0

D

E SU(p, q). Then D is invertible and we have a

decomposition

(~ ~) (i~ ~o,)(~_~o~ o)( i~ 0) C

D

=

0

Iq

0

D

D-1C

Iq

"

Thus the Harish-Chandra embedding G / K '--+ p+ _ M ( p x q, C) is given

by

~((~ ~) c

D

KcP -

) (o ~o~) =

0

0

)+ B D - I

inducing a biholomorphic isomorphism su(p, q)IS(U(p) x u(q))_~ D,.~, where D,,~ := {Z e M(p x q,C) 1 I~ - Z ' Z > 0}. H ~ SU(p, q) on Dp,q is given by ( A \ C

B ~ . Z = (AZ + B)(CZ + D)_, D/

th~ ,ction o~

2.6. S Y M M E T R I C SPACES RELATED TO TUBE DOMAINS

67

G / K is of tube type if and only if p = q. In that case we have

,(0 ,.) ,(o

Y~

0

x~

=

~

c=

I.

v~

c_2

=

(

E

=

/n,

'

o

_i.

0 -I,

'

i"

,

In) 0 '

s = u(.). The involution T = 7"yo is conjugation by 2Yo. Thus r

((A O))(O C

D

=

B

A

"

Therefore H

h ( A , B ) "=

B

A

A * A - B*B = In, B*A = A*B

'

{ h(X, Y) e su(n, n)l X* = - X , Y* = Y, Tr X = 0 }, and

q=i

{

q(X,Y):=

(x Y) _y

-X

*X = X, , y = y )

.

Define ~o+ 9H U [} -4 M(n, C) by

9+ ( h ( A , - B ) ) = A 4- B . We leave the simple proof of the following assertions to the reader: 1) ~o+ 9H -4 GL(n, C)+ is an isomorphism of groups. 2) T •

-+ st(n, C) + R In is an isomorphism of Lie algebras.

Note that the Cartan involution on GL(n, C) is O(a) = (a') -x. By (2.27),

~(oIn I.) 0

~_ = 0 o ~0+. We choose ~

as Yo. Then the ~-module structure

of q is described by q+ = i { q ( X , - X ) I t X = X } ~ i q ( X , - X ) ~'~+ X E H(n, C)

CHAPTER2.

68

CAUSALORIENTATIONS

and

q- = i{q(X,X) [ ' X = X } B iq(X,X) ~'I.~- X e H(n, C). Obviously, both ~o+ and ~o_, are isomorphisms. By (2.27) we get for X E q+ and a E H,

~o•

= ~o•

[~-~~o•

~o=v.(a)*.

In this case C• = ~1 o •--1 (H+(n, C)), cf. Example 2.1.14. r In the bounded realization we have G / K = D,,,. The space ~ j = l C Ej corresponds to the diagonal matrices and E = I0. In particular, ~q = U(n). By Lemma 2.6.19 and the table on p. 58, Ore(Z) = det(Z) m". Thus

SU(n,n)/GL(n,C)+ " ((Z, W) 6 U(n) • U(n) I d e t ( Z - W) ~t 0}.

2.6.5

Sp(n,R)

We realize G = Sp(n, R) inside SV(n, n) as Sp(..R) =

{(A

A ' A - tB-B = I,, t(B*A) = B ' A } .

(2.29)

X , Y e M(n,C), X* = - X , t y = y } .

(2.30)

Its Lie algebra is given by

sp(n,R)= {(Xy xY)

The involution r leaves G and g stable and is also given by complex conjugation. Therefore H, b are just the real points of the corresponding object for SU(n, n). We also note that the above Xo and Yo are in sp(n, R). Thus

{h(A,B) e Sp(n,R) ] A , B e M(n,R)}

H ~+ _'2

b

= ~+

GL(n, R)+, {h(X, Y) e sp(-,R)I X, Y e M(n,R), ' X = - X , t y = Y} gt(..R).

q = q+ =

i{h(X,Y)]X,Y e M(n,R), tX = X, ' Y = Y } , i { q ( X , - X ) I X e M,(R), tX = X}

q

i q ( X , - X ) ~'~+ X e H(n,R), i{q(X,X) l X e M(n,R), tX = X} iq(X, X) ~ -

X e n(n, R).

2.6. S Y M M E T R I C SPACES R E L A T E D TO TUBE D O M A I N S

69

T h e C a r t a n involution on G L ( n , R ) is O(a) = 'a -1. From (2.27) we obtain ~o_ = 0 o ~o+ and, with X E q• and a E H , ~o•

= ~o•

[1 ] ~o+(X)

'T4.(a).

In this case C• = 2i~o_u cf. E x a m p l e 2.1.14. T h e b o u n d e d realization of G / K is {Z e M , ( C ) I In - Z ' Z > O, ' Z = Z } . r

T h e space ~_,j=l C Ej corresponds to the diagonal matrices and E = In. In particular,

s =

{A e U ( . ) I ' z = z } .

L e m m a 2.6.19 and the table on p. 58 imply +m (Z) - d e t ( Z )

,,~(.+1) , .

Thus Z, W s y m m e t r i c det(Z-W)#0 f"

S p ( n , R ) / G L ( n , R ) + "~ ~(Z, W) e O(n) x O(n) [

Notes

for Chapter

2

Cones have been used in different parts of mathematics for a long time and are related to concepts such as the Laplace transform [24, 28], Hardy spaces over tube domains, and Hermitian symmetric spaces [83, 84]. The concept of causal orderings associated to cone fields has also been used for a long time implicitly in the context of Lorentzian geometry and relativity (e.g., in [4, 37, 42]). Group invariant cone fields appear in Segal's book [157]. Vinberg, Paneitz, and Ol'shanskii considered the special case of bi-invariant cone fields on Lie groups in [166, 147, 148]. The first article on invariant cone fields on semisimple symmetric spaces was [138]. A systematic study of invariant cone fields on homogeneous spaces was started in [47] and [50]. Later it was taken up in the work of Lawson [93], Olafsson [129, 130], and Neeb [112]-[115], [52]. The algebraic side of the theory, i.e., a closer inspection of the cones that appear in the study of causal orderings, was also initiated by Vinberg in [166] and then taken up by many authors [48, 50, 129, 137, 148]. The order compactification was introduced in [55], motivated by the study of Wiener-Hopf operators on ordered homogeneous spaces. The results in the last section are taken partly from [136], where further information about this class of spaces can be found. This causal compactification has

70

C H A P T E R 2. C A U S A L O R I E N T A T I O N S

also been considered in [86, 87]. The compactly causal hyperboloids were studied in [107]. in [7] and [6], causal compactifications for a more general class of causal symmetric spaces are given. Compactifications without the causal structure have also been obtained in [98] and [76].

Chapter 3

Irreducible Causal Symmetric Spaces In this chapter we determine the irreducible semisimple causal symmetric spaces. The crucial observation is that the existence of causal structures on .h,4 = G / H is closely connected to the existence of ( H N K)-fixed vectors in the tangent spaces of .hA. With that tie established, one can use the results of Chapter 1 to single out which irreducible non-Riemannian semisimple symmetric spaces admit causal structures. As we have seen already in Chapters 1 and 2, the existence of causal structures may depend on the fundamental group of t h e space. So we include a discussion on how causal structures behave with respect to coverings. In Section 3.2 we give a list of all the irreducible semisimple symmetric pairs (g, r) for which the universal symmetric space admits a causal structure.

3.1

E x i s t e n c e of Causal S t r u c t u r e s

In this section we assume that .A4 = G / H is a non-Riemannian semisimple symmetric space such that the corresponding symmetric pair (g, r) is irreducible and effective. We fix a Cartan involution 0 commuting with r and use the notation introduced in Remark 1.1.15. L e m m a 3 . 1 . 1 Let 0 ~ X E qHnK and C the smallest Ho-invariant convex

closed cone in q containing X . Then C is H-invariant. Prool!. We mimick the proof of L e m m a 2.6.1: If h E H, then h is of the form h = hok with k E H f ' I K and ho E Ho (cf. (1.8)). Further, let 71

72

C H A P T E R 3. IRREDUCIBLE CAUSAL S Y M M E T R I C SPACES

Y = ~ )~j Ad(hj)X e C, )~j >_ O, hj e Ho. Thus Ad(h)Y

=

~_, )~j A d ( h h j ) X

=

~)~j Ad(hhjh-1)Ad(ho)Ad(k)X

=

~ ) ~ j A d ( h h j h - 1 ) A d ( h o ) X ~_ C

since hhjh -1 E Ho. As C is closed and the set of elements of the form )~j A d ( h j ) X is dense, it follows that C is H-invariant. rn L e m m a 3.1.2 If .M admits a G-invariant causal structure, then we have:

1) There exists an H-invariant proper closed convex cone in q. 2) q is a completely reducible H-module with either one or two irreducible components. 3) dim(q HnK) is equal to the number of irreducible H-submodules of q. 4} /fdim(q HnK) < dim(qH*ng), then there exists an element h e H o g such that Ad(h)Y = - Y for all Y e 3(t}).

Proof. 1) Let C be an H-invariant nontrivial closed convex cone in q. Then space qC = C O ( - C ) is H-invariant and not equal to q. According to Lemma 1.3.4, qC is either trivial or otherwise equal to q+ or q-, where q = q+ + q- is the decompostion of q into irreducible b-modules. In each case we find a proper Ho-invariant cone in C. 2) Since Ho is normal in H, the action of H on q maps Ho-submodules to Ho-submodules. Thus q is a reducible H-module if and only if it is a reducible Ho-module and q• are both H-invariant. In this case q• are both irreducible H-modules, which implies the claim. 3) If q is a reducible H-module, then we consider the projection pr: q -+ q+, which is H-equivariant. Therefore pr(C) C_ q+ and 0 o pr(C) C q- are H-invariant proper cones. Thus Theorem 1.3.11.4) implies that dim(q Hng) = dim(q Hong) = 2. If q is irreducible as an H-module, then qC = {0}, i.e., C is proper. Then Lemma 2.1.15 shows that qltnK ~ {0}. It remains to be shown that dim(q Hng) = 1. We have two cases to consider. C a s e 1: dim(q H~ = 1. In this case, obviously, dim(q Hng) = 1 as well. C a s e 2" dim(q HOnK) = 2. Then we are in the situation of Theorem 1.3.11 and, as far as the symmetric pair (g, r) is concerned, Section 2.6. Recall the 0-stable subgroup Hi of index 2 in Ho from Lemma 1.3.14. According to this lemma, we can find an h E H n K, not contained in

3.1. E X I S T E N C E OF C A U S A L S T R U C T U R E S

73

/-/1, such that Ad(h)q + = q-. Since 3(I)) is one-dimensional, we have Ad(h)Yo = rYo with r E R. Moreover, h 2 E HI, whence r 2 = 1. If r = 1, then Xo, Z ~ E q H n K which shows that the Ho-invariant cones generated by X • = Xo=l=Z ~ E q• are H-invariant (Lemma 3.1.1). But this contradicts the H-irreducibility of q. Thus we have r = - 1 . This means that A d ( h ) X o = - X o which, together with h A d ( h ) Z ~ = Z ~ shows that the only H-invariant vectors in qHonK = IR Xo + ]R Z ~ are the multiples of Z ~ This implies dim(q HnK) = 1. Finally, we note that Ad(h)Yo = - Y o by the above, so that assertion 4) follows, too. El T h e o r e m 3.1.3 Let .It4 = G / H be a non-Riemannian semisimple symmetric space. If .A4 is irreducible, then the following statements are equivalent:

1) .All admits a G-invariant causal structure. 2) dim(q HnK) > O. If these conditions hold, then Conen(q) # 0, i.e., A4 even admits a regular G.invariant causal structure. Proof:. L e m m a 3.1.2 shows that the existence of G-invariant causal structure on ,~4 implies that qHng # {0}. Assume conversely that qUng # {0}. We have to consider two cases. C a s e 1" Suppose that q is irreducible as an I)-module. Then by Theorem 1.3.11, Ho is semisimple, and by Lemma A.2.5, dim q u*ng - 1. Thus qHnK _ qHong. Let 0 # X E qHng and Cmin be the Ho-invariant cone generated by X (cf. Theorem 2.1.21). Then L e m m a 3.1.1 shows that Cmin is H-invariant. C a s e 2: Suppose that q is not irreducible as an l)-module. Then we are in the situation of Theorem 1.3.11 and Section 2.6. Let C be the Ho-invariant cone in q generated by Z ~ Then the group case described in Section 2.5.1 shows that C is proper. It is also H-invariant by L e m m a 3.1.1. It remains to show that the existence of proper H-invariant cones imply the existence of regular H-invariant cones. If q is H-irreducible this is obvious, since the span of an H-invariant cone is H-invariant. If q is not H-irreducible, then the cones C+ constructed in Section 2.6 are H-invariant, proper, and generating in q+. Thus C+ + C _ is an H-invariant regular cone. o L e m m a 3.1.4 Let ./t4 = G / H be an irreducible non.Riemannian semisimple symmetric space and C an H-invariant cone in q. If either

cont#O

and

cnp#{o}

74

C H A P T E R 3. I R R E D U C I B L E C A U S A L S Y M M E T R I C S P A C E S

or

c~162

a.d Cn~#{o},

then C contains a line. Proof:. We will only show that the first assumption implies that C contains a line. The second part then follows from c-duality. Assume that C is proper and C ~fl t~ ~ 0 and C I"1p ~ {0}. Let Z~ E C ~ N and X1 E C Iq p, Z1, X1 ~ 0. Define

z "= fA

and

k . Zx dk

X:= L

da(KnH)

do(KNH)

k. X l dk.

Then (cf. Lemma 2.1.15) Z, X are nonzero and (KNH)-fixed. Furthermore, as 0 commutes with Ad(K N H), Z E C ~ N q~ and X E C N qO. Thus we are in the situation of Theorem 1.3.11 and Section 2.6. In particular, Remark 2.6.2 shows that Z E R Z ~ and X E R Xo. Normalize Z and X such that ad Z has the eigenvalues -t-i and 0 and ad X has the eigenvalues +1 and 0. In particular, and

ad(Z)2lp = - i d

~d(X)'l..

= id.

Let Y = [Z, X] E R Yo C } (D) cl p. Then

It, z] = [[z, x ] , z] = - ~ d ( Z ) 2 X = x and similarly, [IF,X] = Z. Thus we have reduced the problem to one on 5[(2,1R). A short direct argument goes as follows:

~,.d ~ ( Z + X)

=

~' (Z + X),

e tad Y ( Z - X )

=

e-t(Z - X).

In particular, we get e ' ad Y Z

"-

cosh t [Z + (tanh t)X],

e t ad Y X

--~

cosh t [(tanh

t) Z + X].

Dividing by cosht and letting t ~ -I-c~ shows that + ( Z + X ) C contains a line.

E C. Thus

cl

T h e o r e m 3.1.5 Let .s = G / H be a non-Riemannian semisimple symmetric space. Suppose that M is irreducible and admits a G-invariant causal structure. Then the following cases may occur: 1) dim(q HnK) = dim(q H*nK) = 1. In this case q is irreducible as H and I)-module. There are two possibilities which are c-dual to each other:

3.1. E X I S T E N C E OF C A U S A L S T R U C T U R E S

75

1.k) q H n r C qk. In this case, .for every cone C 6 ConeH(q) we have c ~ n qHnK # (~,

C n p = {o}.

1.19) qHnK C qp. In this case, for every cone C 6 ConeH(q) we have C~n % _HAg #

2) dim(q HnK) -- dim(q HONK) [}-irreducible and we have

O,

c n ,e = { o } .

2. In this case

ConeH (q) --

q is

neither H- nor

{::ECk,:ECp}

(cf. Remark 2.6.2 ). 3) dim(q HnK) = 1, dim(q HonK) = 2. In this case, q is H-irreducible but not O-irreducible and we have

ConeH (q) "-- {-{-Ck }. Proofi Note first that Theorem 1.3.11 and Theorem 3.1.3 show that no more than these three cases are possible. Moreover, Lemma 3.1.2 shows the claims about H- and [j-irreducibility. 1) Recall that qHng is 0-invariant. This proves the dichotomy of (1.k) and (1.p). Now the claim follows from Lemma 3.1.4 in view of Lemma 2.1.15. 2) In view of Theorem 1.3.11, this is a consequence of Theorem 2.6.8. 3) Theorems 1.3.11 and 2.6.8 show that ConeHo(q) = {-I-Ck,:ECp}. This proves that Ck is the Ho-invariant cone generated by Z ~ Since Z ~ is an (H n K)-fixed point, Lemma 3.1.1 implies that Ck is H-invariant. On the other hand, Lemma 3.1.2 shows that we can find an h 6 H n K with i d ( h ) Y o = - Y o so that also h d ( h ) X o = - X o and hence does not leave Cp invariant. 0

The following corollary is an immediate consequence of Theorem 3.1.5 and Remark 2.6.2. C o r o l l a r y 3.1.6 If M is an irreducible non-Riemannian semisimple symmetric space and C 6 Coneu(q), then we have either c ~

n

#

Cn

p =

{o}

or

C~

_HAK

#0,

C n t = {O}.

76

CHAPTER 3. IRREDUCIBLE CAUSAL SYMMETRIC SPACES

R e m a r k 3.1.7 1) Theorem 1.3.8 shows that in Case (1.k) of Theorem 3.1.5 the Riemannian symmetric space G / K is a bounded Hermitian domain and r induces an antiholomorphic involution on G/K. Moreover, Theorem 1.3.11 implies that G / K is not of tube type in this case. 2) Under c-duality t corresponds to I}a and Hermitian to para-Hermitian structures. Therefore in Case (1.p) of Theorem 3.1.5 the space G / H a carries a para-Hermitian structure. 3) Theorem 1.3.11 shows that in the cases 2) and 3) of Theorem 3.1.5 the Riemannian symmetric space G / K is a bounded Hermitian domain of tube type. 4) Theorem 2.6.8 implies that Case 3) in Theorem 3.1.5 cannot occur if H is connected. El D e f i n i t i o n 3.1.8 Let M be an irreducible non-Riemannian semisimple symmetric space. Then we call ~/f CC) a compactly causal symmetric space if there exists a C E Conen(q) such that C ~ N t # 0, NCC) a noncompactly causal symmetric space if there is a C E ConeH(q) such that C ~ N p # $, and CT) a symmetric space of Cayley type if both (CC) and (NCC) hold. CAU) a causal symmetric space if either (CC) or (NCC) holds. The pair (0, T) is called compactly causal (noncompactly causal, of Cayley type) if the corresponding universal symmetric space M has that property. Finally, (9, r) is called causal if it is either noncompactly causal or compactly causal.

D

R e m a r k 3.1.9 1) It follows directly from the definitions that (g, r) is noncompactly causal if and only if (go, r) is compactly causal. 2) In view of (1), Lemma 1.2.1 implies that a noncompactly causal symmetric pair (g, r) is of Cayley type if (g, r) ~- (0 c, v). 3) If (g, r) is compactly causal, then Theorem 1.3.8 shows that either g is simple Hermitian or of the form gl x gl with gl simple Hermitian and r(X, Y) = (Y, X).

3.1. E X I S T E N C E OF C A U S A L S T R U C T U R E S

77

4) Example 1.2.2 now shows that if (g,r) is noncompactly causal, then either Be is simple Hermitian or of the form ~c with r = a. Note that in both cases g is a simple Lie algebra. 5) If (g, r) is of Cayley type, then 3) and 4) imply that g is simple Hermitian. 6) Let g be a simple noncompact Lie algebra. Then (g, id) is an irreducible symmetric pair. It is not causal since it does not belong to a non-Riemannian semisimple symmetric space (cf. Remark 1.1.15). n D e f i n i t i o n 3.1.10 Let (9, r) be noncompactly causal symmetric pair. An element X ~ e qp is called cone-generating if spec(ad X ~ = { - 1 , 0, 1} and the centralizer of X ~ in g is IJa. t::! P r o p o s i t i o n 3.1.11 Suppose that (g, r) is a noncompactly causal symmetric pair. 1) Cone-generating elements exist and are unique up to sign. 2) Let b be an abelian subspace of p containing a cone-generating element X ~ Then b C qp. 3) Let a be a maximal abelian subspace of qp. Then a is maximal abelian in p and maximal abelian in q. Moreover, a contains X ~ 4) Let a be maximal abelian in q and assume that X ~ E a. Then a C qp. 5) Let X ~ E qp be a cone-generating element. Then R X ~ = ~(~a). Proo~ 1) According to Theorem 3.1.5, the centralizer 3,~([J) of [~k in q is nontrivial. Then, in view of Theorem 1.3.11, it is one-dimensional, say of the form RX. Lemma 1.3.5 says that ~,~(~) = ~ ( ~ ) N q. But then ~0(X) is 0-invariant and contains ~ , so Lemma 1.3.2 implies that ~ = ~0(X). Note that the spectrum of ad(X) is real and pick the largest eigenvalue r of X. Then - r is also an eigenvalue and g ( - r , X) + ~ + g(r, X) is a 0-invariant subalgebra strictly containing ~ , hence equal to g, again by Lemma 1.3.2. Thus there are no more eigenvalues of ad(X) t h a n - r , 0, r and this implies the claim. 2) If b is abelian and contains X ~ then b C ~s(X ~ = h a. Thus b C %. 3) If a is maximal abelian in qp, then X ~ E a is abelian since X ~ E ~([}a). Let b be maximal abelian in p containing a. Then 2) implies b C qp, so b = a. That a is maximal abelian in q follows from 3q(X ~ = %. 4) This again follows from ~q(X ~ = %.

78

C H A P T E R 3. I R R E D U C I B L E C A U S A L S Y M M E T R I C SPACES

5) We have seen already that ba is the centralizer of X ~ Therefore it only remains to show that dim(~(ba)) _< 1. But that follows from Lemma 1.3.10 applied to (g, l~a). El The analog of Proposition 3.1.11 for a compactly causal symmetric pair follows via c-duality. We only record the first part, which will be used later. P r o p o s i t i o n 3.1.12 Suppose that (g, r) is compactly causal. Then there exists an, up to sign unique, element Z ~ E qk such that spec(ad Z ~ = {-i, 0, i} and the centralizer of Z ~ in g is t. Q E x a m p l e 3.1.13 Recall the SL(2,R) Example 1.3.12 for which one has G ~ f'lG ~ = {:I:1} so that qG'nG ~ = q is two-dimensional. This shows that the one-sheeted hyperboloid is of Cayley type. Note that a := qp is abelian itself. The corresponding cone Cp is R + ( X ~ + Z ~ + R + ( X ~ - Z~ cl

The following proposition gives some useful isomorphisms between dual spaces of causal symmetric pairs. P r o p o s i t i o n 3.1.14

1) Let (g, r) be a compactly causal symmetric pair. Fix Z ~ E ~(t)q such that ado Z ~ is a complez structure on p. Let t/,k = ~Ozo (cf. Lemma 1.2.1). Then the following hold:

~) r b) l~k I = ~k

0

0 = 0

0

l[~le,.

c) r o C k = r

~. a) Ck : (g, ~, e) -+ (g, r ~, e) is ~ i, omo,~u,,,. 2) Let (g, 7") be a noncompactly causal symmetric pair. Fix X ~ E 3q~(IJk) such that g = g(O,X ~ ~ g ( + l , X ~ (9 g ( - 1 , X ~ Let Cp = ~ixo. Then the following hold:

~) r

= ~~

b) r o ~p = Cp o e.

c) ~2p defines an isomorphism ~p : (9, e, r) --+ (9, r, O)r. 3) Let (g, r) be a symmetric pair of Gayley type. Fix an element yo E 3(~) ~,,ch that g = g(o, Y~ ~)eg(-z, yo). D~Fne r = ~,vo. Then the following hold:

~) r b) r 1 6 2

c)

r

= r~ or

3.1. EXISTENCE OF CAUSAL STRUCTURES

79

d) Ce defines an isomorphism Ce "(g, r, 0) --+ (ge, 7-, r a ) . Proof:. 1.a) follows from L e m m a 1.2.1. 1.b) and 1.c)" By 1.a), we have r = id. Thus ~_zo = r = r = o O = O o r As r Z ~ = - Z ~ it follows that r o r = r o r. This implies 1.b) and 1.c). 1.d): This is an immediate consequence of 1.c). Parts 2) and 3) can be proved in the same way as P a r t 1). rn Given a causal symmetric pair, it is not clear which, if any, symmetric space A4 associated to (g, 7") is causal (cf. Section 2.5). With the structure theory just established, we are in a position to clarify the situation. The following proposition shows t h a t compactly causal symmetric pairs do not pose any problems in this respect. P r o p o s i t i o n 3.1.15 If ( g , r ) is compactly causal, then every symmetric space associated to (g, T) iS causal.

Proo~ Let M = G / H be a symmetric space associated to (g, r). Choose Z ~ 6 3(~)Nq (cf. Proposition 3.1.12). Then A d ( k ) Z ~ = Z ~ for every k E K. In particular, Z ~ = qHnK which proves the claim in view of Theorem 3.1.3. o Note here t h a t Proposition 3,1.15 does not say that any Ho-invariant cone in q is H-invariant, i.e., gives a causal structure on ~4. As we have seen before (el. e.g. Theorem 3.1.5), in the noncompactly causal case the existence of causal structures is related to the nature of the component group H/Ho of H. The right concept to study in our context is that of essential connectedness. D e f i n i t i o n 3.1.16 Let M = G / H be a non-Riemannian semisimple symmetric space and (g, r) corresponding symmetric pair. Further, let a be a maximal abelian subalgebra in qp. Then H is called essentially connected in G if H = ZKnH(a)Ho.

[3

We note that this definition is independent of the choice of a, since the maximal abelian subspaces in qp are conjugate under Ho fl K. R e m a r k 3.1.17 Let (g, r) be a noncompactly causal symmetric pair. We fix a cone generating element X ~ 6 qp and a maximal abelian subspace a of qp containing X ~ Then a is a maximal abelian subspace of p and we denote the set A(g, a) of restricted roots of g w.r.t, a by A. Further, we set = e a l (x ~ = 0}.

80

CHAPTER 3. IRREDUCIBLE CAUSAL S Y M M E T R I C SPACES

Since t~" is the centralizer of X ~ we get Ao = A(b",a).

(3.2)

Let A+'={aeAla(X

~

and

A_'={aEAla(X

~

(3.3)

Choose a positive system A + in A0. Then a positive system A+ for A can be defined via A + :-- A+ U A + . (3.4) Set

n•

'~'~ ~(xo)=+z

g~,

no= ~

g,~.

~e~+

Then n = n+ + no, [n+, n+] = (0},[n_, n_] = {0} and[[}a, n•

C n•

T h e o r e m 3.1.18 Let A4 = G / H be a symmetric space such that the cor-

responding symmetric pair (g, v) is noncompactly causal. Then A4 is noncompactly causal if and only if H is essentially connected in G. Proof. Choose a cone-generating element X ~ E qp. Then X ~ centralizes Dk and hence is contained in qffonK. Next we choose a maximal abelian subspace a of p containing X ~ If H is essentially connected in G, then obviously X ~ ~_ qHnK and G / H is noncompactly causal by Theorem 3.1.3. Assume conversely that G / H is noncompactly causal. Then, in view of Theorem 1.3.11, Theorem 3.1.5 implies that qHnK = qHonK = R X 0. Let a, A, and A + be as in Remark 3.1.17 and fix some k ~. K N H . Then Ad(k)a is a maximal abelian subalgebra in tip. Since all such algebras axe Ho N Kconjugate, we can find an h 6_ Ho N K such that Ad(hk) normalizes a. But k and hk axe contained in the same connected component of H, so we may as well assume that Ad(k) normalizes a. Since H N K fixes X ~ it leaves A0 invaxiant. Therefore k. A + is again a positive system in A0 and we can find a ko E Ho N K such that ko(kA +) = A +. But then A + is invariant under kok, so that kok E M N H = ZHnK(a). Thus k E ZHnK(a)Ho and H is essentially connected. D We use Theorem 3.1.18 to show that the symmetric space A:t = G / H is noncompactly causal if the corresponding symmetric pair (g, r) is noncompactly causal. To do this we need one more lemma. L e m m a 3.1.19 Let (g, 7) be a noncompactly causal symmetric pair and

Gc be a simply connected complex Lie group with Lie algebra gc. Choose a cone-generating element X ~ E qp. Then

3.1. E X I S T E N C E OF C A U S A L S T R U C T U R E S

81

1) Z c c ( X ~ is connected and equal to G~~ 2) Z a ( X ~ = G

.

3) Z K ( X ~ = K ~, where K = G ~ Proo~ 1) Let ~ on Gc as Gc is groups have the qo = r a - r0. (x ~ =

= fix0 (cf. L e m m a 1.2.1). Then ~ defines an involution simply connected. Obviously Z a e ( X ~ C G~ and b o t h same Lie algebra I~. = ([}k)c 9 (qp)c. This shows t h a t By Theorem 1.1.11, G~ is connected. Hence we have

= ag.

2) Z G ( X ~ = G g f) G = G r* because of 1). 3) Z K ( X O) = G r" N g = g r because of 2) and 0[a- = id. 3 . 1 . 2 0 Let (g, r) be a noncompactly causal symmetric Gc be a simply connected complex Lie group with Lie algebra gc. let G be the analytic subgroup of Gc with Lie algebra g and H a of G r containing Gro. Then H is essentially connected and .M is a noncompactly causal symmetric space. In particular .I(4 = noncompactly causal.

Theorem

D

pair and Further, subgroup = G/H G I G r is

Pro@ Fix a cone-generating element X ~ E qp. Then L e m m a 3.1.19 shows t h a t X ~ E qpG r NK . Now Theorem 3.1.3 and Theorem 3.1.5 imply t h a t A4 = G I G ~ is noncompactly causal. Therefore Theorem 3.1.18 shows t h a t G r is essentially connected. But then all open subgroups of G r are essentially connected as well, so t h a t the converse direction of T h e o r e m 3.1.18 proves the claim. El R e m a r k 3.1.21 T h e o r e m 3.1.20 shows t h a t in the situation of Section 2.6, i.e., for spaces related to t u b e domains, the assumptions made to ensure the H-invariance of the various Ho-invariant cones are automatically satisfied. In fact, one can choose X ~ = Xo in t h a t context, so one sees t h a t Xo ~.

qHnK.

El

The results of L e m m a 3.1.19 can be substantially extended. 3.1.22 Let (g, r) be a noncompactly causal symmetric pair and Gc be a simply connected complex Lie group with Lie algebra ftc. Further, let G be the analytic subgroup of Gc with Lie algebra g and K = G ~ If l1 C qp is a maximal abelian subspace, then

Lemma

1) G ~~ - M(G~:)o, where M = ZK(a). 2) M =

(.) = z u . (.).

82

CHAPTER

3. I R R E D U C I B L E

CAUSAL SYMMETRIC

SPACES

3) Let F = K fl expia. Then F = {g E expia l g 2 = 1} c M , K ~ = F ( K ' ) o , and G " = F ( G ' ) o . ~) H a = F ( H a ) o = F ( G ' ~ (1.11))

= G "~ where H a = ( H M K ) e x p q p (c].

Proo~ 1) Let X ~ E a C qp be as in Proposition 3.1.11. Then L e m m a 3.1.19 implies M C G r~ , since X ~ E a. Conversely, recall from T h e o r e m 3.1.20 t h a t G r is essentially connected, so t h a t G ' ~ r'l K = G ''~ 13 G ~ C [ Z K n e , , ( a ) ( G ' ) o ] 13 G ~ C

M[G"~

2) According to L e m m a 3.1.19, we have Z a . (a) C Z a . ( X ~ C G r f3 G~.~ C G ~ N G ~ C g

which implies ZG. (a) c M . Conversely, M c G ~~ N G o = G ~ N K by 1), so M c Z G . ( a ) . 3) We recall t h a t the involutions 7, a, 7a, and 8, on gc with fixed point algebras [~c,g,g c and t + ip, respectively, all integrate to involutions on Gc and have connected sets of fixed points He, G, G c and U in G c (cf. T h e o r e m 1.1.11). The involution 0, induces C a r t a n involutions on G, G c, and Go. Let K c be the corresponding maximal compact subgroup of G c. Then ( g e ) ~ = (Uar) ~ = U a 13 U r = g *. Now assume t h a t k E K f'l exp ia. Then k = a ( k ) = k -1, so t h a t k 2 = 1. Conversely, if k E e x p i a and k 2 = 1, we have a ( k ) = k -1 = k, i.e., k E G. But we also have 8,(k) = k, whence k E G gl U = K . Note t h a t 7(k) = k -1 = k implies t h a t F C K r. It is clear t h a t F C Z g ( a ) , so it only remains to show t h a t G r C F ( G r ) 0 . To this end we fix h E G r and write it as h = kexpX

E Kexpp.

Now the 7-invariance of the C a r t a n decomposition shows t h a t k E K r -( g c) r and X E bp. Note t h a t (to, v) is a compact Riemannian symmetric Lie algebra and ia is a maximal abelian subspace in t e n qc = i(%). According to [44], C h a p t e r 7, Theorem 8.6, we have g c = ( g r ) o ( e x p i a ) ( g r ) o , so we can write k = lal ~ with a E exp ia and l, l ~ E (Kr)o. Applying r, this yields lal ~ - l a - l l ~ and thus a = a -1 E F . Now the claim follows from k E (gr)oF(gr)o(gr)o

C F(gr)o.

4) L e m m a 3.1.19 implies H ~ = K r Z c ( X ~ o, so 3) proves the first two equalities. For the last equality, note t h a t 3) implies M = F M o and hence McH a. n

3.2.

3.2

T H E C L A S S I F I C A T I O N OF C A U S A L S Y M M E T R I C P A I R S

83

T h e Classification of Causal S y m m e t r i c Pairs

In this section we give a classification of the causal symmetric pairs (9, v). Remark 3.1.9 shows that in order to do that it suffices to classify the noncompactly causal symmetric pairs and then apply c-duality to find the compactly causal symmetric pairs. The same remark also shows that we may assume $ to be a simple Lie algebra. Let g be a noncompact simple Lie algebra with Cartan involution 0 and corresponding Cartan decomposition $ = t + p. As was noted in Remark 1.1.15, to each involution 7 on 9 there exists a Cartan involution 01 on 9 commuting with T. Let 9 = tl + Pl be the Cartan decomposition belonging to 01. According to [44], p. 183, there exists a ~p e Aut($)o such that ~ ( t ) - - ~1 and ~ ( p ) = Pl. But then 0 : ~ - 1 0 0 1 0~0,

and ~-x o r o ~ commutes with 0. Thus, in order to classify the causal symmetric pairs (9, T) up to isomorphism, it suffices to classify those causal involutions on 9 that commute with the fixed Cartan involution 0. P r o p o s i t i o n 3.2.1 Let 9 be a simple Lie algebra with Cartan involution 0 and 7":9 --r 9 be an involution commuting with O. If (9, r) /s irreducible, then the following statements are equivalent: 1) (9, "r) /s noncompactly causal. 2) There exists an X E qp such that

g = gO, x) 9 ~(+1, x) e g(-x, x) and T = 9 n x (cf. Lemma 1.2.1). Pro@ 1) =~ 2) is an immediate consequence of Proposition 3.1.11 and Theorem 3.1.14. For the converse, note that comparing the eigenspaces of a d X and f i x , condition (2) implies that X E ~r and hence X e q~ong for any symmetric A4 = G / H associated to (9, 7). But then Theorem 3.1.3 and Theorem 3.1.5 imply that f14 is noncompactly causal, and this proves the claim, o

P r o p o s i t i o n 3.2.2 Let (g, r) be a noncompactly causal symmetric pair with T = OTix, where X E qp such that

g = g(o,x) ~ g(+1,x) 9 g(-1,x).

84

C H A P T E R 3. I R R E D U C I B L E C A U S A L S Y M M E T R I C SPACES

Further, let a be a maximal abelian subspace of p containing X and A = A(g,a) the corresponding set of restricted roots. Pick a system A+ of positive roots in A such that a ( X ) > 0 .for all a E A+ and denote the corresponding set of simple roots by ~2. Then there exists a unique a x E ~ with a x ( X ) = 1. In particular, a x determines X completely. Pro@ Let Ao = {a E A l a ( X ) = 0} and A + "= Ao Iq A+. Then we have Ao = A({t(0, X), a) = A(0 a, a) and A+ =

e a I

= 1} u Ao+.

Consider the set ~0 C A0 of simple roots for A +. We claim that ]EoC~

and

#(~\IEo)=l.

(3.5)

In fact, let a E A +. Assume that a = f l + 7 with fl,7 E A+. Then a ( X ) = f l ( X ) + 7(X) = 0. As fl(X) > 0 and 7(X) > 0, this implies fl(X) = 7 ( X ) = 0 and hence fl, 7 E A +. Thus we have ~2o C ~. Proposition 3.1.11 shows that R X = 3(I}a) and dim(a t'l [I~~, I}~]) = dim(a) - 1. But Ija = 9(0,X), so dim(a) = # ( E ) and dim(a gl [0a, I~a]) = #(Eo). This proves (3.5) and the claim follows if we let a x be the only root in ]E which is not contained in ]E0. El R e m a r k 3.2.3 Let {I be a noncompact simple Lie algebra with Caftan involution 0. Consider a maximal abelian subspace a of p and A = A(g, a), the corresponding set of restricted roots. Pick a system A+ of positive roots in A and denote the corresponding set of simple roots by ]E. Let J be the highest root of A+ (cf. [44], p. 475) and denote by d(a) the multiplicity of a E A+ in ~. This means that 5 = ~ a e ~ d ( a ) a . Given a E ]E with d(a) = 1, define X ( a ) E a via fl ( X ( a ) ) =

1 for f l = a 0 otherwise.

Suppose that a E ~ and d(a) = 1. We claim that

g = g(0, X(a)) ~ g(-1, X(a)) 9 g(+l, X(a)). In fact, if 7 e A, then 7 = ~ ~ r . m~(7)fl with m~(7) e Z, and 7 ( X ( a ) ) = ma(7). But d ( a ) = ma(6) = 1 and Ima(7)l _< d(a), since J is the highest root. There we have ma(7) = 0,1, or - 1 , and this implies the claim. Now we can apply Proposition 3.2.2 to X ( a ) and obtain a = aX(,~).

3.2.

T H E C L A S S I F I C A T I O N OF C A U S A L S Y M M E T R I C P A I R S

85

We denote the involution on [t obtained from X ( a ) via Proposition 3.2.1 by r ( a ) and its algebra of fixed points by II(a). E! T h e o r e m 3.2.4 Let g be a noncompact simple Lie algebra with Cartan involution O. Let a be a maximal abelian subspace of p and A = A(g, a) the corresponding set of restricted roots. Pick a system A+ of positive roots in A and denote the corresponding set of simple roots by ~2. Then the following statements are equivalent: 1) There exists an involution 7":g --+ it commuting with 0 such that (g, 7") is noncompactly causal. 2) There exists an element X E p, X ~ 0 such that

g = g(-1,X)~

{t(0, X ) ~ g ( + l , X ) .

3) A is a reduced root system and there exists an a E ~ such that the multiplicity d(a) of a in the highest root tf E A is 1. Proof. 1) =~ 2): This follows from Proposition 3.1.11. 2) =~ 1): Given X {5 p as in 2), we apply Lemma 1.2.1 to i X E ip and find that r/x is an involution on g commuting with O. Then 7- := Or/x also commutes with 0 and r ( X ) = - X . Moreover, (g, 7-) is irreducible since It is simple. Thus we can apply Proposition 3.2.1 and conclude that (g, r) is non-compactly causal. 2) =~ 3): Conjugating by an element of K, we may assume that X E a. If A is nonreduced, then [44], Theorem 3.25, p. 475, says that A is of type (be)n, i.e., of the form

a) = •

•

I 1 _< i, j, k < r;i < k}

(cf. also Moore's Theorem A.3.4). But this contradicts the fact that

spec(ad X) = {- i,O, 1 }. Hence A is reduced. Now let a x ~_ ~, be the element determined by Proposition 3.2.2. Then a x ( X ) = 1 and ~(X) = d ( a x ) . As t~(X) E { - 1 , 0 , 1 } , it follows that d ( a x ) = 1. Thus 3) follows. 3) =~ 2): This follows from Remark 3.2.3. El R e m a r k 3.2.5 The maximal abelian subspaces of p are conjugate under K and the positive systems for A = A(g, a) are conjugate under the normalizer NK(a) of a in K. Therefore Theorem 3.2.4, Propositions 3.2.1 and 3.2.2,

86

C H A P T E R 3. IRREDUCIBLE CAUSAL S Y M M E T R I C SPACES

and Remark 3.2.3 show that the following procedure gives a complete list of representatives (g,r) for the isomorphy classes of noncompactly causal symmetric spaces. Some of the symmetric pairs obtained willbe isomorphic due to outer automorphisms of the relevant diagrams. Step 1: List the simple noncornpact real Lie algebras g together with the Dynkin diagrams of the restrictedroot systems and the multipliciesof the highest root ~. Step 2: Given a simple root c~ whose multiplicityin ~ is 1, construct the s y m m e t r i c pair (g, 7(a)). Note t h a t after removing a from the set E of simple roots one obtains the Dynkin diagram for the restricted roots of the c o m m u t a t o r algebra of l}(a) ~ and the corresponding set of simple roots is E0 = YI. \ {a}. If one wants to read of Ij(a) a and I}(a) from diagrams directly, one has to use the full Satake diagram instead of the Dynkin diagram of the restricted root system, t3

E x a m p l e 3.2.6 W e show how the procedure of Remark 3.2.5 works in the case that g has a complex structure. T h e type A,: (s[(n+ 1,C)) 1

1

1

1

Here d(ak) = 1 for all k = 1 , . . . , n. Thus k can be any n u m b e r between 1 and n. Furthermore, Eo = E(ak-1) x Z ( a , - k - 1 ) . In particular,

gc(O, xk)

=

st(k, c) 9 5t(~ - k, C) 9 CXk

~_ su(k)c 9 sa(n - k)c 9 C Xk. Hence I~ = su(k, n - k).

T h e type B,: (so(2n + 1,C)) 1

~1

2

~2. . . .

2

2

"~n--1

In this case k = 1. Then Eo = E ( b . - 1 ) and

g(O, X;)

=

,o(2n - 1, C) @ CX; so(2n - 1)c @ CX1.

Thus IJ - so(2, 2n - I).

3.2. THE CLASSIFICATION OF CAUSAL S Y M M E T R I C P A I R S The type C.:

87

(sp(n, C))

2

2

2

1

Hence k = n, Eo = E ( a n - 1 ) a n d s t ( n , C) ~ R x,,

=

,-, su(n)c (9 R Xr,. Thus tj = sp (n, R). T h e t y p e D n : (so(2n, C)) 1

2

2, art--2 1

O~n--1

Here we can take k = 1, n - 1, n. B u t k = n - 1 and k = n give isomorphic O's. Thus we only have to look at k = 1 and k = n . For k = 1 we get E0 of t y p e ~)n--1 a n d

g(O, Xx)

=

so(2.-

~_ s o ( 2 n -

2,c) e c x , 2)c ~B CX1.

Thus 0 = so(2,2 n - 2). For k -- n we get Eo -- E ( a n - 1 ) and

g(o,x.)

=

9 cx. tiC.

T h u s tJ = so*(2n). T h e t y p e E6:

1

2

2, a2 2

1

~4 Here k = 1, 6. As b o t h give isomorphic ~, we m a y assume t h a t k = 1. T h e n ]Eo = ]E(~5). Thus g(O,X,)

=

ao(lO, C) ~ C X , $0(10)C ~ CX1

9

88

CHAPTER 3. IRREDUCIBLE CAUSAL SYMMETRIC SPACES

Hence [}

=r

T h e t y p e E~: 1

2

3

2, a

2

3

2

I~~4 Thus k = 1, Eo is of type t8 and g(0, XT) = te ~ C X7. Hence b o

= t7(-25)-

E x a m p l e 3.2.7 To conclude this chapter we work out the real groups of type AI, AII, and AIII, i.e., S L ( n , R ) , SV*(2n), and SU(p, q). We fix a maximal abelian subalgebra a of p and denote a set of positive roots in A = A(g,a) by A+. Let E = {c~;,... ,c~r} be the set of simple restricted roots. S L ( n , R ) : . In this case E is of type A . - 1 , so we can take out any or/. It follows t h a t I~" is of type Ap-1 x Ar with p + q = n. In particular, 9(0,Xk) = a[(p,R) • s[(q,R) x R X k . Thus Ok = so(p) x so(q). There are thus two possibilities for lj. Either = g(0,Xk) or Ij = so(p, q). We can exclude the first case, as ~ ,,~ ~" is possible only for n = 2, in which case b o t h a and a" are one-dimensional and abelian. SU*(2n): In this case ~ = E(an-1) and m x = dimgx = 4 for every A e I~. Once again we can take out any one of the simple roots. It follows t h a t I~o = E(ap_l) • E(aq-1) and multiplicities equal 4. It follows t h a t g(0, Xk) = su*(2p) • su*(2q) • RXk. Thus tk = sp(p) • sp(q). We can exclude t h a t l} "~ ~", therefore lj = sp(p, q). SU(p, q): The root system A is nonreduced if p # q. So the only possibility is p = q. In that case I: = E ( c , ) and the multiplicities are m~,~ = 2, j < n and m~,. = 1. The only possibility is to take out 7n, and we are left with ~(a,-1) and all multipicities equal 2. But then g(O,X,)= s[(n,C) x RXn. This leaves us with ~ = s[(n, C) x R except in the case n = 8. In t h a t case - r would be another possibility. But ad Xk is an isomorphism ~p qk, which shows that d i m H / ( H N g ) = d i m K / ( g N H ) . The dimension of e7(7)/su(8) is 70, which is bigger t h a n dim u . = s ( u . x u n ) / s . , o

3.2. THE CLASSIFICATION OF CAUSAL SYMMETRIC PAIRS

89

The procedure described in Remark 3.2.5 yields the following theorem. T h e o r e m 3.2.8 (The Causal S y m m e t r i c Pairs) The irreducible semisimple causal symmetric spaces are given up to covering by the following symmetric pairs. g with Complex Structure

o=~c

oc=~x~

noncompactly causal

compactly causal

s[(p + q, C) so(2n, C)

su(p, q) • su(p, q) so*(2n) • ,o*(2n)

so(. + 2. c) sp(.. c)

~0(2..) • 50(2..) ~p(.. R) • sp(.. R)

r r

r r

X t6(_14 ) X e7(_25 )

su(p, q) so*(2n) so(2, n)

5p(~.nr t6(--14) t7(--25)

g without Complex Structure g gc noncompactly causal compactly causal s[(p + q, R)

su(p, q)

so(p, q)

..(.,.)

..(.,.)

.t(.. c) • R

su*(2(p + q))

su(2p, 2q)

,p(p, q)

so(...)

so*(2.)

so(..c)

s0*(4n) so(V + 1, q + 1) sp(n,R)

s0*(4n) so(2, p + q) sp(n,R)

su*(2n) x R so(p, 1) x so(l, q) s[(n,R) • lit

~p(...)

~p(2.. R)

sp(.. c)

r r r C7(7)

r r 14) C7(--25) r

sp(2, 2) ~4(-20) t6(_26 ) X R SU* (8)

Notes for Chapter 3 Most of the material in Section 3.1 appeared for the first time in [129, 130]. Definition 3.1.16 is due to E. van den Ban [1]. Theorem 3.1.18 can be found in [131, 136]. The idea of using a simple root with d(a) = 1 for classifying the causal symmetric spaces was pointed out to us by S. Sahi, cf. [85]. There are other ways

90

C H A P T E R 3. I R R E D U C I B L E CAUSAL S Y M M E T R I C SPACES

of classification. In [129] this was done by reducing it, as in Theorem 3.2.4, to the classification of para-Hermitian symmetric pairs. The para-Hermitian and para-K~ihler spaces were introduced by Libermann in 1951-1952 [95], [96]. The para-Hermitian symmetric spaces were classified by S. Kaneyuki [75] by reducing it to the classification of graded Lie algebra of the first kind by S. Kobayashi and T. Nagano in [80] (see [77] for the general classification). A list may be found in [78], [111], and also [138]. Another method is to use Lemma 1.3.8 to reduce it to the classification of real forms of bounded symmetric domains. This was done by H. Jaffee using homological methods in the years 1975 [69] and 1978 [70]. Compactly causal symmetric spaces were also introduced by Matsumoto [103] via the root structure. These spaces were classified the same year by Doi in [21]. Later, B. Orsted and one of the authors introduced the symmetric spaces of Hermitian type in [133] with applications to representation theory in mind. The connection with causal spaces was pointed out in [129, 130].

Chapter 4

C l a s s i f i c a t i o n of Invariant Cones Let A4 = G/H be causal symmetric space with H essentially connected. In this chapter we classify the H-invariant regular cones in q, i.e., all possible causal structures on A4. Because of c-duality, we can restrict ourselves to noncompactly causal symmetric spaces. The crucial observation is t h a t regular H-invariant cones in q are completely determined by their intersections with a suitable C a r t a n subspace a. We give a complete description of the cones in a which occur in this way. Further, we show how H-invariant cones can be reconstructed from the intersection with an a. An important fact in this context is t h a t the intersection of a cone with a is the same as the orthogonal projection onto a. In order to prove this, one needs a convexity theorem saying t h a t for X in an appropriate maximal cone Cmax in a, p r ( A d ( h ) X ) e conv(W0.X) +Cmax. We also prove an extension theorem for H-invariant cones saying t h a t these cones are all traces of GC-invariant cones in ig c. An i m p o r t a n t basic tool in this chapter is s[(2, R) reduction, which is compatible with the involution T.

4.1

Symmetric SL(2, R) Reduction

Let A4 = G/H be a noncompactly causal symmetric space with involution T. In this section we describe a version of the usual SL(2, R) reduction t h a t commutes with the involutions on G and SL(2,R). Recall the decomposition A = A_ U A0 U A+ associated to the choice of a cone-generating element X ~ E qp and the corresponding nilpotent 91

92

C H A P T E R 4. CLASSIFICATION OF I N V A R I A N T CONES

subalgebras n• from Remark 3.1.17. L e m m a 4.1.1 Let (g, 7") be a noncompactly causal symmetric pair. Then we have 7 " = - 0 on n+ $ n_.

Proof:. From g(0, X ~ = IJk ~ qp and ad(X ~ (qk 9 Ijp) C qk 9iJp we obtain n+ @ n_ = qk @ Op,

(4.1)

and this implies the claim.

C]

Recall that a C qp is maximal abelian in p. Therefore the Killing form and the inner product (. I') := (" I')0 agree on a. We use this inner product to identify a and a*. This means that

B ( X , A ) = ( X I A ) = A(X) for all X E a and A E a*. For A ~t 0 we set XX._

A

.-lal=

e.,

where l" ] denotes the norm corresponding to ( - I ' ) . :~(x ~) = 1.

(4.2)

We obviously have

L e m m a 4.1.2 Let a ~. A and X ~_ g~. Then [X, 0(X)] =

-Ixl~.

Proof:. Note first that 0(X) fi g_~. Hence [X, 0(X)] E ~,(a)Iq p = ,. Let Y E a. Then

B(Y,[X,O(X)])

=

B([Y,X],O(X))

=

a(Y)B(X,O(X))

= =

-Ixl2a(Y) B(Y,-lxl2a), E!

and the claim follows. Let a E A+ and choose Ya E ga such that 2 IY~I2= la12.

(4.3)

We set Y_= "- r(Y~). By Lemma 4.1.1 we have Y_= = T(Y,~)=-O(Ya). Thus Lemma 4.1.2 implies

[Y., Y_.] = 2 x -.

(4.4)

4.1. SYMMETRIC SL(2, It) REDUCTION

93

Finally, we introduce Y':' := ~1 (y~ + y_~) E l}p,

(4.5)

Z~, . - 1~ ( Y - ~

(4.6)

- Y~) e , k ,

and X•

:= X ~ 4 - Z a E q.

(4.7)

E x a m p l e 4.1.3 Let ~ = S L ( 2 , R ) / S O o ( 1 , 1 ) . We use the notation from Example 1.3.12. Then a := qp = R X ~ is abelian and the corresponding roots are A = {cz,--c~}, where a ( X ~ = 1. We choose a to be the positive root. As root spaces we obtain g~=g(+l,X

~

and

g_a=g(-1,X

~

The Killing form on s[(2, R ) i s given by B(X, Y) = 4tr(XY). In particular, we find I z X ~ + zZ~ = = 2(x 2 + z =)

and Ig+l = = 4,

I x ~ = = 2.

This shows t h a t a = 89 ~ and la] 2 = 89 Now we obtain Y•

= Y•

X•

= X:t:,

and

X ~=x

o,

y~=yO,

Z ~=Z ~

o

R e m a r k 4.1.4 Note that we can rescale the inner product without changing X a. Also, the norm condition on Ya is invariant under rescaling. On the other hand, the construction of y a , Z ~ and X • depends on the choice of Y~ (recall t h a t in general dim(g~) > 1). D Define a linear map ~a : s[(2, R) --+ g by

5

0-

~X~'

0

0

~Y~

and

(00) 1

0

~Y-~"

(4.8)

Then ~a is a Lie algebra monomorphism such t h a t 0 induces the usual Cartan involution X ~ - ~ X on 5[(2,R), whereas r induces the involution on

C H A P T E R 4. C L A S S I F I C A T I O N OF I N V A R I A N T C O N E S

94

s[(2,R) described in Example 1.3.12 (denoted also by ~" there). In particular, sa := Im ~o~ is r and 0-stable. As a consequence, we have ~o~(Y~ = Ya and ~oa(Z ~ = Z ~. Of course, ~oa exponentiates to a homomorphism of SL(2, R), the universal covering group of SL(2, R), into G. Similarly, ~oa defines a homomorphism ~oa : SL(2, C) ~ Go, since SL(2, C) is simply connected. The notation in Example 1.3.12 was set up in such a way that X ~ E s[(2,R) plays the role in the noncompactly causal space (~[(2,R), v) that it should play in Proposition 3.1.11. Unfortunately, this property is not carried over by the maps ~pa just constructed. In other words, X a is in general not a cone-generating element. All we have is the following remark: R e m a r k 4.1.5 Let X ~ e }([}a) with a ( X ~ = 1. Then x ~ - x ~ e ~

= {x

e ~ I~(X)

= 0}.

[-i

T h e o r e m 4.1.6 Let A4 = G / H be a noncompactly causal symmetric space and C E ConeH(q) the cone defining the causal structure. 1) There exists a unique cone-generating element X ~ E C N qp _Hng C

s(~o). 2) Let a be a maximal abelian subspace of qp and A = A(g, a) the corresponding set of restricted roots. Then for all a E A+ = {fl E A I /~(X ~ = 1} and for any choice of Ya e ga satisfying (4.3), we have X ~ , X ~ , X - ~ e C. ..HNK Proo~ 1) According to Corollary 3.1.6, we can find an element of X E 4p in the interior C ~ of C. But then Proposition 3.1.11 shows that a multiple of X satisfies the conditions of 1), since q~nK is one-dimensional and contains

s(~o). 2) Note first that [X ~ - X ~, ya] = 0 for all c~ E A+ since Y~ E g~ + g-~. Hence by 8[(2, R)-reduction we have Ad(exp t Y ~ ) X ~ =

=

Ad(exp t Y ~) [~ (x~ + x _ ~ ) + ( x ~ - x~)]

! (~'x~ + ~-'x_~) + ( x ~ - x~). 2

Thus 2 lim e -t A d ( e x p t Y a ) X ~ = X a E C $-+oo

and 2 lim e t Ad(exp t Y a ) X ~ = X _ a e C . t-+--oo

(4.9)

4.1. S Y M M E T R I C SL(2, It) R E D U C T I O N

95

As 2X a = Xa + X - a , the lemma follows,

ra

Let (g, r) be a noncompactly causal symmetric pair. We choose a cone generating element X ~ E a C qp. According to Remark 3.1.9, (go, 7") is compactly causal and either simple Hermitian or the product of a simple Hermitian algebra with itself. In either case we can choose a Cartan subalgebra t c of tic containing ia and contained in ~c = [~k + iqn- Note that Z c := i X ~ e ~(~c) and the centralizer of Z c in i~c is ~2 (cf. Proposition 3.1.12). Let (pc)• be the =i:i-eigenspaces of ad Z c in p~ = (0p + iqk)c. Then

( ~ ) ~ ~ ~ = n~.

(4.~0)

In addition to the notation from Remark 3.1.17, we use the following abbreviations /~ := A(gc, t~),

~,• := A((pC) • t~),

and

/~o "= A(t~, t~).

(4.11)

Then we obtain

and

ao := {alo l a e Ao,alo # o}. Moreover, we can choose a positive system /~+ for /~ such that A + "= A + f'l Ao is a positive system in Ao and

~o+ = {~1. l a e ho+,~1~ # 0}

and ~+ = {~l~ I~ e h +,alo # 0}. (4.12)

L e m m a 4.1.7 Let & E A+ be such that - v & ~ &. Then & and - v & are strongly orthoyonal. Proo~ Let & E A+ be such that - z ~ # ~. Then -~'~ E A+ and & - r ~ is not a root. Assume that 7 := & + v& is a root. Since "YIo = 0, we have (gc)-r C ~gc(a) C ac ~ (IJk)c. It follows that (gc)-y C ~c N lJc because (gc)~

is ~-inv~ri~nt. Let X ~ (gc)~, X # 0 . As ~ is a C~rta~ sub~lgebra of gc, it follow~ that dimc(gc)~ = 1 and 0 # IX, ~(X)] e (gc)~. But then Ix, ~-(x)] ~ qc gives a contradiction,

o

R e m a r k 4 . 1 . 8 The ~--invariance of t e shows that also A is invariant. Let ~c = ~0 be the complex conjugation of ~c w.r.t, ge. Then t~ is cre invariant. Since the elements of A take real values on t e, it follows that ae~ = - ~ for all ~ e A. Therefore we can c h o o s e / ~ e (gc)~ such that

/~_~ = / ~ o ~ = ~r~

and / ~

= r/~

96

C H A P T E R 4. C L A S S I F I C A T I O N OF I N V A R I A N T C O N E S

for all & E /X. Moreover, the normalization can be chosen such that the element = IBm, E- l = E- l = satisfies & ( / t ~ ) = 2 (cf. Appendix A.3). Fix a maximal set - {71,..., 7r,} C/X+

(4.13)

of strongly orthogon~ roots (cf. Appendix A.3). In fact, Lemma 4.1.7 shows that we may assume F to be invariant by - 7 simply by adding --7(~k) after each inductive step. Recall from (A.27) and Lemma A.3.3 that the space

where 17"$ = - i ( / ~ - / ~ _ $ ) , is maximal abelian in pC. Note that /X+ is invariant under - r and renormalize Y~" Y~ "= r~I?~ e It~,

(4.14)

where 7 = 71,. Here we choose r$ in such a way that Y$ satisfies the condition (4.3), i.e., 2 ,-~

= l~,121i,. 12

Now we see that Y~ := ~1 (y# + rye) e l}p fl a~ and

r'] ~re~ L e m m a 4.1.9 The space a~ is maximal abelian in Ijp. Pro@. Let Gc be a simply connected Lie group with Lie algebra gc and G, G c, H, K, K c, A c, A~,, etc., the analytic subgroups of Gc corresponding to g, gc, Ij, t, tc, ac, a~, etc. Recall the Cartan decomposition KCACK c = G c from [44], p. 402. The AC-component is unique up to a conjugation by a Weyl group element. Let a: Gc -+ Gc be the complex conjugation with fixed point set G. Then G c, K c and A c are a-invariant. Thus we have

(G I"1 KC)(G fl AC)(G I"1 K c) = G I"1 G ~.

4.1. S Y M M E T R I C SL(2, It) RED UCTION

97

The Lie algebras of G N G e, G N K c, and G N A c are [~, [}k, and a~. Since a~ is an abel/an subspace of [}p and [}k is maximal compact in I}, this shows that ( g N H ) A ~ ( K N H) = H (4.15) is a Cartan decomposition and hence a~ is maximal abel/an in [}p.

D

R e m a r k 4.1.10 We set r .=

e

(4.16)

c a+.

View ~ as an element of (itc)" and write ~ = 7 + 7' with 7 --- 71o and 7 ~ - ~l(/tc)n~. Then, under the identification of dual spaces via (. I'), the restriction means orthogonal projection to the respective space. Note that --T~ -- 7 - 7 ~ E F, so the orthogonality of the elements of r implies also that their restrictions to a are orthogonal. r actually consists of strongly orthogonal roots. To see this, suppose that 7i = 7/Io and 7j = 7j ]o with 7 i - T j E A. Since 7i and 7 / a r e orthogonal we have sTj ( 7 i - 7 j ) = 7i+7j, where sTj is the reflection in a* at the hyperplane orthogonal to 7i- But sTj is an element of the Weyl group of A and hence leaves A invariant. Therefore we have 7i q- 7j E ~, a contradiction. Suppose that 7o E A+ is strongly orthogonal to all 7 E r . Then there exists a 7o E A+ such that 70 = 7o[o and 0 ~ 7 j - 70 E A whenever ~j - % E A. Therefore % is strongly orthogonal to all ~j, in contradiction to the maximality of r . Thus r is a maximal set of strongly orthogonal roots in A+. Note that the orthogonality of the elements of r together with 7 = I(~--T~) shows that the only elements o f f restricting to a given 7 = 7]o E r 2 on a are ~ and - r ~ . The definition of a~, shows that each element Y E a~ can be written in the form Y = E r , Y" (4.17) 7EF

constructed via (4.5) from pairwise commuting elements Ya E fla satisfying the normalization condition (4.3). Moreover, the images of the corresponding 8[(2, R)-embeddings ~pa commute, since the images of the 8[(2, C) embeddings corresponding to the different elements of G (cf. Appendix A.3) commute. [::! L e m m a 4.1.11

2) I] Y = ~ E r

1) L := X ~ - ~ e r

X~ E ~ e r

ker 7.

t'rY7 E a~ is chosen as in Remark 4.1.10, then we have ~= L +

cosh(t,lX, + 7Er

i 7EF

h(t,lZ'.

98

CHAPTER 4. CLASSIFICATION OF INVARIANT CONES

Proof:. 1) Fix 7 E F and note t h a t 7 ( X ~ - X ~) = 0 because of the normalization of X ~. On the other hand, 7 ( X ~) = 0 for all 7 # fl E F by R e m a r k 4.1.10. This proves 1). 2) Using 5[(2, R) reduction we calculate ead YX 0 7EF

=

L+ Z

[e'" (X" + Z ' ) + e -'~ (X" - Z')]

7EF as X • 7 = X ~ 4- Z ~. From this the claim now follows.

4.2

The Minimal and Maximal Cones

In this section we study certain convex cones which will turn out to be minimal and maximal H-invariant cones in q, respectively their intersections with a. D e f i n i t i o n 4.2.1 Let A~ = G / H be a noncompactly causal symmetric space and (g, r) the corresponding symmetric pair. Further, let X ~ E qp be a cone-generating element. Then the closed convex cones

Cmin(X O) "----Cmin "-- conv [Ad(Ho)(R+X~

(4.18)

and

Cm~x(X~ "= Cm~x := { x ~ q lVY ~ Cmin" B(X, Y) > O}

(4.19)

in q are called the minimal and the maximal cone in q determined by the choice of X ~ A reference to the space fl~ = G / H is not necessary, since the definitions depend only on the group generated by e ~d ~ in GL(q). El D e f i n i t i o n 4.2.2 Let (g, r) be a noncompactly causal symmetric pair and a C qp a maximal abelian subspace. Choose a cone-generating element X ~ E a and recall the set A+ of restricted roots taking the value 1 on X ~ Then the closed convex cones

Cmin(X O) := Cmin := aEA+

Ro

~

aEA+

Ro

(4.20)

4.2. T H E M I N I M A L A N D M A X I M A L CONES

99

and Cmax(X 0) 1-- Cmax -'--~ t x e I1 I V'~ e

A+ : ,~(x) >__o} - C'in

(4.21)

in a are called the minimal and the maximal cone in a determined by the choice of X ~ E! It follows from Proposition 3.1.11 that there are only two minimal and maximal cones. R e m a r k 4.2.3 It follows from the definition of Cmin that 0(Cmin) -- -Cmin. This shows that we can replace the definition of Cmax by Cm~x = {X ~ q IVY e

Cmin

:

(X I Y) _> 0}.

L e m m a 4.2.4 Let .M = G / H is a noncompactly causal symmetric space. Then Cmin /8 minimal in ConeH(q) and Cmax i8 maximal in ConeH(q). Proofi. Note first that, by duality via the Killing form, we only have to show the assertion concerning Cmin. Theorem 4.1.6 implies that X ~ is (HMK)-invariant, so that Cmin is indeed H-invariant by Lemma 3.1.1. Since any element of ConeH (q) contains either X ~ or - X ~ again by Theorem 4.1.6, it only remains to show that Cmin is regular. If q is [}-irreducible this follows from Theorem 2.1.21. If q is not l} irreducible, then the only elements in ConeH(q) intersecting p nontrivially are =t=Cp (cf. Section 2.6). But the contruction of C n shows that it contains X ~ This implies Cp = Cmin and hence the claim. D

The following proposition is an immediate consequence of Theorem 4.1.6. P r o p o s i t i o n 4.2.5 Let .M = G / H be a noncompactly causal symmetric space and (g, T) the corresponding symmetric pair. Suppose that the causal structure is given by C E ConeH(q) and let X ~ be the unique cone generating in C. Then Cmin(X 0) C C .

i-1

L e m m a 4.2.6 Assume that .M is a noncompactly causal symmetric space. Let a be maximal abelian in qp. Let Ao = A(l;a,a) and Wo be the Weyl group of the root system Ao. Then

Wo NHNK(fl)/ZHNK(fl). =

100

C H A P T E R 4. C L A S S I F I C A T I O N OF I N V A R I A N T CONES

Proof. Recall from [44], p. 289, that W0 = NHonK(a)/ZHonK(a). But Theorem 3.1.18 implies that H N K = ZunK(a)(Ho gl K) and that proves the lemma, o

P r o p o s i t i o n 4.2.7 Cmin and Cm&x are regular Wo-invariant cones in a. Furthermore, Cmin (Z Cmax and = IX e a lW e

:

> 0}.

Proof. As the Weyl group W0 fixes X ~ it follows that W0 permutes A+. We have w ( X ~) = X wa and hence Cmin is W0-invariant. By duality also Cmax is W0-invariant. If c~,/~ E A+, then a ( X ~) = (a ] ~ ) / ( ~ ] ~) >_ 0 for otherwise ~ + would be a root. Hence Cmin (= Cmin. The equality C~ = {X E a i r a E A+ : a ( X ) > 0} is an immediate consequence of the definitions. It shows that Cmax is generating and hence that Cmin is proper (cf. Lemma 2.1.3 and Lemma 2.1.4). It only remains to show that Cmin is generating. If < Cmin > # ft, then we can find a nonzero e l e m e n t X E a w i t h a ( X ) = 0 f o r a l l a E A+. Thus i X E ia C ge centralizes (n+ @ n - ) c = (qk (9 ~p)c = P~, which is absurd. [::]

L e m m a 4.2.8 X ~ E Cmin(X~ ~ Proot. Let X E Cmin(X~ ~ and define 2 := [1/#Wo] EweWo w - X . Then .~ # 0 is Wo-invariant and contained in Cmin(X~ ~ Let c~ E Ao and s~ E Wo the reflection at the hyperplane orthogonal to c~. Then =

2

i.e., a ( X ) = 0 for all c~ E Ao. Therefore 2 E ~(I}a) and hence X is a multiple of X ~ On the other hand, X ~ E Cm~(X~ ~ so that Cmin(X~ ~ C_ Cm~x(XO) ~ implies the claim. [::] For later use we record an application a convexity theorem due to Kostant (cf. [45], p. 473) to the Lie algebra I}a. P r o p o s i t i o n 4.2.9 Let M = G / H be a noncompactly causal symmetric space and pr: qp ~ a the orthogonal projection. Then for X E a we have pr(Ad(K NHo)X) = conv(Wo. X ) . P r o p o s i t i o n 4.2.10 Let .s = G / H be a noncompactly causal symmetric space. Choose a maximal abelian subspace a of qp and a cone-generating element X ~ E a. Denote the orthogonal projection q -+ a by pr. Then

4.2. T H E M I N I M A L A N D M A X I M A L CONES

1)

101

pr(Cmin) C Cmin.

2) pr(Cmax) C CmaxProo~ 1) Let X = Ad(h)X ~ for some h e Ho. The Cartan decomposition (4.15) Ho = (Ho N K)A~(Ho N K ) shows that we can write h = k(exp Y ) k l with k, kl E K N Ho and Y E a~. We can write Y = ~ T e r t'yY'r as in Remark 4.1.10 and then Lemma 4.1.11 shows that pr(Ad(h)X ~

=

pr(Ad(k)eadVx ~

=

It follows that pr(Ad(h)X ~ E C~ as p r ( A d ( k ) X ' ) e cony Wo. X ~ C Cmin by Proposition 4.2.9. Since Cmin is the closed convex cone generated by Ad(Ho), the claim follows. 2) Let Y E Cmin. By Lemma 4.2.5 we have Y E Cmin. Thus (Y[ pr(X)) = (Y[X) > 0, which implies that pr(X) e Cmin ~- Cmax. ["1 Recall the intersection and projection operations from Section 2.1. We set I := I," and P := P2. P r o p o s i t i o n 4.2.11 Let .h4 = G / H be a noncompactly causal symmetric _Hng . Then space with cone-generating element X ~ E qv

1) Cmin "- I ( C m i n ) - P(Cmin). 2) Cmax -- / ( C m a x ) =

P(Cmax).

Proo~ 1) According to Proposition 4.2.5 we have Cmin C I(Cmin) C P(Cmin). Proposition 4.2.10 now shows that P(C'min)

=

train.

102

C H A P T E R 4. CLASSIFICATION OF I N V A R I A N T CONES

2) Lemma 2.1.8 and part 1) show I(Cmax) = P(Cm.x)* = P(Cmin)* = c'in = Cm&x and hence the claim again follows from Proposition 4.2.10. D Recall that for any noncompactly causal symmetric pair (g, r), the symmetric pair (gc, he), where ac: gc --+ gc is the complex conjugation w.r.t. go, is either noncompactly causal (if g carries no complex structure) or the direct sum of two isomorphic noncompactly causal pairs. In particular, any symmetric space G c / G c associated to (gc, a c) admits a causal structure. We assume for the moment that g carries no complex structure. Then any cone-generating element for (g, 7") is automatically a cone-generating element for (gc, ac). Fix a Cartan subalgebra t e of t e containing a, which then is also a Cartan subalgebra of {~e. Further, we choose a cone-generating element X ~ E qp C i{}k + qp and a positive system/~+ for/~ = A(gc, t c) as in (4.12). The corresponding minimal and maximal cones in the maximal abelian subspace it c of i[}k + qp are then given by Cmin =

E

(4.22)

R0+ a

aes and

Z'm~x = {X E it e IV6 E ,'x+: &(X) > O} = E'min. ,,,,

P r o p o s i t i o n 4.2.12 The Moreover, 1) I/'r

cones

(4.23)

,,,,

train(A+) and cmax(A+)

are--r-invariant.

P/t'(cmin)= Cmin;

~) Iaire (C'm&x) -" P~ t~(Cmax) = Cmax.

Proof:. The -v-invariance follows from the -v-invariance of A+. In view of duality and Lemma 2.1.9, it only remains to show that P/tr = Cmin. But that is clear, since P_dt" (a) = &].. [::] Lemma

4.2.13 Let X E C~

. Then

I) 2) Let h E H be such that Ad(h)X E a. Then h E K fl H. Proof. 1) Let Y = ~ a e a + [ L a have

[x, Y] = ~

aEa+

o r(La)] with La E g~. As X e Cma x we

a(X)(L. + r(La)) # 0

4.2. T H E M I N I M A L A N D M A X I M A L CONES

103

which implies the claim. 2) Let Y E a. Then

[ A d ( h - 1 ) y , x ] = Ad(h-1)[Y, Ad(h)X] = O. Thus Ad(h - 1 ) Y e [~Nq = qp. In particular, Ad(h-X)a is a maximal abelian subalgebra of qp. Thus there is a k E K N H such t h a t A d ( k h - 1 ) a = a. This implies t h a t kh -1 ~. NH(a) C K r Thus h E K N H as claimed,

ra

Let G c / G e be any symmetric space with corresponding symmetric pair (go, a*). Then the maximal and the minimal cones in ill c are given by

0min(X 0) -" 0rain ~-~COnY[Ad(a,~)(lR+X~

(4.24)

and

Om~x(X ~ = Om~x = {X e q l V r e emin "B(X, Y) > 0}.

(4.25)

R e m a r k 4.2.14 In order to be able to treat the cases of complex and noncomplex g simultaneously, we make the following definitions. Suppose that g = [c and r the corresponding complex conjugation, so t h a t ge = [ x [ and r c is the switch of factors. More precisely, gc = [c x [c with the opposite complex structure on the second factor and the embedding g = tc ~ X ~+ (X, r ( X ) ) ~ tc x tc = go.

The involution ar ge -+ ge is given by

a c ( x , Y) = ( r ( X ) , r(Y)). The algebra [ is Hermitian and we can choose the maximal abelian subspace of qp to be a = it, where t is a Cartan subalgebra of [ N t. Then ff := t • t is a Cartan subalgebra of {tc. Choose a cone-generating element X ~ ~- (X ~ r(X~ = (X ~ - X ~ ~ a. We set/X "= A([C • [C, tC • iC) and

s

:= {(~,/~) e ~ x ~ l ~ ( X ~ = 1 = / ~ ( - x ~

= ~ + • ~_,

where A = A([c, it) and A+ = {a E A [ a ( X ~ = 1}. Now the formulas (4.22) and (4.23) make sense and yield

C-inin(X0, - X 0) --Cmin(X 0) • Cmin('-X 0) C ~{:c and ~'max(X ~ - X ~ ~-Cm&x(X ~ x Cmax("-X ~ C it c.

104

C H A P T E R 4. C L A S S I F I C A T I O N OF I N V A R I A N T C O N E S

Note that both cones are - r - i n v a r i a n t . It follows directly from these definitions and the embedding of q = i[ --} i[ x i[ = ig e that

/'-/re(Cmin) "- Cmin,

I/t" (Cmax) ~- Cmax.

In particular, we see that the conclusions of Proposition 4.2.12 stay valid. Let Lc be a simply connected complex Lie group with Lie algebra fc and L the analytic subgroup of Lc with Lie algebra [. Then the involution a ~ integrates to an involution of Gc = Lc x Lc, again denoted by a ~, whose group of fixed points is G ~ := L x L. Now also the equations (4.24) and (4.25) for the minimal and maximal cones in ig e make sense and yield

Cmin(X~

0) --" Cmin(X O) X Cmin(iX 0)

and Cmax(X~

~ ~-~ Cm.x(X ~ • Cmax(-X~

Note t h a t both cones are invariant under - r and qi ' e

(Groin) "~ Cfmin

Tise (Cmsx) = Cmax.

[[]

L e m m a 4.2.15 Let .M = G / H be a noncompactly causal symmetric space and X E C~ 9 1) X is semisimple and ad X has real eigenvalues.

2) If G c / G ~ is any symmetric space corresponding to ""0 exists a g E Geo such that Ad(g)X E Cmax.

(gc, a~),

then there

3) If X E q, then there exists an h E Ho such that A d ( h ) X E Cmax. Proof:. By Theorem 2.1.13 the centralizer of X in G~ is compact. Since every compact subgroup in Geo is conjugate to one contained in Koe, we may assume that Z a i ( X ) C K e. But then 3 , , ( i X ) C ~c. As i X e 3 , , ( i X ) , it follows that i X is semisimple with purely imaginary eigenvalues. Hence X is semisimple with purely real eigenvalues. Note that ah @ ia is a Caftan subalgebra of {~e, so there is a k E Koe such that A d ( k ) ( i X ) E ah @ ia. This proves 1) and 2). Now assume that X E q. According to Theorem 1.4.1, we can find an h E Ho and a 0-stable A-subspace b = bk ~B bv in q such that A d ( h ) X E b and bv C a. Proposition 4.2.11 implies that prit,(Ad(h)X) E

~O

Cmn x.

Then Proposition 4.2.12 shows t h a t p r , ( A d ( h ) X ) E c~ and hence L e m m a 4.2.13 implies that bk = {0}. Thus A d ( h ) X is actually contained in C~ [3

4.3. T H E L I N E A R C O N V E X I T Y T H E O R E M

105

Theorem 4.2.16 (Extension of Cmin and Cmax) Let~ .M = G / H be a noncompactly causal symmetric space. Then Cmin and Cmax are -r-stable and satisfy [i'~ ---~ C m a x [/lle -q -q (Cmin) = Cmin, Proof:. We may assume that g carries no complex structure, since the case of complex g was already treated in Remark 4.2.14. Let G c / G e be any symmetric space corresponding to (go, ae) 9 Then G c / G c is noncompactly causal. Let H c be the analytic subgroup of Gc with Lie algebra Ij and X ~ the cone-generating element in Cmin. Then H ~ C G r and Cmin C C'min, since they are the He-invariant, respectively GC-invariant, cones generated by X ~ But then Cmin

C / i-q 'e(Cmin)

C

~oilte(Cmin) 9 -q

By Lemma 4.2.15 we can find an h E H e such Let now X E I~~9 (Cmin). ~o that A d ( h ) X E a. But then, by Proposition 4.2.11, and Proposition 4.2.12:

Ad(h)X E a

n

Cmin

C

ll n

(iff

n

Cmin)

--"

11n Cmin --'-- Cmin.

Consequently, the HC-invariance of Cmin proves /~'C(Cmin) ----- Cmin. By duality we get p~ge (Cmax) "- Cmax. Now Lemma 2.1.9 implies the claim. E!

4.3

T h e Linear Convexity T h e o r e m

This section is devoted to the proof of the following convexity theorem, which generalizes the convexity theorem of Paneitz [147] and which is an important technical tool in the study of H-invariant cones in q. T h e o r e m 4.3.1 ( T h e L i n e a r C o n v e x i t y T h e o r e m ) Let M = G / H be a noncompactly causal symmetric space and a a maximal abelian subspace of qp. Further, let pr: q --~ a be the orthogonal projection, X E Cmax and h E Ho. Then pr (Ad(h)X) E conv(Wo. X) + Cmin. L e m m a 4.3.2 Let (g, T) be a noncompactly causal symmetric pair. Then 1) Op = Im(id +T)I. + and qk = I m ( i d - ' r ) l . + . 2,) dim Ilp = dim n+ = dim qk.

106

CHAPTER 4. CLASSIFICATION OF INVARIANT CONES

Proof:. 1) Recall from Lemma 4.1.1 that r = - 0 on n+. Thus X + r ( X ) E Op for X E n+. Conversely, let X E [)p. Then we can find L~ E g~, a E A+ such that X = ~ a [La + r(L~)]. In order to show that La E n+ for all a E A+, we only have to show that La ~ [~a = it(0, X~ where X ~ is a cone generating element in qp. But for L~ E Oa we have L~ + r(L~) E ~ N t, so X E p implies La + r ( L a ) = 0. Therefore these La can be omitted in the representation of X. The second statement is proved in the same way. 2) ker(id + r ) = q and q N n+ = {0}, since no element in q can be an eigenvector of ad(X ~ E q. Similarly, k e r ( i d - r ) = [~ and tj N n+ = {0}. Now the claim follows from 1). D L e m m a 4.3.3 Let L E C~

and X E Ad(Ho)L. If X ~ qp, then there is

a Z E Ad(Ho)L such that I) I pr,~ ZI < Ipr,~ XI, ~) pr(X) E conv(Wo, pr(Z)) + train.

Proof:. Assume that X = Ad(h)L and let Y := pr(X) = pr(Ad(h)L) E pr(Cmax) = CmaxO. Assume for the moment that Y - prqp (X), i.e., X E Y + qk. Proposition 4.2.7 shows that a(Y) > 0 for all a E A+. By Lemma 4.3.2 we may write X as a linear combination, X = Y + E

[Y~ - r(Y~)],

aE~,+

with Y~ E ga. As X # Y, there is a/~ E A+ such that Y~ # 0. Define 1 W := -o,y,p~j (Y~ + rye) E ~p

and

Now a simple calculation gives

[W,Yl

=

1 - fl{,y-----~([Y~, Y] + [r(Y~), Y])

=

e qk.

4.3. THE LINEAR C O N V E X I T Y THEOREM

107

From Lemma 4.1.1 and Lemma 4.1.2 we derive [W, Ya - r(Ya)]

= =

1 fl(y) ([Ya + r(Ya), Ya - r(Ya)]) A X ~,

where A = [Y~I2I~I2/~(Y) > 0. Furthermore, W2 "= [W, Wx] e qp f3 a x. Let Zt = Ad(exp t W ) X ~. gd(Ho)L. It follows from the above calculations that

Zt

= =

x - t[w, x ] + o ( t ~) ( Y - tAHa) + (1 - t)(Y# - rY#) + Wx - tW2 + O(t2).

Thus Ipr.~(Z,)l 2

t)21Ya

rYal ~ + IWx I= + v t =

<

(1 -

=

I pr,, X l = - t ((2 - t)IYa - rYal = - vt)

<

I prq~ XI 2

-

for t > 0 sufficiently small. We claim that for t > 0 small enough, Y - pr Zt = t)~H# + O(t 2) e Cmin = Cmax. To see this, let V E Cmax. Then V = 7H# + L, with fl(L) = 0 and -y > 0. Thus (Y - pr(Z,)lV) = t,X'ylHal2 + O(tZ), and this is positive for small t. This proves the lemma if prqp X E a. Assume now that Xp := prq~(X) # Y. There exists a k ~_ Ho gl K such that Ad(k)Xp E a, since a is maximal abelian in qp. On the other hand, A d ( K gl Ho) is a group of isometries commuting with prqp and prqk. Therefore we get pr,~(Ad(k)X) = Ad(k)Xp and [pr,~ ( g d ( k ) X ) l = I Ad(k) Prqk (X)l = I prqh (X)[. In particular, A d ( k ) X r qp. By the first part of the proof we may find a Z E A d ( H o ) A d ( k ) X = Ad(Ho)X such that I Prqk (Z)I < I prqk (Ad(k)X)l = I prqk (X)I.

108

C H A P T E R 4. CLASSIFICATION OF I N V A R I A N T CONES

Now Proposition 4.2.9 shows that Y

=

pr(Ad(k-l)Ad(k)Xp)

6_ convtW0. Ad(k)Xp]

C

conv [Wo. (pr(Z) + Cmin)]

=

conv[pr(Z)] + Cmin,

and this implies the claim.

El

Recall that adjoint orbits of semisimple elements in semisimple Lie algebras are closed according to a well-known theorem of Borel and HarishChandra (cf. [168], p. 106). The fonowing 1emma, taken from [20], p. 58, is a generalization of this fact. L e m m a 4.3.4 Let G / H be a symmetric space with G semisimple. II X 6_ q is semisimple, then the orbit A d ( H ) X is closed in q. El Define a relation ~ on q via

Ipr,, (Y)I < I pr, k (X)l X ~ Y : .-'---y. pr(X) 6_ conv[Wo, pr(Y)] + Cmin L e m m a 4.3.5 Let X 6_ q. Then the set { Y 6_ q l X _ Y} is (Ho fl K ) invariant and closed in q.

Pro@ The (Ho N K)-invariance follows from Proposition 4.2.9 and the (Ho N g)-equivariance of prqk. Now assume that Yj 6_ {Y 6_ q l x Y}, j 6_ N, and that l~ --r Y0 6_ q. As [prqh (l~)l ~ I Prqk (X)I, it follows that I prqk (Y0)l _ [prqk (X)l. Furthermore, there are Zj 6_ conv[W0 9pr(l~)] and Lj 6_ Cmin such that pr X = Zj + Lj. But the union of conv [W0. pr(l~)], j > 0 is bounded, so {Z j} has a convergent subsequence and one easily sees that the limit point is in conv(W0. Y). Thus we can assume that {Zj } converges to Z0 6_ conv(WoY). Therefore Lj = p r ( X ) - Z j ~ p r ( X ) - Z ~ 6_ Cmin, since Cmin is closed. El o x and let L E Ad(Ho)X. Then the set L e m m a 4.3.6 Let X 6_ Cm~ M x ( L ) := {Y E Ad(Ho)X I L ~_ Y} is compact. Proof. By Lemma 4.3.4 and Lemma 4.3.5 it follows that M ( L ) = M x ( L ) is closed. Thus we only have to show that M ( L ) is also bounded. Let h d ( h ) X 6- M(L). Write h = k e x p Z with k 6_ K I 3 H o and Z 6_ [jp.

4.3. THE LINEAR CONVEXITY THEOREM

109

As Ad(k) is an isometry, I Ad(h)XI = I Ad(exp Z)X I. Thus we may as well assume that h = exp Z. Since adq(Z) is symmetric, we may write X = ~ x e R Xx, with Xx 6 q(A,Z). Thus Ad(h)X = E

eXXx"

Now O(q(A, Z ) ) = q(-A, Z) as Z 6 1~.. From O(X) = - X we get

X = Xo + E [Xx + X_x I = Xo + E [ x x - o(xx)], A>0

(4.26)

)~>0

with Xo 6 qp. Thus Ad(h)X

= Xo + E

[exXx -e-XO(Xx)]

X0 + E

sinh(A)[Xx + O(Xx)l+ E

~>o

cosh(,k)[xx - 0(xx)].

A>o

In particular, sinh(A)(Xx + O(Xx))

pr,k (Ad(h)X) = E A>0

and pr,p(Ad(h)X) = Xo + E

cosh(,k)(Xx - 0 ( x x ) ) .

,k>O

The eigenspaces are orthogonal to each other and Ad(h)X 6 M(L), so we find I pr, h (Ad(h)X)l ~

sinh('X)2lXx + O(Xx)l ~ _<

I pr.~ (L)I 2

<

ILl 2 .

Fhrthermore, IXx 4-0(Xx)l 2 7 IXxl 2 + I~ 2. Hence IXx - 0 ( X x ) l 2 = IXx + O(Xx)l 2 . From cosh(t) 2 = 1 + sinh(t) 2 we now obtain I pr,, (Ad(h)X)l 2

-

IXol2 + ~

eosh(A)21Xx + O(Xx)l2

A>O

=

IXol2 + ~

IXx - o(xx)l 2 + ~

~>o

=

IXl ~ +lprqk(Ad(h)X)l 2

<

1212 + 1512.

A>o

sinh(A)21Xx + O(Xx)l2

110

C H A P T E R 4. CLASSIFICATION OF I N V A R I A N T CONES

Thus [ Ad(h)XI 2 < 2ILl 2 + IXl which proves the claim, o Now we are ready to prove Theorem 4.3.1: Let X E Cmax. As conv(W0 9 X) + Cmin is closed, we may assume that X E COax. Let L = h d ( h ) X E A d ( H ) X . Since H is essentially connected, we have A d ( H ) X = Ad(Ho)X, so we may assume that h E Ho. Since M x ( L ) is compact, the map

M x ( L ) ~ Y ~ ]pr, h (y)[2 e R attains its minimum at a point Y - Ad(a)L, a E H. By Lemma 4.3.3 we must have Y 6 qp. Moreover, prCL) E conv(W0, pr(Y)) + Cmin. Because of Y E qp, we can find a k E K FI Ho such that Ad(k)Y = A d ( k a h ) X (F. a. By Lemma 4.2.13, part 2), it follows that kah E K F'l Ho. But then ah E K N H. Hence Proposition 4.2.9 shows that pr(Y) E conv(Wo. X ) , which in turn yields

pr(Ad(h)X) -- pr(L)

E

conv[Wo- pr(V)] + Cmin

conv[Wo. (cony Wo" X)] ~- Cmin conv(W0. X) + Cmin and therefore proves the theorem.

4.4

T h e Classification

Let .M = G / H be a noncompactly causal symmetric space and a a maximal abelian subspace of qp. Recall the orthogonal projection pr: q --+ a and the corresponding intersection and projection operations for cones. Theorem 4.1.6 implies that any C e ConeH(q) contains one of the two minimal cones and consequently is contained in the corresponding maximal cone. Then Proposition 4.2.11 implies that

Cmin C

I(C)C P(C)C Cmax.

Clearly I(C) is W0-invariant, but Theorem 4.3.1 shows that P ( C ) i s also W0-invariant. The goal of this section is to show that any W0-invariant cone between Cmin and Cmax arises as I(C) for some C e ConeH(q) and

4.4. THE C L A S S I F I C A T I O N

111

that C is uniquely determined by I(C). We start with a closer examination of the projection pr. Let ff be a Cartan subalgebra of tc containing ia. Then t c is a Cartan subalgebra of g c since the simple factors of g c are Hermitian and the analytic subgroup T c of e ad(Bc) with Lie algebra ad(ff) is compact. Consider the closed subgroup {~ E T c [ 7"~7" = ~-z} of T c. Its connected component Ta :-- e ad(ia) then is a compact connected subgroup of T c with Lie algebra ad(ia). We normalize the Haar measure on Ta in such a way that it has total mass of 1. L e m m a 4.4.1 Let X e q. Then p r ( X ) = fT. ~ ( X ) d~.

ProoJ!. Write X - pr(X) + ~"~-aeA+ [La -- 7"(La)], with La E ga. Then ~(X) = pr(X) + E

[~a(La) - ~-aT"(La)] ,

aEA+

where (ead Y) a -- e a(Y) for Y E ac. As Ta ~ ~ ~ ~a E C* is a unitary character, it follows that fTo ~~ d~o = O. Hence

/To ~ ( X ) d ~

=

pr(X)§ ~+

=

pr(X).

[(/T. ~ p a d ~ ) L ~ - ( / T o

~p-ad~p)7"(L~)]

This implies the claim.

E!

For g E G define the linear map (I)g 9gc -4 gc by

~g(X) :-"/To ~pAd(g)~-Z(X) d~p

(4.27)

and set n := {(I)h e End(a) I h e Ho}. Then Lemma 4.4.1 implies that ~ / = {(I) e End(a) I 3h e Ho 9~) = pro Ad(h)}. L e m m a 4.4.2

(4.28)

I) Let Y E a and g E G. Then ~ g o ad Y = ad Y o (I,g.

2) T o r = ~ ( g ) 0 T for all g E G. In particular, 7"o ~h = ~h o 7" for all hEH.

3) Cb(3,(a)) C 3u(a). If h ~. H and X ~. a, then ~ h ( X ) = pr(Ad(h)X). 4) Let k E NKnH(a) and h E H. Then A d ( k ) ~ h ( X ) = ~ k h ( X ) .

112

C H A P T E R 4. C L A S S I F I C A T I O N OF I N V A R I A N T C O N E S

Proof. 1) Let Y E a. Then e ~d tiY ~. Ta for all t E R. H e n c e r t i a d Y = eti nd Y r Differentiating at t = 0 yields @g o ad Y = ad Y o ~g. 2) Since r ~ r = ~o-1 for all ~ ETa and Ta is urdmodular, it follows that f

:=

1•_ r ~ A d ( g ) ~ - l r ( X ) d~

=

f T ~-1 g d ( r ( g ) ) ~ ( X ) d ~

=

r

3) From part 1) we get ~g(~e(a)) C 3,(a). If X E a, then ~o(X) = X for all qo ~. Ta. Hence P

9 h ( X ) = / ~o( a d ( h ) X ) d~. JT

and this equals pr(Ad(h)X) by Lemma 4.4.1. 4) Note first that pr is NKnH(a)-equivariant by Lemma 4.2.6. Now the claim follows from part 3). t:] L e m m a 4.4.3 Let h E H. Then ~

= ~O(h-l). In particular, 7-l* = 7-l.

Proof:. Note that Ad(g)* = Ad(0(g) -1) with the usual inner product on g. Thus for X, Y E a:

(~h(X)IY)

=

(pr(Ad(h)X)lY)

=

(Ad(h)XIY)

=

(X I A d ( O ( h - x ) ) Y )

=

(Xlprgd(O(h-~))Y)

=

(Xl~o(h-,)(Y)).

El

From this the lemma follows. R e m a r k 4.4.4 Theorem 4.3.1 shows that a convex cone c with is W0-invariant if and only if it is 7/-invariant.

Cmax

Cmi n C C C

[:l

L e m m a 4.4.5 Let c be an 7-l-invariant cone in a. Then 1) c* is 7-l-invariant. 2) If c is closed and regular, then we can choose a cone-generating element X ~ E c so that

~ C C Cm.x(XO).

4.4. T H E C L A S S I F I C A T I O N

113

Proof:. Let X E c \ {0}. Let X := [ l / # W o ] ~wEwo w . X . By Remark 4.4.4, )~ E c. Since c is proper, we see that X ~ 0. The Wo-invariance of shows that all ~ E Ao vanish on X and hence )~ E }(1)a). Thus we may choose a cone-generating element X ~ E c. Set Cmia := Cmin(X ~ etc. Then the ~-invaria~ce of c shows that Cmin -- p r ( C m i n ) -- p r (conv [Ad(Ho)(R+X~

C 7 4 . c = c.

The regularity of c is equivalent to the regularity of c*. But c* is 7/-invariant by Lemma 4.4.3. Applying what we have already proved to c* implies that Cmin C c* hence by duality c C Cmax. r'l We now turn to the question of which cones in a occur as intersections I(C) with C E ConeH(q). Recall the extension operators for cones from Remark 2.1.12. We set E "= E~ ,H. Then we have C m i n ( X 0) ~---E ( R + X ~

and

P (E(U)) = cone {74(U)}.

(4.29)

T h e o r e m 4.4.6 ( E x t e n s i o n , I n t e r s e c t i o n a n d P r o j e c t i o n ) Let j~4 = G / H be a noncompactly causal symmetric space and a a maximal abelian subspace of qp. Then for any c E Conew0 (a) we have E(c) E ConeH(q) and

c = P ( E ( c ) ) = I(E(c)). Proo~ It follows from Lemma 4.4.5 that there exists a cone-generating element X ~ E c. As Cmin :---- Cmin(X 0) is generated by the H-orbit of X ~ E Cmin :'- Cmin(X 0) (cf. Lemma 4.2.8), it follows that Cmin C E(cmin). On the other hand, Cmin C Cmin SO that S ( c m i n ) C Cmia and hence E(cmin) = Cmin. Now Cmax C Cmax (CI. Proposition 4.2.11)implies, E ( c m ~ ) C Cmdr. So far we know Cmin C E(c) C Cmax. But E(c) is generated by Ad(H)c Ad(Ho)c (cf. Theorem 3.1.18), so E(c) E Cone~/(q). Now let X E c and h E H. Then Lemma 4.4.2, part 3, implies pr(Ad(h)X) = (~h(X) 6_ 7-l. X C c. Moreover, P(E(c)) C c, since E ( c ) i s generated by Ad(H)c = Ad(Ho)c and c is 7/-invariant by Remark 4.4.4. But clearly c C E(c), which implies c C P(E(c)). Thus P ( E ( c ) ) = c. Finally, c C I(E(c)) C P(E(c)) = c proves I(E(c)) = c. rn T h e o r e m 4.4.7 Let .s = G / H be a noncompactly causal symmetric space and a a maximal abelian subspace of qp. Then ,for a closed cone c in a the following conditions are equivalent:

C H A P T E R 4. C L A S S I F I C A T I O N OF I N V A R I A N T CONES

114

1) c is Wo-invariant and

Cmi n C (2 C

Cmax for a suitable chosen minimal

cone. 2) c is regular and 7-l-invariant. 3) There exists a cone C 6 Conen,(q) such that P ( C ) = c. ~) There exists a cone C E ConeH(q) such that I ( C ) = c. Proof: If 1) holds, then by the convexity theorem 4.3.1, Ch(X) 6 c for all h 6 Ho and X 6 c. Thus 2) follows. 3) and 4) follow from 2) by Theorem 4.4.6. If 3) holds, then Cmin C I(C) C c C Cmax, which implies 2) and 1). Similarly, 4)implies 2 ) a n d 1). rl

The last step in our classification program is to show that H-invariant regular cones in q are indeed completely determined by their intersections with any maximal abelian subspace of qp. In order to do this we again have to use the structure theory provided by the fact that ge has Hermitian simple factors. Definition 4.4.8 Let .M - G / H be a noncompactly causal symmetric space and G c / G c be any symmetric space corresponding to ~gc, he). A cone C 6 ConeH(q) is called Ge-extendable if there exists a cone C 6 Coneo~ (i9 c) such that C - I~" (C). rl If C 6 ConeH(q) is G~-extendable, we can find C e ConeG~(ig e) such that C - I~" (C) and - r ( C ) - C. In fact, if 6'z 6 ConeG,(ig e) satisfies C - -Tqi'' (Cz) , we simply set C : - C1 N ( - T ( C z ) ) 9 We have seen in Theorem 4.2.16 that the minimal and the maximal cones in q are Ge-extendable for any symmetric space G c / G c corresponding to (go, he). We will see in the next section that this is indeed true for all cones in ConeH(q). What we need now is a much weaker statement. L e m m a 4.4.9 Let C 6 ConeH(q). Then El '~'~ ( C ) e Coneo~(ige). Proo~ As Cmin C C for a suitable cone-generating element, it follows that

F~ioe'Gr( G r o i n )

G r o i n ~- - q

Ei'e'Gr (C)

C --q

But Cmax is Ge-invariant, so C C Cmax C Cmax implies

c Cm,x. Therefore E - - qi~

(C) is regular.

9

115

4.5. E X T E N S I O N OF C O N E S

T h e o r e m 4 . 4 . 1 0 Let M = G / H be a noncompactly causal symmetric space and C E Coneu(q). If X E C ~ then X is semisimple and a d X has real eigenvalues and there exists an element h E Ho such that Ad(h)X E

x(co).

Proo~ According to Lemma 4.4.9, we know that C C Cmax for a suitable choice of a cone-generating element X ~ Since X ~ is contained in the in~O teriors of C and Cmax, we see that C ~ C Cmax. But then Lemma 4.2.15 shows that there exists an h E Ho such that Ad(h)X E a f'l C ~ = I(C~ Q

From Theorem 4.4.10 we immediately obtain the following theorem, which completes our classification program: T h e o r e m 4.4.11 ( R e c o n s t r u c t i o n of C o n e s ) Let .Azf = G / H be a noncompactly causal symmetric space and C E ConeH(q). Then C ~ = Ad(H)I(C ~ and C = E 'Ho,{1} (I~(C~

= A d ( H ) I ( C o) .

r~

C o r o l l a r y 4.4.12 Let .hi[ = G / H be a noncompactly causal symmetric space and C e ConeHo(q). Then C e ConeH(q). Proo~ According to Theorem 3.1.18, we have H = HoZnnr(a) and Z u n K ( a ) acts trivially on I(C~ Theorem 4.4.11.

4.5

Therefore the claim follows from El

E x t e n s i o n of C o n e s

Assume that j~4 is a noncompactly causal symmetric space and G c / G e any symmetric space corresponding to (go, ac). The goal of this section is to show that any C E Conen(q) is Ge-extendable. We note first that Remark 4.2.14 shows that this is the case if g carries a complex structure, since then g* is Hermitian. So we may assume that G c / G c is noncompactly causal and by Corollary 4.4.12 we only have to show G~-extendability. In particular, we may assume that Gc is simply connected and that G e and G are the analytic subgroups of Gc with Lie algebras gc and g. Fix a Cartan subalgebra t c of [~e as before and consider

116

C H A P T E R 4. C L A S S I F I C A T I O N OF I N V A R I A N T C O N E S

where K c is the analytic subgroup of G c corresponding to the Lie algebra to. Then K c is compact and Wo is the Weyl group for the pair (to, tc). We choose a cone-generating element X ~ E qp C ibk + qp and a positive system /~+ for/~ = A({tc, t e) as in (4.12). Let

c(a0+) := { x e , I W e

at'.(x)

> 0}

be the corresponding open Weyl chamber in a. Similarly, let C ( s chamber in it c. We define C := C(A +) and C "= C(/~+).

be the

R e m a r k 4.5.1 Let C' be the closure of the Weyl chamber in a' := a M [b a, b a] corresponding to A +. Then C = C' @ R X ~ and C* = (C')* C a', where (C')* is the dual of (Y in a ~. If X = X1 + X2 E C with X1 E C' and X2 E R X ~ then

~(x) = ~(Xx)+ x= for all s E Wo. Similar things hold for C and W0. L e m m a 4.5.2

[:3

1) Let a E Ao be such that al, i~ O. Then ~ is positive ~, po,iti~e.

if , , d o,ly if-~(~)

2) C = C N a = pr(C), where pr: iff --+ a is the orthogonal projection.

3) C* = pr(C*). Proo~ 1) This follows from (4.12). _

2~ Obviously C C C gl a C pr(C). pr(C) C C. Let X E (~. Then

Thus we only have to show that

pr(X) = ~1 I X - r(X)l Let a E A + and choose fl E s

such that f~[, - a . 1

= ~ (z-

Then

~z).

Then 1) shows - r f l E A+. Thus

1 [x - ~(x)])

a(pr(X))

= Thus pr(X) E C as claimed.

1 5 ( ~ ( x ) + [-r~(X)]) > 0.

4.5. E X T E N S I O N OF CONES

117

3) This follows from 2) and L e m m a 2.1.8. Consider the following groups (cf. Theorem 4.2.6)"

~r er Lemma

"= ( ~ e ~r I ~ o w = ~ o ~}

(4.30)

"= ( ~ e r162 I~1, = id}.

(4.31)

4.5.3 The restriction to a induces an exact sequence

{1}

~r162162

"$0(~)

~Wo

,{1}.

Proof:. We only have to show t h a t the restriction to a induces a surjective m a p Wo(r) --~ Wo. So let w e Wo and k e NgnH(a) be such t h a t Ad(k)l, = w. Note t h a t Ad(k)(iff N it) is a maximal abelian subspace of ira = i~,(a). Therefore there exists an h e Mo := ZK(a)o = ZlcnH(a)o such t h a t A d ( h ) [ A d ( k ) ( i f f n im)] = i f f N ira. But then hk e NgnH(a)M NKnH(iteN ira) C NKnH(it e) and therefore hk corresponds to an element of W0. As t~ leaves a and ~ N ira invariant, it commutes with r. Thus fro E Wo(r). Finally, we note t h a t ~. X = Ad(hk)(X) = Ad(h)Ad(k)(X) = Ad(h)(w. X) = w. X for X E a, which implies the claim. Remark pr~ ot~.

D

4 . 5 . 4 Let ~ e tTv'0(r) and w = t~l,. T h e n we have w o pr, = rn

In view of R e m a r k 4.5.1, the following lemma is a consequence of L e m m a 8.3 in [45], p. 459: Lemma 4.5.5

1) Let X ~. C. Then

conv(l~0 .X) =

U w~Vr

= n

wEl~o

2) Let X E C. Then

cony(W0. X) =

U wEWo

w [C n ( X - C')l =

N

~ (X - C').

o

wEVr

4.5.6 Let (g, v) be a noncompactly causal pair, a a maximal abelian subspace of qp, and t e a Cartan subalgebra of t e containing ia. Further, let Wo and Wo be the Weyl groups of (g, a) and (t c, re), respectively. Then pr(conv 1~0 .X) = conv(W0. X ) for all X E a, where pr: iff ~ a is the orthogonal projection.

Theorem

118

C H A P T E R 4. CLASSIFICATION OF I N V A R I A N T CONES

Proof: Set L = conv(l?Vo .) C iV, L = conv(Wo. X) C a and F := pr(L). Fix w E W0. According to L e m m a 4.5.3 we can choose a ~ E W 0 ( r ) such t h a t t~l. = w. Then Remark 4.5.4 and the Wo-invariance of L imply that w(F) = w. [pr(L)] = pr(,~L) = F. Therefore F is convex and W0-invariant. As X E F , it follows that L C F. To show the converse, we choose w E W0 such that e c c c.

Choose ~ E I~V0 with ~1, = w. Using L e m m a 4.5.5 and L e m m a 4.5.2, part 3), we find f C p r ( ~ ( X ) - (~*) = w ( X ) - C*. This together with L e m m a 4.5.5, part 2), shows F N C C [w(X) - C'] N (~ C conv[W0 9w(X)] = conv(W0 9X) and hence the claim. C o r o l l a r y 4.5.7 Let c be a Wo-invariant cone contained in a. Then 5 := conv{l~Vo(C)} is a ITVo-invariant cone in ~ with p r ( 5 ) = c = 5Na.

Prooj!. We obviously have c C 5 Na C pr(5). Let X E c. Then pr(conv W o ( X ) ) = conv Wo. X C c and hence also pr(5) C c. T h e o r e m 4.5.8 ( E x t e n s i o n of Cones)

Suppose that . M = G / H is a non-compactly causal symmetric space. Let G c / G c be any noncompactly causal symmetric space corresponding to (go, ae). If C e Coneu(q), then 1) E - - q i'',G" (C) e convG,(i{t e) J 2) - r ( E ~ '''a~ (C)) = E~'~

(C), and

3) E~'''~ (C) N q = pr,(E~ '''~ (C)) = C. In particular, every cone in ConeH (q) is Ge-extendable. Proof: We may assume that g carries no complex structure and all groups are contained in the simply connected group Gc. 2) is obvious, as - r ( C ) = C. We prove 1) and 3) together. Fix a C a r t a n subalgebra t e of te containing a maximal abelian subspace a of qp.

4.5. E X T E N S I O N OF CONES

119

Let C E ConeH(q). We may assume that Cmin (~ C n a (~ Cmax. Now we apply Corollary 4.5.? to find a W0-invariant cone ~ in it r such t h a t pr(~) -- C n a -- ~ n a, where pr: it c --+ a is the orthogonal projection. Replacing ~ by ~ + C-min if necessary we m a y assume t h a t Cmin (: c" (: Cmax. By Theorem 4.4.7 applied to G c / G c there is a cone C E convG,(i{~ c) such that (7 n it c = P_~tc (s = 8. In particular, we have C n a = C n a. Thus n q and C are b o t h cones in Coneuo (q), and their intersections with a agree. Thus T h e o r e m 4.4.11 proves t h a t C n q = C. Since C is GC-invariant E iO~'G"(C) n q and contains C, it also contains -E- qi~ (C). Therefore C = _q and then Theorem 4.4.11 implies the claim, since E~'',G` (C) is regular by L e m m a 4.4.9. n

Notes

for Chapter

4

The material in this chapter is taken mainly from Chapter 5 in [129]. The relation between the strongly orthogonal roots in Section 4.1 is from [130]. A more algebraic proof of Lemma 4.1.9 can be found in [133]. The convexity theorem in the group case was proved by Paneitz in [147]. The linear convexity theorem, which was proved in [129], can also be viewed as an infinitesimal version of the convexity theorem of Neeb [116]. It can be derived from general symplectic convexity theorems applied to suitable coadjoint orbits (cf. [62]). The proof presented here is based on the proof of the convexity theorem of Paneitz by Spindler [158]. The classification for simple groups is due to Ol'shanskii [139], Paneitz [147, 148], and Vinberg [165]. Their results were generalized to arbitrary Lie groups by Hilgert, Hofmann, and Lawson in [50]. The extension theorem for invariant cones was proved in [48] for the classical spaces and in [129] for the general case using the classification. The idea of the proof given here is due to Neeb [116]. The invariant cones in the group case have been described quite explicitly by Paneitz in [147] for the classical groups. Thus Theorem 4.5.8 can also be used to obtain explicit descriptions in the general case.

Chapter 5

The Geometry of Noncompactly Causal Symmetric Spaces If A4 = Gc/G is a causal symmetric space, then G/K is a bounded symmetric domain. It is well known that in this case G/K can be realized via the Harish-Chandra embedding as a complex symmetric domain in p - , which in our notation for noncompactly causal symmetric spaces corresponds to n_. We generalize this embedding to the general case in the first section. More precisely, we show that if C:I/H is a noncompactly causal symmetric space, then H/H f3 K is a real symmetric domain ft_ in n_ which can also be realized as an open subset O in a certain minimal flag manifold X = G/Pmax of G. The importance of this observation lies in the fact that the semigroup S(C) associated to the causal orientation of A4 = G/H is essentially equal to the semigroup S(G r, Pmax) of compressions of O. This semigroup consists of all elements in G mapping O C X into itself. We show that S(C) = (exp C)H is homeomorphic to C x H. With this information at hand, one easily sees that noncompactly causal symmetric spaces are always ordered and have good control over the geometric structure of the positive domain A4+ of A4 which consists of the elements greater than the base point with respect to the causal ordering. In particular, we prove that intervals in this order are compact. Finally, we give a proof of Neeb's non-linear analog of Theorem 4.3.1 which turns out to be extremely useful in the harmonic analysis of noncompactly causal symmetric spaces. 120

5.1.

T H E B O U N D E D R E A L I Z A T I O N OF H / H n K

5.1

The Bounded

121

Realization of H/H n K

In this section we fix a noncompactly causal symmetric space A4 -- G / H and assume that G is contained in a simply connected complex group Gc with Lie algebra go. Recall the abel• subalgebras n+ C g which form the irreducible pieces of the [ja-module q = n_ + n+ (cf. Remark 3.1.17 and L e m m a 1.3.4) and let N• - exp n• be the corresponding analytic subgroups of G which are automatically closed. Similarly, we have a closed nilpotent subgroup No - exp no in G. Since H a centralizes X ~ E qp, it also normalizes n• and N• Therefore Pm~x = H aN+ = H a N

(5.1)

defines a maximal parabolic subgroup of G (cf. Appendix A.1 and Lemma

1.3.4). Consider the involutive anti-automorphism ~" G -+ G,

g ~+ r(g) -~

(5.2)

and denote its derivative at 1 also by ~ 9 From r~[. = id we obtain the following. L e m m a 5.1.1 Both Pmin and Pmax are ra.stable. Furthermore, 8 ( P m i n ) -" T ( P m i n ) ---- M A N ~

and 8(Pm~x) = r ( P m ~ x ) = HaN~+ = H a N - = H a N ~. Recall the maximal set F of strongly orthogonal roots in A+ = A(n+, a) from Remark 4.1.10 and the corresponding maximal abel• subspace a~ = ~ e r RY~ of t)p. We set A~ = exp a~. (5.3) L e m m a 5.1.2 Let the notation be as above. Then HPmin = HPmax = (G~)oPmax and H n Pmax = H n K . Furthermore, HPmax is e ~P,ITI~X open in G. Proop. L e m m a 3.1.22 implies that H a - (H n K ) M A N o so H -

( e x p i } p ) ( H n K ) (cf. (1.8)) proves the first part.

CHAPTER

122

5.

THE GEOMETRY

Let h 6 H N Pmax. Write h = k l a k 2 with kj 6 K N H and a 6 A~. From K N H C Pm~x it follows t h a t a E H N Pm~x. But then v(a) = a -1 E Pmax N P-"~ax = M A . But A~ f3 M A = { 1 }, which implies t h a t a = 1. A simple calculation shows that the differential of G r • A • N 9 (h,a,n) ~ han 6 G

is bijective everywhere. Thus G r A N = GrPm~x = HPm~x is open in G, cf. [99]. El The generalized Bruhat decomposition (cf. [168], p.76) shows that both N-Pmax and N~Pmin are open and dense in G and the maps N _ x H a x N+ 9 ( n _ , g , n + ) ,--+ n _ g n + 6. N - P m ~ I and N lI x M x A x N 9 ( O n l , m , a , n2) ,-.-r Onxman2 6. G

are diffeomorphisms onto their images. E x a m p l e 5.1.3 Recall the situation of the Examples 1.3.12 and 3.1.13, i.e., G = SL(2,R). Then M = {=hl} and

c

d

=

Moreover, we have

Pmin = Pmax =

{(o 0

c)

-b

Y

d

"

0)1 y 6 R }, ,eR,~eR\{0}},

U- 1

and

b o 0} d) # "

if(ab) e

d

=

(1

y

1)(o

0

e

e = a

0

(o x)

y - c/e

1

1

and

(5.4)

6 N_Pmax, then

x - b/e.

o

L e m m a 5.1.2 implies that the set O = ( G ~ ) o . o x in the real flag manifold X = G / P m a x , where o x := 1Pmax/Pmax, is open. Moreover, we have n/H n K

~_ n P m . ~ / P m , x = G~Pm~x/Pm~x = =

(Gr)oPmaxlPmax = ( G r ) o . O x - O.

(5.5)

We will now describe 0 in more detail using the symmetric SL(2, R)reduction from Section 4.1.

5.1. THE BO UNDED REALIZATION OF H / H 13 K

123

L e m m a 5.1.4 H C N-Pmax. Furthermore, exp E

t~Y~

~EF _

Proo~ The polar decomposition for H gives H = (H 13 K)ACh(H 13 K). As H 13 K C G *~ and H f3 K normalizes N_, we only have to show that A~ C N - G *~ But Example 5.1.3 implies that

expE

tTY7

=

-yEt

This shows that A~ C N-Pm~x and the lemma follows. Define a map to- n_ ~ A' by

(5.6)

~(x) = (exp x ) . Pro,,.

Using H / H n K = HPmax/Pmax C N-Pmax/Pmax ~- N - ~ u_ we see that tr is injective and find for h E H:

hPmax = exp(t~-l(h(H 13 K))Pmax)Pmax. Let

(5.7)

f~_ := ~ - ~ ( O ) c . _ . By L e m m a 5.1.4 we have the following. L e m m a 5.1.5 Let a = exp ~ r

t~ Y~ ~- A~. Then r-i

g-l(aPmax) = E tanhtvY_v. -~EF

L e m m a 5.1.6 Let h E H and X E I~_. Then h. X = A d ( h ) X .

Proof" Let X E ~ - and h E H. Then h exp X claim follows from A d ( h ) X E n_.

=

[exp Ad(h)X]h and the [::1

CHAPTER 5. THE GEOMETRY

124 L e m m a 5.1.7 Let Y E n_.

Then there exists a k G K I"1Ho such that

r

Ad(k)Y e Ej=x RY_ . Proof. v - - 8 on n_ (cf. Lemma 4.1.1) implies that Y + v(Y) G [Jp. Hence there exists a k G K n Ho such that

Ad(k)[Y+ r(Y)] =

Z t'(Yv + Y-v) fi ah.

VEI ~

But then Ad(k)Y = ~ T e r tTY_7 and the claim follows. T h e o r e m 5.1.8 Let the notation be as above. Then

f~_= Ad(K n H){ verZYvY-v -I
~-~(h(HNK))

nH

and a = exp ~ E r

t~ Y~ G A~.

= Ad(k')~-x(expZtvYv(Pmax)) =

Ad(kl) Z

tanht~Y_~ E f~-.

~EF

On the other hand let X = A d ( k ) ~ e r arctanh(y~). Then

Y~Y-~ be in f2_ and t~ =

~(X) = ( k exp ~ _ t , y ~ ) p~,~. This proves the theorem.

[:3

R e m a r k 5.1.9 As r interchanges N_ and N+ in such a way that r(Y_~) = Y7 and YV E [}p we see that for f~+ := r(f~) C n+ we have

Hv(Pmax)lv(Pmax) ~- Ad(K n H) { ~erZY'rY'r

er}

- l
=a+

0 E x a m p l e 5.1.10 Let G be a Hermitian Lie group, i.e., G is semisimple and G / K is a bounded symmetric domain. Then Gc/G is a noncompactly causal symmetric space. In this case a -- it, where t C ~ is a Cartan

5.1. THE BOUNDED REALIZATION OF H / H N K

125

subalgebra of [~ and of g. Furthermore, A+ = A , is the set of noncompact roots, A0 = A k the set of compact roots, and , + = p+ in the notation of Appendix A.3. Hence the above result states that G C P - K c P + and G / K can be realized as an bounded symmetric domain in p - In other words, Theorem 5.1.8 is a generalization of the Borel-Harish-Chandra realization of Hermitian symmetric spaces as bounded domains in p - . [3 Let a and ~ be the involutions on Gc with fixed point groups G and G e (cf. Section 1.1). Then Ge/K e is a bounded symmetric domain. We want to relate fl_ and 69 to the Borel-Harish-Chandra realization of Ge/K e. Recall t h a t 9 ~ = ~ + iq is a Hermitian Lie algebra. Moreover, let t c = t m + ia be a compactly embedded Cartan algebra in 9 e and [e = [Jk + iqp the uniquely determined maximal compactly embedded subalgebra of ge containing t c (cf. [50], Theorem A.2.40, and Section 4.1). As ~bef~ we write A = A(gc, t~) for the corresponding set of roots. Then A0 denotes the roots of [~ which we call compact and /i+ the corresponding set of noncompact roots. The Cartan decomposition of t]e with respect to Oa is ge = [~+pC with pC = ~p+iqk. Note that ([~e)C = ([~a)C and (pe)C = Z g~"

For

we have

(pc)+=

ae~p

e+ (~max)C ---- ([C)C "~" (~})C"

The theory of Hermitian symmetric spaces (cf. Appendix A.3) says t h a t

GC/K c embeds as an open Ge-orbit (PC into the complex flag manifold Xc = Gc/(Pmax)C and then as a bounded symmetric domain (f/_)c in (pc)- (and (~2+)c in (pc)+). The complex parabolic (Pmax)C is stable under the conjugation a. Hence a yields a complex conjugation on Gc/(Pm,,x)C which we still denote by a. We write (Gc/(Pm~x)C)" for the set of a-fixed points. Lemma

5.1.11 ,I'~. = X.

Pro@. ( N - ) c . o x is open dense in Xc and invariant under a. If x E A'~, then there exists a sequence of nj E ( N - ) c such t h a t n j . o x converges to x. But then a(nj).ox converges to x as well, whence n j a ( n j ) - l . o x converges to ox. Thus n j a ( n j ) -1 converges to 1 in ( N - ) c . Now, the fact t h a t ( N _ ) c is a vector group shows that the imaginary part of nj converges to zero (identifying ( n - ) c and ( N - ) c ) and we can replace nj by its real part without changing the limit. This proves x 6 X and hence the claim.O

C H A P T E R 5. T H E G E O M E T R Y

126

L e m m a 5.1.12

1)

O c N X = O.

2) (a_)c nn_ = a_. Proofi 1) Let g ~. a c with a(g.o) = g.o. Since K c C (Pmax)C, we may w.l.o.g, assume that g = exp X with X E pC. Then the hypothesis implies t h a t exp ( a ( X ) ) ~. e x p ( X ) g c, so that the Caftan decomposition of G c yields t h a t a ( X ) = X . Therefore X E Ijp and consequently g E G r. 2) follows immediately from 1). El E x a m p l e 5.1.13 We continue the SL(2, R)-Example 5.1.3. In that context we have Xc = CI? 1 and X = R P 1. Under the Harish-Chandra embedding we find Oc = (fZ-)c =

{(0 0) 0

Izl < 1, z e c }

0)0

Irl < 1, z e R } .

z

and

o o

The action of Gc on (fl_)c is given by

(o c

5.2

The

"

Semigroup

z

o) (o o) 0

=

~bz+a

0

"

S(C)

If G is an arbitrary Lie group, then the differential of the exponential map at the point X E g is given by 1 -- e - a d X

d x exp = (d 1texp x )

---- ( d l t e x p X )

adX

f(adX)

where f ( t ) = ~,,~176 + 1)! and t g : G ~ G denotes left multiplication by g. We derive similar formulas for arbitary symmetric spaces G / H . Define

fh(t)

:=

1 - cosht t =

~ t 2n-1 (2n)!

(5.8)

n----1

fq(t)

:--

sinh t oo t - E n=0

L e m m a 5.2.1 plane.

t2 n

(2n-I-

1)!"

(5.9)

1) The Junctions fq and .fh are analytic in the complex

5.2. THE SEMIGROUP S(C)

127

e) /h (0)= 2 iZ \ {0}.

s) 1(1(0) =

\ {0}.

4} f = fh "t- A.

El

L e m m a 5.2.2 Let the notation be as above and X, Y G q. Then

f ( a d X ) Y = h(adX)Y + fq(adX)Y a~d A ( ~ d X ) Y e ~, fq(~dX)Y e q. Proo~ This follows immediately from Lemma 5.2.1 and ad(X)kq C I} if k is odd and ad(X)kq C q if k is even. El L e m m a 5.2.3 Define ~ : q x H -r G by ~o(X, h) = (exp X)h. Then for all X, Y G q, Z G ~, and h G H the following holds:

d(x,h)~(Y, Z) = [Z + Ad(h-1)(fh(adX)Y)] + Ad(h -1) [fq(adX)Y] . Here we identify ThH with I} and TgG with g via the left multiplication. Proof. It is clear that d(x,h)~(O,Z) = Z. Let F ~ Cr162 exp X h t e x p ( - Ad(h-X)X]. Then ~F(expd (X + tY)h),-o

Let a(h,X) :=

=

-~F(expd

Xh[h -1 exp(-X)exp(X + tY)h]),=o

--

d--F(a(h X)exp(Ad(h -x)X + t Ad(h -~)Y)),=o dt

from which the lemma follows. El If we identify To(M) with q, then the exponential map Exp : q -~ G / H is given by E x p X = ( e x p X ) H = ~r(~o(X, 1)), where 7r : G --+ A4 is the canonical projection. Identify Tx(q) with the vector space q in the usual way. Then, using that dlg~ : TI G --+ TaG is an isomorphism, we have

dx Exp = dl~expX o fq(adX),

X e q.

(5.10)

Hence Exp is a local diffeomorphism for all X such that spec(ad X) Iq (~riZ \ {0}) = 0. We will actually need more than this. Define for A > 0

U(A)

:=

{X G q[

V(A) W(A)

:= :=

Exp(U(A)) {exp(X)h [ X G U(A),h e H} = r - l ( Y ( A ) )

max

#Espec(ad X)

< A}

Let g act on V(A) • H by h. (X, k) = (Ad(h)X, hkh -1).

(5.11) (5.12) (5.13)

CHAPTER 5. THE G E O M E T R Y

128

L e m m a 5.2.4 Let the notation be as above. Then the following hold:

1) U(A) is an open H-invariant O-neighborhood in q. e} I l A < ~, then W()~) is an open H-invariant 1-neighborhood in G and ~ " V() 0 • H ---r W ( ) 0 is an H-equivariant diffeomorphism, where g acts on G by conjugation.

3) If A < ~, then V() 0 is an open H-invariant o-neighborhood in .M and Exp 9U(A) --+ V()~) is an H-equivariant diffeomorphism. Proof:. The first part is obvious. To prove 2) assume that A < ~. We first show t h a t ~ is a local diffeomorphism. This will imply that W(A) is open. By L e m m a 5.2.3 we have d(x,h)~(Y, Z) = [Z + A d ( h - ' ) ( f h ( a d X ) Y ) ] + Ad(h-1)[fq(adX)Y]. If d(x,h)~(Y~ Z) = 0, then Z + Ad(h -1) ( f h ( a d X ) Y ) ) = 0

and

Ad(h-1)(fq(adX)Y) = 0

according to L e m m a 5.2.2. But then Y = 0 as ~ q ( a d X ) 9q --r q is an isomorphism for X E U(A). Therefore Z = 0, too and it follows that d(x,h)T is an isomorphism. Thus - by the implicit function t h e o r e m is a local diffeomorphism. Now we only have to show that qo is injective. Assume that g = exp(X)h = exp(Y)k,

X, Y e U(A), h,k e H .

Then gr(g) -x = exp 2X = exp 2Y. By [163], p. 193, it follows t h a t X = Y. But then also h = k. The H-equivariance follows from

~o(Ad(k)X, khk -x) = e x p ( A d ( k ) X ) k h k - l = k~o(X, h)k -1. 3) As dx Exp is a local diffeomorphism for X E U()~), we only have to show t h a t Exp is bijective. Assume that E x p ( X ) = Exp(Y) for X, Y e U(A). Then exp X = (expY)h for some h E H. By (2) this shows that

X=Y.

Q

Theorem V.4.57 in [50] says 5.2.5 Let G / H be a symmetric space. Let C C q be a regular H-invariant cone in q such that s p e c a d ( X ) C R for all X e C. If (exp C ) H is closed in G, then S(C) := (exp C ) H is a semigroup in G with L ( S ( C ) ) = C+b. t~ Theorem

5.2. T H E S E M I G R O U P

S(C)

129

From now on we will always assume t h a t a~/[ is a noncompactly causal irreducible semisimple symmetric space. T h e o r e m 5.2.6 Let .A4 -- G / H be a noncompactly causal semisimple symmetric space. Let C E ConeH(q). Define

S = S(C)= (expC)H = ~(C,H).

(5.14)

Then S is a closed semigroup in G. Furthermore, the following hold:

1)

S n S -~ = H.

2) C x H 9 (X, h) --~ e x p ( X ) h 6 S is a homeomorphism. 3) S ~ = e x p ( C ~ morphism.

and C ~ x H 9 ( X , h ) -+ e x p ( X ) h 6 S ~ is a diffeo-

4) S = H ( S fq A ) H . Proofi As C C U(A) for all A > 0, 2) and 3) follow from L e m m a 5.2.4. Assume t h a t s = e x p ( X ) h E S Iq S -1. Then s -1 = ( e x p Y ) k for some YECandkEH. Hence (exp Y ) k = h - ' e x p ( - X ) = e x p ( - A d ( h - 1 ) X ) h -1 As - A d ( h - 1 ) X E U(A), it follows that Y = - A d ( h - 1 ) X E C N - C = {0}. Hence Y - 0 and s E H. This implies 1). As C x H is closed in U(A), it follows t h a t ( e x p C ) S is closed. Now Theorem 5.2.5 shows t h a t S(C) is a semigroup. The last assertion now follows from the reconstruction theorem 4.4.11. O The cone C defines a G-invariant topological causal orientation _ on j~4. From Theorem 2.3.3 we obtain T h e o r e m 5.2.7 Let .h4 be a noncompactly causal symmetric space, C E ConeH(q) and ~_ the corresponding causal orientation on .A4. Then ~_ is antisymmetric and

S(C) = {s e g lo_~ s.o},

i.e., S(C) is the causal semigroup of .M.

[3

In particular, Theorem 5.2.7 shows t h a t ~ and the order <_s(c) defined in Section 2.3 agree (cf. Remark 2.3.2), so t h a t the positive cone is simply the S(C)-orbit of o" M+

= s(C)

. o.

We conclude this section with the observation t h a t one can view the noncompactly causal space .M as a subspace of Gc/(~ (cf. Section 1.1).

C H A P T E R 5. T H E G E O M E T R Y

130

P r o p o s i t i o n 5.2.8 A/" := G c / G e is a causal symmetric space and the canonical map

is a G-equivariant homeomorphism onto its (closed) image and preserves the causal orientation. Proo]!. It follows directly from the definitions that the map is well defined, (~-equivariant, and injective. Let C E ConeH(q) and C be the minimal G~-invariant extension to ige, cf. Section 4.5. As C is H-stable, it follows t h a t (7 is in ConeGo (ige). In particular, Af is causal and ~r is a causal map. Theorem 5.2.6 implies that ,~1+ and Af+ is homeomorphic to C, respectively C. But then homogeneity and G-equivariance show that 7r is a proper map. In particular, it is closed, which implies the claim. [::]

5.3

The

Causal

Intervals

In this section we will show that the causal intervals [x, y], x, y E A4 are compact. Fix x and choose g E G such that g . x - o. Since gg 9.M --+ .M is an order-preserving diffeomorphism, it follows that

and [x, y] is compact if and only if [o, g. y] is compact. Thus we may assume t h a t z = o and y e [o, oo) = S(C) 9o. Let A < ~; then Exp" U()~) ~ V(),) is a diffeomorphism. In particular, we may define Log" V (),) --r U(),) to be the inverse of Exp. T h e o r e m 5.3.1 L o g ' [ o , c~) -~ C is order-preserving.

We prove this theorem in several steps. Consider the function x

=

sinh x Then ~ = 1/fq, with fq as in (5.9). L e m m a 5.3.2 Let ~ be as above. Then

e ixy ~(x) = 4

oo

In particular, ~ is positive definite.

dy. cosh ~ ( r y / 2 )

5.3.

THE CAUSAL INTERVALS

131

Proof. It is well known that fo r sinh(x/2) sin y z d~o

=

r tanh r y

As sinh is an odd function, the integral on the right-hand side equals

1

ei=U

F

r162

1 sinh('x/2)

dx.

Differentiating with respect to y gives

~r2 1 cosh2(~ru) = 2

F

ei=U

oo

x dx. sinh(x/2)

Taking the inverse Fourier transforms now yields:

x

sinh(x/2) = lr

F

1

oo e -'=u cosh2 (lrY)

dy.

Finally, we replace x by 2x to obtain x

sinhx

e--izY 4

r162

co h

dy.

L e m m a 5.3.3 Let C e ConeH(q). Then adX

------CcC sinh X

for every X 6 C. Proof. Let X and C be as in the lemma. We may assume t h a t X E C ~ as ~p is continuous. Then ad X has only real eigenvalues. Let Gc be the complex Lie group generated by e x p ( a d X ) , X E g c . Further, let G c be the closed subgroup of Gc generated by e x p ( a d X ) , X E ge _ [} ~ iq. Consider the minimal extension D of C to a Ge-invariant cone in ig c = iIj ~ q (cf. Section 4.5). Then D I"1q = pr, D = C. As i X E ge, it follows t h a t

eiU(ad X) C C D for all y 6 R. But 1/cosh2((lry)/2) > 0 for all y 6 R, so

eiuadX

1 cosh(lry/2)

C C D

CHAPTER 5. THE G E O M E T R Y

132

for all y E R and L e m m a 5.3.2 shows that ~ ( a d X ) C C D. But obviously qo(adX)C C g. Hence

~ ( a d X ) C c D n g = D n q = C, which proves the lemma.

13

Proof of Theorem 5.3.1 : We have to show that dExp x Log(Cv.xp x ) C C for all X E C. Using (5.10) we calculate dExp X Log

=

(dx Exp) -1

--

(dxtexpX o f q ( a d X ) ) -1

=

~ , ( a d X ) o (dlt~xp x) -1 9

The claim now follows from L e m m a 5.3.3, since CExp x = dttexp x(C).

13

D e f i n i t i o n 5.3.4 Let G / H be an ordered symmetric space. Then G / H is called globally hyperbolic if all the intervals [m, n], m, n E G / H are compact. D 5.3.5 Let .M be a noncompactly causal symmetric space. Then .M is globally hyperbolic.

Theorem

Proot. Let 7 : [0, a] --+ :t4 be a causal curve with 7(0) = o. By Theorem 5.3.1, Log 07 is a causal curve in el with Log oT(0 ) = 0. In particular, Log(7(t)) e C o [C - Log(7(a))] for all t e [0,a]. It follows that Log([o, Exp X]) C C fl (C - X), As C fl ( C - X) is compact and Log is a homeomorphism, it follows t h a t [o, Exp X] is compact, rl

5.4

Compression Semigroups

In this section we show how closely related the semigroups of type S(C) are to the semigroup of serf-maps of the open domain O in X. Recall t h a t we assumed G to be contained in a simply connected complexification Gc. L e m m a 5.4.1 Let Fr C F C M be a set of representatives for ( H o n E ) \ F .

Then the group multiplication gives a diffeomorphism Ho x F~ x A x N -+ HoPm~x

5.4. COMPRESSION SEMIGROUPS

133

Proo~ L e m m a 3.1.22 implies that multiplication gives a diffeomorphism Ho x Fr ~ G ~, since F is a finite subgroup of G r normalizing Ho. According to L e m m a 5.1.2 we have

G" n A N C K N A N =

{1}.

This shows that the map is injective. The surjectivity is also a consequence of Lemma 5.1.2. Finally, we recall that the bijectivity of the differential has already been observed in the proof of Lemma 5.1.2. [:3 Recall the cones Cmax,C-max, and Cmax from (4.21), (4.23), and (4.25). We know that

~m.~ = Ad(a~

= ..~,(,)~"~176

is a closed convex GC-invariant cone in ig c whose intersection with and projection to 5 is 5max (cf. Theorem 4.4.6). One associates the Ol'shanskii semigroup

S(Cm.x) := a ~ exp((:=.x)

(5.16)

with Cmax and observes that it is closed and maximal in a c (cf. [52], Corollaries 7.36 and 8.53). D e f i n i t i o n 5.4.2 Let A' be a locally compact space on which a locally compact group G acts continuously. Further, let O be a nonempty subset of A'. Then S(O) is defined by

S(O) := {g e G i g . 0 c 0}.

o

From this definition and Proposition B.1.3 we immediately obtain the following. L e m m a 5.4.3 S(O) is a subsemigroup of G. If G acts transitively on 2(,

then the interior S(O) o of S(O) is given by s ( o ) ~ = {g e e l g . - 5 c o } .

u

Recall the special situation from Section 5.1. If g e S(O) and X e ~_, then we define g . X by

g. x = ,,-1 ( g , , ( x ) ) .

(5.17)

This turns ~ into a S(O)-equivariant map. We note that H C S(O). For any group G and any pair of closed subgroups L, Q of G we write

S(L, Q):= {g E G: gL c LQ} = {g E G: gLQ c LQ} = S(LQ/Q)

134

CHAPTER 5. THE G E O M E T R Y

and called it the compression semigroup in G of the L-orbit in G/Q. Then it follows from [52], Proposition 8.45, that S(Cmax) coincides with the subsemigroup S(Ge, B), where B is the Borel subgroup belonging to lJc and /k+. It is important for the cases we are interested in to observe that [52], Lemma 8.41, implies S(Ge, B) = S(Ge,(Pmax)C) (this also follows from Lemma 5.1.2 applied to Gc/Ge). Thus we have S(Cmax) = S(G e, (Pmax)C). L e m m a 5.4.4 Fix any parabolic subgroup Q between Pmin and Pmax- Then

s(v',Q) = S(Ho, q) = S(Ho, Pm,.) = S ( G ' , P ~ , . ) . Proof. This follows from Lemma 3.1.22, Lemma 5.1.2, and the observation that S(Gr, Pmin) C S(Gr, Q) C S(Gr, Pmax) as well as the corresponding relation for Ho.

D

Consider Cmax = Cmax 13 q and S(Cmax) = H exp(Cmax). R e m a r k 5.4.5 The cone Cmax is Gr-invariant because G" = G~:r n GE = G~:~ n ~n~-r c =GEr n G ~ = (G~) ~

and Cmax is GC-invariant. Note also that =

G~ n S(~:=..)

=

G" exp(Cmax) = FS(Cmax).

=

(GO)a exp(Cmax)

This is sometimes helpful to reduce questions concerning the semigroups H exp(Cmax) to similar problems for S(Cmax). r'] The closed subsemigroup S(Cmax) is called the real maximal Ol'shanskii semigroup. L e m m a 5.4.6 S(Cmax) C S(Gr, Pmax).

Proof. Let g E S(Cmax) and x E O = G r . o x C X C XC. Then

S(Cmax) C S(Cmax)= S(G e, (Pmax)C) implies

g.ox E Ge . ox 13 X~ = 0 = G r . ox (Lemma 5.1.11 and Lemma 5.1.12). This shows that S(Cm,x) = G n S(~m,x) r

S(G', Pm,x).

135

5.4. C O M P R E S S I O N SEMIGRO UPS Theorem

5.4.7

1) S(C) C H A N n N~AH.

~) S(C) = H ( S ( C ) n A N ) = (S(C) n N~A). 3) G ~ exp(C) n A N = S(C) n A N = S(C)o n A N is connected. Proof:. 1) L e m m a 5.4.6 shows t h a t S(C) C HPmax. In particular, we have S(C)o C (HPm~x)o, which is equal to ( G r ) o A N = H o A N by L e m m a 5.4.1. But then S(C) - H S ( C ) o C H A N . Since S(C) - r ( S ( C ) -1) we also have S(C) C r ( N ) A H . 2) This follows from 1) in view of g C S(C). 3) Since Ho C S(C)o, L e m m a 5.4.1 shows that S ( C ) o A A N is connected. Now the claim follows from G r exp(C) = GrS(C)o, S(C) = HS(C)o, and G" n A N = {1}. !:3 T h e o r e m 5.4.8 1) S(Cmax)n B ~ is a generating Lie semigroup in B~ with the pointed generating tangent cone

L(S(Cmax) n B ~) = (Cmax -~" ~) n (11.~ + ill.) ::) Cmax.

~) S(Cmax) n B ~ C

N ~ exp(Cmax).

Proof:. 1) Let G1 "- N~A x H act on G by (t, h ) . g - tgh -1. Then the orbit of 1 is the open subset N ~ A H of G. Moreover, the stabilizer of 1 is trivial. We define the field (9 of cones on N ~ A H by 6)(g) "= dlAg(Cmax q" ~)

Vg e N~AH,

where Ae : G ---} G, x ~ gx denotes left multiplication by g. We claim t h a t 6} is invariant under the action of G1, i.e., t h a t d,p(,,h)O(1) =

e(th-'),

where P(t,h) " N ~ A H --} N~AH, g ~ tgh -1. To see this, we first note t h a t P(t,h) -- At o ph-1 = Ath-1 o Ih,

where Ih : g ~ hgh -I. Therefore dp(t,h)(1)O(1) = dAth-~ (1)Ad(h)(Cmax + I~)= O(th -1)

is a consequence of Ad(H)(Cm~x % I))= Cmax % I). The semigroup S(Cmax)o is the set of all points in N ~ A H o for which there exists a E)(1)-causalcurve. Since this set is closed, the inverse image of S(Cmax)o under the orbit mapping ~ " G1 --~ G,

(t,h) ~-~ th -1

CHAPTER5.

136

THE G E O M E T R Y

is a Lie semigroup whose tangent wedge agrees with d(I)-l(1) (O(1)) = d(I'-l(1)(Cmax 4" I)) = D h" (Cmax q- t)) n (a -t- It~)

([1141, p. 471). We know that S(Cmax)o = (S(Cmax)n BI~)Ho and therefore r

(S(Cm.x)o n Bt) x Ho

=

is a Lie semigroup. We conclude with Theorem 5.4.7 that S(Cm~x n B~) = S(Cmax)o n B tl is a Lie semigroup with L(S(Cmax n BI~)) = L(S(Cmax)) n

(ll -~" nil) -" (Cmax q- ~))n (ll 4- nlI). 2) The mapping p: B ~ --+ A, nta e-r a is a group homomorphism because N~ is a normal subgroup of Bt. Therefore p(S(Cm,x) O Bt) is a subsemigroup of A which is contained in the Lie semigroup generated by

V "= dp(1)L(S(Cm~x) O B ~)

=

=

n,

(Cm.~ + b + ,") n , .

This cone is the projection of Cmax along Ij + n t onto a. Let w E Cmax C q. Then there exists X E a and Y E n lI such that w = X + Y - r ( Y ) . Hence

w e X + 2Y - [ r ( Y ) + Y] E X + n t + [j. Therefore V is the orthogonal projection of Cm~x in a. Thus V = a O Cmax = Cmax and hence p(S(Cmax) O Sil) = exp(cmax). Consequently, we find S(Cmax) n B ~ c N ~ exp(cmax). [3 L e m m a 5.4.9 S(Gr, Pmax) O e x p a = eXpCmax.

Proof:. The inclusion exp Cmax C S(G r, Pmax) is clear. To show the converse direction, we let X E a \ Cmax. Then there exists an c~ E A+ such t h a t a ( X ) < 0. Consider the subalgebra 5a = ~a(~[(2,1R)) described in (4.8). Then exp(RY ~) C (Gr)o, and it suffices to show t h a t e x p ( X ) e x p ( R Y ~ ) 9o cannot be contained in (9. L e m m a 5.4.1 shows that the map an" G r A N = GrPmax ~ a,

han ~ log(a)

is well defined and analytic. We call it the causal Iwasawa projection. When we restrict this map to the group generated by exp(~a), a simple SL(2, R)-calculation shows that

an ( e x p ( t X ~ ) e x p ( s Y ~) = t + ~1 l o g ( l + ( 1

-

e -zt )sinh2(~)) s ]X(,.

5.4. COMPRESSION SEMIGROUPS

137

Now choose X ' := X - toX ~, where to = a ( X ) . Then X' commutes with s~ and hence we calculate

an (exp(X)exp(sY~))

=

aH (exp(toX ~) exp(sYa)exp(X'))

=

an (exp(toX a) exp(sY~ + X' [ 1 ( (e2'~ - 1) sinh2(~))] Xo" x ' + t o + ~ l o g 1+ ~,:

=

But this function in the parameter s is not extendable to all of R so that exp(X) exp(sY~) 9o cannot be contained in O for all s e R. D So far we know that the semigroup S(G r, Pmax) contains the Ol'shanskii semigroup S(Cmax) and that the intersection of A with S(G r, Pmax) is not bigger than the intersection with the Ol'shanskii semigroup. The remainder of this section will be devoted to the proof of the equality S(G r, Pmax) =

S(Cmax). We start with a description of the open H-orbits in the flag manifolds

G/Pmi.. L e m m a 5.4.10 Let a', C p be 7-invariant maximal abelian subspaces such that a' O qp and a" n qp are maximal abelian in qp. Then there exists k G ( g r ) o such that Ad(k)a' = a".

Proofi (cf. also Lemma 7 in [99], p. 341.) Since the maximal abelian subspaces of qp are conjugate under K~ ([44], p.247), we even may assume that a' n qp -- a" n qp. Set go := ~0(a,,). We consider the symmetric Lie algebra (go, r) which is invariant under 0 and therefore reductive ([1681, Corollary 1.1.5.4). Then go n qp = a" is central in g0 and a' = (a" n qp) @ (a' n ljp). Hence a' n [jp c g0 n [~p is maximal abelian in I]p n g0. The same holds for a" n I] c l j p n go. The pair ([j n g0,0) is Riemannian symmetric, hence a" n lip and a' n tjp are conjugate under exp(bkn0 ~ ([44], p.247). We conclude that a' is conjugate to a" under (Kr)o. [3 Note that H-orbits in G/Pmin correspond to H-conjugacy classes of minimal parabolic subalgebras of g. According to [99], p. 331, each minimal parabolic subalgebra of g is (Gr)o-conjugate to one of the form m + a + n for some 7-invariant maximal abelian subspace a of p and some positive system in A(g, a). Let as, a 2 , . . . , as be a set of representatives of the K r conjugacy classes of maximal abelian v-invariant subspaces of p. [99], w Proposition 1 (cf. also [156], Proposition 7.1.8), among other things, says: L e m m a 5.4.11 Let A + be a positive system for h j := h(g, ai). Denote the corresponding minimal parabolic subalgebra by p(aj, A+). Then the H-

CHAPTER5.

138

THEGEOMETRY

+ conjugacy class of p(aj, Aj ) corresponds to an open H-orbit in G/Pmin if and only if the following conditions are satisfied: I) aj N q is maximal abelian in qp. 2} A + is q-compatible, i.e., the set A + \ (aj fl q)a. is --r-invariant.

0

We call a r-invariant maximal abelian subspace a' of p a q-maximal subspace if tt~ fl q is maximal abelian in qp (cf. [156], p. 118). Note that according to Lemma 5.4.10, q-maximal maximal abelian subspaces of p are conjugate under (Kr)o. This shows that only one of the ttj can be q-maximal; i.e., condition 1) can be satisfied only by one of the aj. The part of [99], w Proposition 1 we have not yet stated here concerns the number of open H-orbits. For a fixed r-invariant maximal abelian subspace a of p we consider the Weyl groups

w(,)

w.(,) w0(,)

= NK(II)/ZK(II), := {, e w ( , ) I ,(* n = , n := NKnH(a)/ZKnH(a).

Then W0(,) C W , ( , ) C W ( , ) , and [99], w Proposition 1 says that the number of open H-orbits in G/Pmin is the number of cosets in Wr(a)/Wo(a). R e m a r k 5.4.12 Let a be a r-invariant q-maximal abelian subspace of p and A+ a positive system for A(g, a). Then A+ is q-compatible if and only Pl,nq r 0 implies - r ( p ) E A+ for all p E A+ (cf. [156], p. 120, and [99],

p. 355)

o

L e m m a 5.4.13 1) Let a be a r.invariant q-maximal abelian subspace of p and A+I, A + two q-compatible positive systems.for A(g, a). Then the corresponding minimal parabolic subalgebras belong to the same open H.orbit if and only i.f there exists a s E Wo(a) such that s. A+x = A+2 .

2) Let a be a r-invariant q-maximal abelian subspace of p. Assume that A+(g,a) is q-compatible. Then the open H-orbits in G / P are precisely the HsPmin/Pmin ttrith 8 ~. WI.(I1). The orbits H$1Pmin/Pmin and Hs2Pmin/Pmin agree if and only if there exists an 8 G.. Wo(ll) with ssl = s2. In particular, I-IffPmin/emin i8 open if and only if g E HW~(a)Pmin. The union of open I-I-orbits is dense in G/Pmin.

5.4. COMPRESSION SEMIGROUPS

139

Proof:. 1) The Weyl group Wr(a) acts simply transitively on the set of el-compatible positive systems (cf. [156], Proposition 7.1.7]). Clearly, two q-compatible positive systems belong to the same H-orbit if they are conjugate under Wo(a). Now the formula for the number of open orbits implies the claim. 2) Only the last claim remains to be shown. But that follows from the fact that there are only finitely many H-orbits (cf. [99], Theorem 3, and [52], Proposition 8.10(ii)), since each orbit is an immersed manifold. Cl L e m m a 5.4.14 If P1 C P2 are parabolic subgroups of G, ~i = G./Pi the corresponding flag manifolds, xi E .T'i, and 7r:Jrl --~ ~'2 the natural projection, then the following assertions hold: ~) if H . ~1 i~ ope~ i , 71, the, ~ ( H .

~ a ) = H . ~(~a) - ope, i , J=2.

2) If H . x2 is open in J:2, then 7r-X(H 9x2) contains an open H-orbit in J:l .

Proo~ 1) follows from the fact that lr is open and G-equivariant. 2) r - l ( H 9x2) is open by continuity. Suppose for a moment that P1 is a minimal parabolic. Then Lemma 5.4.13.2) says that the union of open H-orbits is dense in ~'1 and therefore lr -1 (H. x2) intersects, hence contains, an open H-orbit. If we now apply 1), this argument shows that for any flag manifold the union of the open H-orbits is dense, and we can prove our claim for arbitrary P1. D L e m r n a 5.4.15 Let pi be an arbitrary parabolic, x E G / P ~, and P~ the stabilizer of x in G. Then the following statements are equivalent:

1) H . x is open in G/P'. 2) There exist a q-maximal maximal abelian subspace a~ of p and a elcompatible positive system A~ of A(g, a~) such that P~ is H-conjugate to a standard parabolic associated to A~. Pro@. In the case where P~ is a minimal parabolic, our claim is just Proposition 5.4.11. "1) =~ 2)': For the general case recall that G / P ~ can be identified with the set of parabolic subgroups of G conjugate to P~ and the natural projection rr: G/Pmin ~ G / P ' maps a conjugate gPming -1 of Pmin to gp,g-1. Identifying x and P~, one has that 7r- t (x) consists of all minimal parabolics contained in P~. Lemma 5.4.14 says that there is one such minimal parabolic P~ that lies in an open H-orbit. But then Proposition 5.4.11 shows that p~ is of the form p(a~, A~) for suitable a~ and A~, and thus P~ must have the right form since it contains P~.

C H A P T E R 5. THE G E O M E T R Y

140

"2) =~ 1)": For the converse we invoke Lemma 5.4.14.1) to see that the H-orbits of the parabolic group associated to a q-compatible system A~ are always open. E! L e m m a 5.4.16 Recall the flag manifold X := G/Pmax with base point o x .

Then the following assertions hold: 1) The open orbit H . o x

is contained in the open Bruhat cell N_ . o x .

2) Every other open H-orbit is not entirely contained in N_ . o x . 3) H . o x is the largest open subset of the open cell which is H-invariant. Proof. 1) This is a consequence of Lemma 5.1.12. 2) Let y E &' and suppose that the H-orbit of y is open and different from the H-orbit of the base point. Then it follows from Proposition 5.4.15 that there exists a point in this H-orbit which is fixed by the subgroup A. Since the base point is the only A-fixed point in the open Bruhat-cell, we conclude that H . y cannot be contained in the open cell. 3) Since the set of all elements in ,~' whose H-orbit is open is dense ([52], Proposition 8.10(ii)), this follows from the fact that H . o x is the interior of its closure, c] L e m m a 5.4.17 The following assertions hold:

1) HPmax = exp(f~-)HaN+ is the largest open H-lefl-invariant subset of N_HaN+ = N-Pm,x. 2) There exists an open bounded subset f~+ C n+ such that P~maxH = N _ H a exp(fl+). This set is the largest open H-right-invariant subset of Y _ H a N + . 3) Every H.biinvariant open subset of N_HaN+ is contained in the open set exp(f~_)H a exp(f~+). Proof:. 1) Lemmab.l.12 shows HPmax = e x p ( f t - ) H a N + and HaN+ = Pmax follows from Lemma 3.1.22. Now the claim follows from Lemma 5.4.16.3). 2) This follows from 1) by applying the automorphism T. 3) This is a consequence of 1) and 2). rl L e m m a 5.4.18 S(Gr, Pmax) ~ is the largest open H-biinvariant subset of

N_HaN+. Proo~ It follows from Lemma 5.4.16.1)that S(Gr, pmax) C HPmax C N_HaN+. Moreover, for every s E S(Gr, Pmax) the double coset H s H is contained in S(Gr, Pmax) and therefore in N_HaN+. This shows that

5.4. COMPRESSION SEMIGRO UPS

141

S(Gr, Pmax)~ is an open H-biinvariant subset of N_HaN+. Now suppose t h a t E C N - H a N + is an open H-biinvariant set. Then we first use L e m m a 5.4.17 to see t h a t E H C E C exp(~2-)H a exp(~+) C exp(f~-)Pmax = HPmax. It follows in particular that E C S(G~,Pmax). Thus S(G~,Pmax) is the unique maximal open H-biinvariant subset of N_HaN+. D C o r o l l a r y 5.4.19 S(Gr, Pmax) is invariant under the involution s ~ s~.

Proof. This is a consequence of L e m m a 5.4.18 because the set N_HaN+ is invariant under this involution and therefore the same is true for the maximal H-biinvariant subset of this set. o 5.4.20 Let .s = G / H be a noncompactly causal symmetric space and 7" the corresponding involution on G. Assume that G embeds into a simply connected complex group Gc and let Q be a parabolic subgroup between Pmin and Pmax = Ha N+. Then

Theorem

S(H, Q) = G ~ e x p ( C m , x ) = S(G~,Pmax). Proof. It follows from Lemmas 5.1.2 and 5.4.4 that we may w.l.o.g, assume that Q = Pmax and that H = G r. First we apply Theorem 1.4.2 to obtain further information on the semigroup S(G~,Pmnx). Recall the notation from Section 1.4 and let aq C q be a 0-invariant A-subspace. Suppose t h a t 7r-'(A~) O S(Vr, Pmax) ~ # 0. In view of Corollary 5.4.19, the semigroup S(Gr, Pmax) is invariant under the mapping rr and therefore we find s E Sr~ O Aq'. Next we recall that Aq = (Aq O K)AP, where A~ := exp(aq O qp). We consider the semigroup SA := S(G r, Pmax) ~ n Aq. Then the semigroup SAAPq/A~ is an open subsemigroup of a compact group, so t h a t it must contain the identity element (cf. [52], Corollary 1.21). We conclude t h a t SA intersects the subgroup A p. This subgroup is conjugate to a subgroup of A (Lemma 5.4.10). Suppose that Aq O K is nontrivial. Then aq O p is not maximal abelian in qp and the description of the W0(a)-conjugacy classes of A-subspaces given in [143], p. 413 shows that the conjugate of Aqp in A lies in the exponential image of the set U a e u + ker a. It follows t h a t this set contains interior points of S(Gr, Pmax). On the other hand, we know already t h a t S(Gr, Pmax)O A = exp(Cmax) (Lemma 5.4.9). This is a contradiction because every element in Cmax which is in the kernel of a noncompact root lies on the boundary. Thus we have shown that the only A-subspace Aq for which the open subset H e -1 (Aq) intersects S(G r, Pmax) ~ is A := Z~(G)(a). Let s e S(Gr, Pmax)n H r Then =

e Ji.

CHAPTER 5. THE GEOMETRY

142

We have to get a better picture of the set A. So we first remark that A Zr (A) - (MNr (Theorem 1.4.2). Let k 6. r Then k~ - k and on the other hand 7(k) = k by L e m m a 3.1.22. Therefore k = k -1, i.e., k 2 - 1. Moreover, the surjectivity of the exponential function of the Riemannian symmetric space g i g ~ ~- r yields that r ( g ) = exp(qk). We write ss~ = k exp(Z) with k 6. r N M and Z 6. a. Then we we find Y 6. q~ with k - exp(2Y). We set k ~ " - exp Y. We claim t h a t Ad(k~)a C qp. To see this, pick X 6. a. Then r (Ad(k')X) = A d ( k ' ) - I r ( X ) = - A d ( k ' ) - ' X = - A d ( k ' ) X and similarly

O( Ad(k')X) = Ad(k')O(X) = - Ad(k')X. This proves our claim. Now we find that k' exp(~lead Y Z)] [k' exp( lead Y z)] ~

k' exp(e ad r Z)k' -

k exp(Z)

--

88 ~

.

We conclude that kexp(

Z)(k')- 1 = =

k ' e x p ~l'l se

adyZ )

6. C~_1(s)

sH C S(Gr, Pmax).

We have already seen that Ad(k')a C qp. Hence there exists k" 6. (Kr)o such that Ad(k")Ad(k~)a = a (Lemma 5.4.10). This means t h a t k"k ~ 6 NK(a). Multiplying with k" on the left, we find that

k"k' exp(Z) 9x0 6. Ng(a).xo, so t h a t this point is an A-fixed point in G/Pmax. On the other hand, the semigroup S(Gr, Pmax) is contained in the set N-Pmax, so x0 is the only A-fixed point in the set S(G r, Pmax) " x0. Thus k"k' 6.

Pmaxn NK(a)

C Ha

nK

= Z K ( e ) C H C G ~.

It follows that k"k ~ 6- H and therefore that k ~ E H. Thus exp(Z) E A 13 S(G r, Pmax) C G r exp Cmax and hence s 6. G r exp Cmax, which finally shows that S(G r, Pmax) is contained in the semigroup G r exp Cmax. [3

5.5.

THE NONLINEAR CONVEXITY THEOREM

C o r o l l a r y 5.4.21 Recall the open domain 0 = G g . o x Then

S(Cm,,x) = {" e

a

143 in ,u = G/Pmax.

18. o c o } and S(Cm,,x) o = {8 e a l "" ~ C 0 } .

m

R e m a r k 5.4.22 One can show that G r exp(Cm,x) is actually a maximal subsemigroup of G (cf. [58], Theorem V.4). E! E x a m p l e 5.4.23 In the situation of SL(2,R)-Example 5.1.13 we have

{(OC

0) Vl~l
a-1

<1

.

An elementary argument shows that the condition on c and a can be reformulated as

S'Cm.x' O = { (OC 5.5

0) Icl < a-a-X;0 < a} .

a-1

T h e Nonlinear Convexity T h e o r e m

In this section we again consider a noncompactly causal symmetric space jr4 = G / H such that G is contained in a simply connected complexification Gc. We will prove Neeb's nonlinear convexity theorem, which says that a l l ( a l l ) = conv(W0, log a) + Cmin for all a E exp(Cmnx). This will be done first for the special case .M" = G c / G e (cf. Section 1.1). Then the general result can be obtained via the suitable intersections with smaller spaces. Note that in our situation G = (~. We write G c for GO. Recall the situation described in Section 4.5. In particular, let t e be a Cartan subalgebra of t c containing ia and A = A(gc, ~ ) . We set

:= Z (gc)~,

(5.19)

w h e r e / ~ + is chosen as on p. 95. Further we set fi := it r A := expfi a n d N "= exp ft. Then (5.18) yields a causal Iwasawa projection ao," G e A N --4 ft. The derivative dx av," gc -4 a is simply the projection along ge + ft. L e m m a 5.5.1 Let ~" gc ~ t~ denote the projection along the sum of the root spaces. Then ~[ig~ = dl aG, lio'.

C H A P T E R 5. T H E G E O M E T R Y

144

Proof:. Let X E ig e. Then we can write X = Y + Z - Z, where Y E ~ and Z e ~. Therefore ~ ( X ) = ~ ( Y + Z - a t ( Z ) ) = Y and dlaac(X)

=

dxae,, ( Y + Z - a t ( Z ) )

=

Y - dxac,, (at(Z))

=

Y - dxac,, ( g + a"(Z)) = Y .

From this the lemma follows. We note that for g E G e, a E A and n E/V we have a G r 0 ,,kg --" CtGr

and

aa, o Pan = a a , + log a,

where as usual )~g(x) - gz and pg(x) - xg denote left and right multiplication. We briefly recall the basic definitions concerning homogeneous vector bundles. Let L / U be a homogeneous space of L and V a vector space on which U acts by the representation r: U --+ GL(V). Then we obtain an action of U on L x V via u.(l, v ) : = (lu -1, r(u).v) and the space of U-orbits is denoted L x v V and called a homogeneous vector bundle. We write [l, v] for the element of L x v V which corresponds to the orbit of (l, v) in L x V and note that L acts from the left on L X v V by l.[l', v] := [ll', v]. If L is a complex group, U is a complex subgroup, and the representation 7" is holomorphic, the corresponding vector bundle is holomorphic. Fix a linear functional w E iE'max = /~'min (cf" (4.22)) such that iw integrates to a character X of T* = exp(tc) and w(i[ac(X), X]) _> 0 for X e (gr with & E s We put E := {& E A I(VX E (gc)~) w(i[ae(X),X]) >_ 0}. Then the subalgebra

~6E is a (complex) parabolic subalgebra of Oc. Let /~ be the corresponding parabolic subgroup of Gc and G~ the stabilizer of w in G c w.r.t, the coadjoint action. Then B N G c - G~ by Theorem 1.3 in [56], and we obt.ain a complex structure on the coadjoint orbit G c. w ~- GC/G~ by embedding G c. w as the open orbit GC.oi~ of the base point o h in the complex homogeneous space G c / B . We find a holomorphic character X " /3 -+ C with x(exp X ) = e i~(x) for X E b, where we set w ( X ) = 0 if X belongs to the sum of root spaces. Thus we obtain two homogeneous holomorphic line bundles: the line bundle E "= G c x a~ Cx and the line bundle E ~ -= Gc x ~ C x. The bundle E embeds as the open subset of E ~.

E'lo~

5.5. THE N O N L I N E A R C O N V E X I T Y T H E O R E M

145

Let q 9 G c x C -+ E denote the quotient mapping which identifies the elements (g, z) and (gh -1, x(h)z) for h E G c~. We define a function h on E by h([g, z])"= Iz[2 for g E. G c, z e C. We have already seen t h a t the bundle E inherits a complex structure by its embedding in the complex bundle E ~. We write I for the tensor field denoting multiplication by i in each tangent space. For a 1-form a on a complex manifold y we define a 1-form I a by (Ia, v) := ( a , - I v ) on each tangent space Tp ( Y ) . Let G c'~ := G c x C* and G~c := Gr x C*. Then G c'~ acts transitively on the complement E0 of the zero section in E and similarly G~ acts transitively on E~ by (g, ( ) . [g', v] = [gg~, ~v]. L e m m a 5.5.2 The 1-form a = I(dlog h) = ~Idh on Eo is invariant under the action of G c,~.

Pro@ Since the action of G ~,~ on E0 is holomorphic and G c preserves the function h, the GC-invariance is clear. On the other hand, we have for z E C* and pz([g,x]) = [g, zx] t h a t h o pz = Izl2h. Hence log(h o pz) = log h + log Iz]2. Thus ~,:(dlog h) = dlog(h o m ) = d(log h + log [zl 2) = dlog h. This proves the assertion.

El

To calculate the 1-form a, we have to calculate its pull-back q*a to the group G c x C* which is a left-invariant 1-form on this group. Its value in the unit element (1, 1)is given by ( q ' a ) ( 1 , 1 ) = a([1, 1])d(1,1)q = - d ( l o g h)Id(1,1)q. To calculate this expression, we have to pass from q to the mapping q~ 9Gc x C --+ E ~, which restricts to q on G c x C. The calculation of q~*a on Gc x C* in the unit element is easier since q~ is a holomorphic mapping: (q'*a)(1, 1) = - d ( l o g h)Id(1,1)q' = - d ( l o g h)d(1,1)q'I = -d(1,1) log(h o q')I. The function h o q' is given on the subset GCAN x C* of Gc by

h

o

q'(gan, z)

= --

h([gan, z]) = h([an, zl) = h([1, x(an)z]) h([1, x(a)z]) - Ix(.)121 l 2

and therefore

h o q'(s, z) = e 2('~'aac(s)) Izl

(5.2o)

for s E GCfiN. This entails t h a t log(h oq')(s,z) = 2(iw, aac(S)) + log]z[ 2 and permits us to compute the differential of log(h o q~) in (1,1): d(1,1) log(h o q')(Y, () = 2iw o dlaG~

+ 2 Re(().

C H A P T E R 5. THE G E O M E T R Y

146

Finally, we use Lemma 5.5.1 to calculate the form q*a in (1,1):

(q'.)(z, z)(x, r

= = = =

- 2 i w o dxaa~ - 2Re(ir - 2 i w o ~(iX) + 2 Im - 2 i w ( i X ) + 2 Im 2w(X) + 2 I m p . r-i

This proves the assertion. L e m m a 5.5.3 For X E ge, ~ E C, and z E C* we have

c,{[g, z])d(g,z)q(dzpg(X), z~) = 2(Ad* (g).w, X) + 2 Im ~. Proof:. We write [g, z] = (g, z). [1, 1] = #(g,z) ([1,1]). Therefore Lemma 5.5.2 yields that c~([g, z])d(g,z)q(dspg(X), zr

zr

=

=

(q'a)(g, z)(dx)~g(Ad(g - s ) x ) , zr

= =

(q*a)(1,1)(Ad(g-X)X,r 2(Ad*(g).w,X) + 2 I m r

Now the Lemma follows. C o r o l l a r y 5.5.4 For X E ig c, g E G ~, ~ E C, and z G C* we have - d l o g h([g, z])Id(g,z)q(dzpg(iX), z~) = 2(Ad*(g). w, i X ) + 2Imp. P r o p o s i t i o n 5.5.5 For X G ige let m x := sup(Ge" w, iX) and define the vector field &(iX) on E by (r(iX)(p) := d/dt e x p ( - t i X ) . Plt=o for p E E. Then (d v log h, Hr(iX)(p)) < 2rex for art p ~ Eo. Proo~ Let p = [g, z] E E0. Then d/dt e x p ( - t i X ) . [g, zll,=o d / d t [ e x p ( - t i X ) g , ~]1,=o

=

d(g,z)q( -- dlPg(iX), O) and therefore (dr (log h ), I&(iX) (p))

= =

(dvlogh, I d ( g , , ) q ( - d s p g ( i X ) , O ) 2(Ad*(g). w, i X ) <_ 2rex.

5.5. THE NONLINEAR CONVEXITY THEOREM

147

F r o m this the proposition follows. [] Consider the compression semigroup S "-- S(G c,B) in Gc. T h e n S acts holomorphically on the bundle E ~ and since S leaves G e . o B invariant,~the action on E ~ leaves the subbundle E invariant. Note that S C GOB = G C A N since i~ C gc. Therefore we can write each s E S as s - gan with g E G c, a E A, and n E N and we find with (5.20) that ..

~

log h([s, 11) = 2(iT, aa, (s)). ~

It follows in particular that log h([a, 1]) = 2(iw, loga) for a e A f3 S = exp(fim~) (cf. L e m m a 5.4.9). Fix g E G c and X E C ' m a x - - (/~+)*. We set F(t)"= logh(exp(tX).[g, 1]). Then exp R + X C S and therefore exp t X . [g, 1] = [exp tXg, 1] e E0 for all t _> 0. Hence we can use Proposition 5.5.5 to see that

F'(t) = (d(logh),Ib(iX))([exptXg, l]) < 2rex. Therefore

2(iw,aa~(expXg)) --logh(expX.[g, 1]) -- F(1) _< 2rex.1 - 2rex. (5.21) W e want to use the linear convexity theorem (Theorem 4.3.1) to calculate m x for X E Cmax- To this end we recall that our assumptions on w, say in particular that w E /train - /Cmax, cf. (4.23). Let pr: igc ._~ fi be the orthogonal projection. Then Theorem 4.3.1 implies that

pr(G ~. [-iw]) c conv[~o. (-i~)] + CmJn, where l~0 is the Weyl group for (t c, t c) (cf. Section 4.5). Since X E -* it follows that Cmin ~

m x = sup(iX, G c. w) = sup(iX, conv(W0- w)).

Cmax - -

(5.22)

Now we obtain with (5.21) and (5.22) (iw, ao, (exp Xg)) < sup(X, l/Vo. (iw)) for X e 5m~x. Recall the cone (/~+)* - C from Section 4.5 and consider the set

:={

ea*lWes +.

ez}

of integral weights in fi*. Then

:~+ "= {,,, e a*lV,~ e s is the set of dominant integral weights.

I ~) ('~ I ~ ) e ~ }

2(iw

(5.23)

C H A P T E R 5. THE G E O M E T R Y

148

L e m m a 5.5.6 The cone iCmnxN(-iC) is generated by its dominant integral elements which integrate to a character of T e.

Proofi Let T := {50, &x,- ~ , &k} be a basis for /~+ such that To := { & l , . . . , &k } is a basis for A 0 . Recall that there is an element X ~ e i3(t ~) such that 51 (X ~ = 1. Let )~0 e (t~)* be determined by )~0(X ~ = 1 and )~0(X) = 0 for all X e t e gl [t ~, re]. According to [79], p. 85, each dominant integral element of the lattice k j=l

integrates to a character of T e. Let d be the maximal distance between elements of T~'. Then, given e > 0 and an element w in the interior of iCmax CI ( - i C ) , we can find an n e N and a u e T~' such that [nw - u I < d and ~d < e. Thus Iw - 1 u I < , n

and v E iCmax gl ( - i C ) for e small enough. Therefore it suffices to show that v is dominant. But that is clear since v e - i ( / ~ + ) *. [] P r o p o s i t i o n 5.5.7 Let X e ( s

Then a a , ( e x p ( X ) G e) C X +

C-rnin

--

C*.

Proof:. Let w (5 iCmax fl ( - i C ) be dominant integral_and such that it integrates to a character of T e. Then Wo 9(iw) E iw C* by Lemma 4.5.5, so that sup(X, l'Vo" (iw)) = iw(X). (5.24) ,,~

Combining this with (5.23) yields (iw, X Lemma 5.5.6 proves that

a a o ( e x p X a e ) ) C R +. Now

X -aGr (exp(X)G e) C [ic'~nin f'l (--iV)]* = --C'min + C*, i.e., aG,(exp X G ) C X + Cmin- C*. P r o p o s i t i o n 5.5.8 Let X E Cmax and a = exp(X). Then the set aG, (aG c) is invariant under the Weyl group Wo. Moreover, if Y E aG,(aGe), then

conv(Wo. Y)c aa,(aG ). Proo~ Set F : - aa,(aGe). The Weyl group ITV0is generated by the reflections sa, where & is a root contained in the set To simple roots in ,~+. We

5.5.

THE NONLINEAR

CONVEXITY

149

THEOREM

claim that the line segment {Y, sa(Y)) between Y and s a ( Y ) is contained in F whenever Y e F (cf. [45], p. 477). Let & e To and denote the complex image of the homomorphism ~oa:s[(2, C) --+ 9c by s~. Note that s~ C t~. We set ~es Note that & E To implies sa (/~+ \ (&)) C ~,+ since s Therefore

is IYV0-invariant.

According to [45], pp. 440, 477, we have the semidirect decomposition N = /~, • where N' - exp fi' and ~ a _ exp(gc)a. Let Y E F and b = exp(Y). Then there exist g,v E G c and n E N such that av = gbn. We decompose Y = Ya q- Y~, where Ya E RXa and Y~ E ker &. Then s a ( Y ) = sa(Ya) + Y ~ = - Y a + Y :

and {Y,s~(Y)} = [-1, llY~ + Y#. We put ba := exp(Ya), b~ :--- exp Y# and write n = n a n ' in accordance with N =/f/a/f/,. Then g - x av = bn = bab arian • ' = ba nab-kn '.

(5.26)

Let ca e exp([-1,1]Ya) and let S'~ be the group generated by exps~. Then S'~ C K~. Using Lemma 10.7 in [45], p. 476, we find elements ka, va E S~.~ fl K c and n 0a E / ~ a such that o k a b a n a v a = cana.

(5.27)

Now [Y~ fq a, (gc)a] = {0} and (5.26) imply that kag -1 avva = can 0a v a--1 b a• n v, a =

caba• n a0 y a--1 n'va.

We use (5.25) to see that n oa vyaXWv a ~. na0 N" , C N . Thus a a . ( k a g - X a v v a ) = a a . ( a v v a ) = log(cabS)= logca + Y : .

Since ca was arbitrary in exp([-1, 1]Ya), we conclude that {Y, s a ( Y ) ) C aao(aGC).

This proves the I~o-invariance of aa, (aG c) because l~o is generated by the reflections sa for 5 simple. Let/3 E Ao. Then there exists w E Wo such that w. fl E T and we have for each Y E F that w { w - X Y , s$w -x 9Y ) = {IF, w s $ w -x . Y} = {IF, s,o.~ 9Y} C F.

C H A P T E R 5. T H E G E O M E T R Y

150

Now L e m m a 10.4 in [He184, p. 474] implies that conv(l~V0. Y) C F for every element Y E F. [::] P r o p o s i t i o n 5.5.9 Let a E exp(cmax). Then ~

ago

~ C co

(W0, log

+ min.

~

Proo~ Applying a suitable element of W0, we may assume that X : - log a E (s because the sets on the right- and left-hand sides do not change if we replace X by w. X for w e W0 (Proposition 5.5.8). Now Proposition 5.5.7 entails that ~

aG,(aG c) C X + Cmin - C* ~ and since the set on the left-hand side is invariant under l~0, again by Proposition 5.5.8, we conclude with L e m m a 4.5.5 that

since C'min is l~'0-invariant.

D

Recall that G ~ C G c, A C ,4 and N C N are the aC-fixed points of the respective groups. Therefore aGr commutes with a c and the map aG," G r A N --+ a is simply the restriction of aG~ to G r A N . In view of Theorem 4.5.6, this implies that GG.

(aG v) C fl N [conv(l~0. log a) + c-rain] "- coIIv(W0" log a) + Cmin (5.28)

for a E exp(cmax). In order to prove the converse inclusion we need some additional information. Note first that Proposition 3.2.2 yields the following lemma. L e m m a 5.5.10 Let T be a basis of the system A+ . Then To := T n Ao+ is a basis of A + and T contains exactly one root not contained in A0. [:] L e m m a 5.5.11 Let C = (A0+)*. Then the highest root 7 in A + satisfies Cmin C R + 7 -

C*.

Pro@ We note first that the considerations in Section 4.1 show that A is an irreducible root system. Further, we note t h a t the highest root automaticaUy is contained in A+. Let T - {c~0, a l , . . . , ak} be the simple system for A + such that To = { a l , . . . , ak } is the simple system for A0+. Now suppose t h a t / 3 = ~jk=o m j a j E A+ and 7 = ~jk=o njoLj. Then no = m0 = 1 and 7 - fl = ~jk=l (nj - m j ) a j e A+o C C*. Therefore/~ e 7 /~ E A+ which implies the claim since Cmi n --'-- cone(A+).

C* for all [::]

5.5.

THE NONLINEAR CONVEXITY THEOREM

151

L e m m a 5.5.12 Let log a - - X E Cmax and c~ E A+ be such that ~ ( X ) > O. Then X + R + X '~ C aa. (aGr). Pro@ hi(2, R)-reduetion yields aG. (exp(X) exp ]RYa) = X + P.+X a

and this implies the claim (cf. Lemma 5.4.9 and its proof).

O

T h e o r e m 5.5.13 ( T h e N o n l i n e a r C o n v e x i t y T h e o r e m ) Let .h/[ be a noncompactly causal symmetric space, a C qv a maximal abelian subspace, and all" H A N ~ a the corresponding projection. Then

all(all)

=

cony(W0, log a) + Cmin

for 1 # a E exp(Cm~x). Pro@. Note first that Lemma 3.1.22 implies that we may assume H = G ~. In view of (5.28), we only have to show the inclusion D. Replacing X = log a by a suitable W0-conjugate, we may also assume that X E C = (A+) *. Since X # 0, there exists a c~ E A+ such that t~(X) > 0. Let 7 E A+ be the highest root of A+. Then 7(X) _> a ( X ) > 0 and hence Lemma 5.5.12 implies that X + R + X 7 C a l l ( a l l ) . Now Proposition 5.5.8 implies that it suffices to show

cony [W0. (X + R + X ' ) ] = conv(W0. X) + Cmin-

(5.29)

To do this, note first that R + X 7 = R+7 C C m i n and that both sides of (5.29) are closed, convex, and W0-invariant. Thus it remains to verify [conv(Wo. X) + Cmin] [7 C C conv (Wo. (X + R+XT)) rl C.

(5.30)

According to Lemma 5.5.11 and Lemma 4.5.5 we have cony(W0 9X) + Cmin C ( X - C*) + (R+') ' -- C*) -'~ ( X + R+') ') "- C*. Note that ( Y - C*)gl C C conv(W0. Y) for all Y E C by Lemma 4.5.5. But [8], w Proposition 25 implies that 7 E C. Thus for any r > 0 we have [(x + rT) - C*] N C C conv [W0. (X + rT)] N C. This implies (5.30) and hence the claim.

El

C o r o l l a r y 5.5.14 I) Let a E exp(Cmax) and n ~ E N ~ N H A N . an ~a-1 C H A N and

aH(a~t~a -1) -- aH(~t ~) E

train.

Then

152

C H A P T E R 5. T H E G E O M E T R Y

2) aH (N 1~n H A N ) C iCmin. Proof:. 1) Let PH " H A N ~ P be the projection onto the H-factor. Then =

+ ,,,(u).

,,,

Therefore

al.I(anta -x ) = ai.t (apI.i(nt)) + aH(n t) -- log a. Now Theorem 5.5.13 shows that

aH (apl-i(n~)) ~. a l l ( a l l ) C log a + Cmin and this implies the claim. 2) Let X E (Cmax N C) ~ and n~ ~. N ~ gl H A N . lim e x p ( t X ) n ~ e x p ( - t X )

Then =

1

t--~oo

and therefore

---aH(n ~1 = lim aH ( e x p ( t X l n ~ exp(--tX)) -- aH(n ~) E. Cmin. t .-+ o o

E x a m p l e 5.5.15 For G = SL(2,R) the nonlinear convexity theorem can made very explicit. In the situation of Example 5.1.3 we have W0 - {1} and Cmax = Cmin = R + X ~ The causal Iwasawa projection is given by

e

d

whenever it is defined.

5.6

The B

[:3

-Order

Let .M - G / H still be a noncompactly causal symmetric space such G is contained in a simply connected complexification Gc. We write S for the maximal real Ol'shanskii semigroup S(Cm~x). Then Theorem 5.2.7 implies t h a t .M+ = S . o = ( S N B ~ ) . o . This shows that many questions concerning the positive cone of j~4 can be treated via B ~, which has a fairly simple structure. R e m a r k 5.6.1 Theorem 5.2.7 implies that S n B~ = {b ~ E B ~ [ o __ b~. o}. Therefore the restriction of _<s, cf. (2.14), to B ~ defines an order. On the other hand, S N B~ defines an order <_snB~ on B~ via

b <_snBu b'

:r

b' ~. b(S I'1 B~).

5.6.

THE B~-ORDER

153

We claim that the two orders agree. This follows from

Sb ~ O B ~ = b~S -x O B ~ = b~ (S -1 n B ~) = b~ (S o B~) -1

This means that in particular we can use the notation S b~ without any ambiguity. H P r o p o s i t i o n 5.6.2 The map IBm" Y#(G) -+ Y# ( B ) ,

f ~ f n B~

is B~-equivariant, continuous, and surjective. It is injective on the closed set { F e ~'~ ( G) I F~ C N ~A H } . Proof. The equivariance is obvious. Let F e Jr,(G) C ~'(G) H (cf. Lemma 2.4.1). Then IBu (F) = F n B ~ is closed and for s e S O B ~ we have that ( f O B ~) s -1 C F ( S n B~) -1 n B ~ = f o B ~, whence f n B~e ~

(B~).

Let F , -+ F in ~'$(G) and assume that Fn O B ~ ~ E. To see that E = IBm(F), let e e E and $, e F , O B ~ with ] , --+ e (cf. L e m m a S.l.1). Then e = limfn E limFn = F. On the other hand, for f E F N B ~ there exists a sequence $, E F , with f , --+ f. Since F n H = Fn, we have that F , = ( F , n B ~) H, so we find that bn e Fn n B ~ and hn e H with f,, - b n h n . According to L e m m a 5.4.1 we get that bn "+ ~f and hn --+ 1. Thus f = lim bn e lim a(Fn) = E. It follows that E = Io~ (F). For E E ~$ (B ~) we set ~ ( E ) := E l l . Let s - gh e S, where g e B ~ O S and h E H. Moreover,

Hg-' C S -I = (S -I n B ~) H by Theorem 5.4.7. Thus

~ ( E ) s -1 = E H h - l g -1 = E H g -1 C E (S -1 n B~) H = ( $ s n B ~ E ) H = E H

shows $/3(E) = ~(E). The inclusion E C IB, (~(E))= E H n B"

C H A P T E R 5. T H E G E O M E T R Y

154

is clear. Let bII ~. E H Iq B 1I and en E E, hn E H with e , h n -r b~. Then en -+ b and b E -E = E. It follows t h a t / ~ ( E ) gl B ~ = E, and hence IBu is surjective. If F E ~',(G) with F ~ C NII A H , then

F

=

F-'Z= [ F o N B ~ ] H

=

((FNB~)~

=

(F fl BI~)H = IB, (F)H.

Here we used that FFIB tl E Y , (B~), so Ff3B~ has dense interior by L e m m a 2.4.7. Now we see that IBa(F') = I B , ( F ) and (F') ~ C N~IAH imply F ' = F. It remains to show that the set { F E Jr~(G) l F ~ C NI~AH} is closed. We let Fn ~. ff:$(G) with Fn --r F and F~ C BIIH. We have to show t h a t F ~ C B~H. For f E F ~ there exist an f ' E (t f)o N F ~ and no E N with

F.n(T/)~176

Vn>no.

Pick fn in this set. Then

f E ($ f,,,)o C F~ C BIIH which proves that F ~ C. BIIH. L e m m a 5.6.3 Consider the order compactification map r/B," B ~ ~ Y,(B~),

g ~ g(S fl B~) -1

(cf. L e m m a 2.4.2).

1) II x e (L(S n

the,, nm,-,oo

( e x p ( - t X ) ) - O.

Pro@ 1) We consider the projection p : B ~ _~ N ~ x A--r A

(5.31)

and set X ' := dxq(X). Note that r/B, ( e x p ( - t X ) ) i s decreasing in t, so it has a nmit by L e m m a B.I.1. Suppose t h a t g E limoBu ( e x p ( - t X ) ) , t - r c~. Then there exist tn E R and sn ~. S FI B I~ with tn -+ C~ and

5.6. THE B~-ORDER

155

g = lim,-+oo e x p ( - t . X ) s ~ x. Thus

P(g)

=

lim e x p ( - t . X ' ) v ( s . ) -1 I'1--I, o o

-=

lim e x p ( - t n X ' ) exp(-Cmax)

n--I, oo

exp ( - lim[t.X' + Cm.x]) = 0

because of Theorem 5.4.8.3) and the fact that t , X ' + Cmax --~ q) whenever

x' e ~o

= alp (L(S n B~) o) and t. - . oo.

2) In view of 1), this is just a special case of Lemma 2.4.3.3).

t3

L e m m a 5.6.4 The restriction of I m to ri(BII) C Y~(G) yields a homeomorphism ri (BI~) --r riB, (B~) C 2=4(B~).

Pro@. Remark 5.6.1 implies that IB, (ri(g)) = riB, (g) for all g E B ~. Using the continuity of IB, we find that IB,(ri (BII)) C IB, (ri(BlI))= ribs (BII). Since IB, is B~-equivariant and ri is even G-equivariant we have

I0, (o(B~)) = B~. O,,(1) and hence IB~ (rl(Btl)), which is closed because of compactness, contains the closure of the B~-orbit of rim (1), i.e., all of rim (B~). Recall that for F E ri (B~) we have

F = Sg = gS -1 C BtI(B~H) = BIIH and therefore also F ~ C BIIH since B~H is open in G. Now again by the continuity of IB, we get F ~ C BIIH for all F ~. ri(BIIH), and then Proposition 5.6.2 shows that IB~ restricted to ri(B~) is injective. Finally, compactness yields the claim. [:1 L e m m a 5.6.5 We have ri(A) = 0 O A . r i (exp(cmax)).

If ri(a.) --+ F # 0, then the sequence an E A is bounded from below with respect to the restriction of the ordering <s to A.

CHAPTER5.

156

THEGEOMETRY

Proof. Using Lemma 5.6.4 and Lemma 5.6.3, we see that _

_

= =

[lira r/B, ( e x p ( - t X ) ) H ] lim riB, ( e x p ( - t X ) ) H

t--}oo

limo(

t--+~

xp(-tx))

for all X e L ( S I"IB~) ~ Suppose that O(an) -} F # O. Then Lemma 5.6.3 and Lemma 5.6.4 yield

r/B, (a.) --4 F n B r 0. Let f E IntB, F and a "= p ( f ) E A. Then there exists no E N with f ~ s a , for all n > no. H e n c e a = p ( f ) <_s an for all n >_ no. Pick to with e x p ( - t o Y ~ _~ a. Then

exp(toY~

e S n A = exp(Cmax)

(cf. Lemma 5.4.9) and

O(an) ---} exp(-toY ~ l i r n 7/(exp(toY~

e A . r/(exp(cmax)).

This proves the claim. T h e o r e m 5.6.6

D

1) .A,4opt = G. ,.,+ /tzcPt u

2) .A,4~pt has only finitely many G-orbits. Proof. 1) follows from Lemma 5.6.5 and Lemma 2.4.3. 2) is a consequence of 1) and (6.8).

El

The point of Lemma 5.6.5 and Theorem 5.6.6 is that they will enable us to derive the G-orbit structure of f14~nt from the structure of o(S n A) c cpt j~/l+. We conclude this section with the useful observation that the projection p: B~ ~ N~ x A ~ A defined in (5.31) is proper. This is an immediate consequence of the following more general lemma. L e m m a 5.6.7 Let B be a connected Lie group and N a closed normal subgroup such that A := B I N is a vector group. Suppose that C C b is a pointed closed convex cone such that C N n = {0} and S the closed subsemigroup of B generated by C. Then the homomorphism r B --} A, b ~-} bN induces a proper semigroup homomorphism zr: S ---} r

5. 7. T H E A F F I N E C L O S U R E OF B~

157

Pro@ Let D " - d1r Then the condition C N n - {0} shows t h a t D is a pointed cone in the abelian Lie algebra a. Since A is a vector group, we can identify c~ with A. Let w e Int{v e a* J V X e D " w(X) >_ 0}. T h e n w c a n b e v i e w e d a s a function on A and then the function f : - w o r satisfies the hypothesis of Theorem 1.32 in [Ne911 because it is a group homomorphism, hence has biinvariant differential. So we find that the order intervals sS -1 N S in B are compact. Let K C r be compact and L the maximal value of w on g . Then ~ r - l ( g ) C f - I ( [ 0 , L ] ) N S. Now Theorem 1.32 in [114] implies also t h a t there exists a left invariant Riemannian metric d on B such that the length L(7) of 7 is not bigger than L for all curves 7 [0, T] -+ B with 7(0) = 1 and f ( 7 ( T ) ) _< L, which are monotone w.r.t. _<s. Therefore d(x, 1) < L holds for all x e r - x ( g ) . Finally, the theorem of Hopf-Rinow shows t h a t these sets are compact. Q

5.7

The Affine Closure of

Retain the hypotheses and notation from Section 5.6. In this section we realize S N B ~ as a semigroup of affine selfmaps and in this way find a suitable compactification which helps us to make the abstract order compactification much more concrete.

1) B ~ is a twofold semidirect product B~ ~- N_ >4(N~o x

Lemma 5.7.1

A). 2) Let a e A+, fl e A+, and X O e gO" Then [X0,ga+,0] # {0} whenever c~+ (n + 1)/3 E A. In particular, if (~ [ [3) ~k O, we find that

5:, z

# {0}.

3) The mapping B ~ --+ N_ >~ A u t ( N _ ) ,

I..oO

(n_, n~0,a) ~ (n_, I , ,0a),

._

-',

,..

injective homomorphism. Pro@ 1) The first assertion follows immediately from the fact t h a t n = n_ >4 n~0, which is a consequence of n~0 = 3,, (y0). 2) Recall the algebra s o ~- ~[(2,R) from Section 4.1. The space

V~,~ := ( ~ ga+n~ nEZ

C H A P T E R 5. T H E G E O M E T R Y

158

is an s~-module. The above decomposition of Va,~ is precisely the H aweight decomposition. Suppose that a + (n + 1)fl E A. Then there exists a simple $~-submodule V of V~,~ with

v n

# {o}.

But now the classification of $[(2, R) modules says that m+

V =

~

V ~+m~,

t71~ 171_

where V '~+m~ = ga+m~ CIV is one-dimensional. If fla+,# C ker ad Xf~, then n = m _ - 1, since ft,+(,+1)~ f'l V ~ {0}, and [X~, V "+"a] = V ~ for m = m _ , m _ + 1, ..., m+. Thus

(a + n~)(Ha) < (o~ + m_13)(Ha) < O. Now [10], Chapter VIII, w no. 2, Proposition 3(iii), yields a contradiction to ft~+,# C ker ad X#. 3) It follows from the fact that Cmax is pointed and generating that the kernel intersects A trivially. Now 2) shows that the kernel also intersects No~ trivially and the assertion follows as Inmoa = Inn~Ia is the Jordan decomposition when we identify n_ with N_. [3 Recall the flag manifold Xc = Gc/(Pmax)C and its base point o x = 1 (Pmax)C from Section 5.1. L e m m a 5.7.2 The mapping ~ " n_ -+ ( N _ ) . o x , X ~-~ exp(X) 9o x is an equivariant mapping of B~-spaces. Here the action of B ~ on n_ is given by

(N_)N~oA x n_ --r n_, (exp(X) exp(Y) exp(Z), E) ~-~ X + e -d Ye ad ZE. For E = ~ a e a _ Ea with Ea E ga we have that end Z E = E

e~(Z)Ea'

VZ E a

aE~_

Proofl Let X, E E n_, Y E a~, and Z E a. Using that n_ is abelian, we have

xp(X) =

exp(Y)exp(Z) exp(E)(Pmax)C

-

exp(X)exp(Y) exp(Z) exp(E) e x p ( - Z ) .

=

9e x p ( - r ) exp(V) exp(Z)(Pmax)C exp(X + e ~l Ye ~1 z E ) exp(V) exp(Z)(Pmax)C

=

exp(X + e aa Ye *d ZE)(Vmax)C

5.7. THE AFFINE CLOSURE OF B~

159

because e x p ( Y ) e x p ( Z ) E H a C (Pm,x)C. Recall the domain f~_ C n_ from (5.7) on p. 123. We set

(5.32)

Aft(N_) := N_ ~ E n d ( N _ ) = N_ ~ E n d ( a _ ) and

Affeom(N-) := {(n_, ~/) e

(5.33)

Aft(N_) I n_7(~_) C ~-}.

Here we identify N_ and n_ via the exponential function of N_. We refer to these semigroups as the affine semigroup of N_ and the a,ffine compression semigroup of f~_. P r o p o s i t i o n 5.7.3 Affeom(N_)OB ~ = S O B ~, where B ~ is identified with a subgroup of Aft(N_) via L e m m a 5.7.1.

Proof. In view of L e m m a 5.7.2, the claim follows from S O B~ = {b~ E B ~ I b1~. 0 C O} and 0 = exp(f~_) 9ox. [3 P r o p o s i t i o n 5.7.4 Affcom(N-) is compact.

Proo~ Note first t h a t Affeom(N-) is closed in Aft(N_). Now let (n_, ~,) e Aff eom ( N - ). Then n_ = (n_,-),). I e ~_ c N_

so t h a t c

c

Since f~_ is a compact neighborhood of 0 in n_, we can find a norm on n_ and a constant c > o such that ll3'll _< c for all (n_,-y) E Affeom(N-). In other words, Aft,ore(N-) C {(n_, 7) e A f t ( N - ) I n e ~ _ ,

11711 <

c}

is relatively compact, hence compact. m

L e m m a 5.7.5 The action of Affcom(N-) on l~_ extends to a continuous action of Affcom (N_) on :7:(f~_).

Proo~ This follows from the more general fact t h a t E n d ( N _ ) acts continuously on C(N_), the set of compact subsets of N_ equipped with the Vietoris topology for the one-point compactification of N_. To see this, let Kn ~ K in C(N_) and sn ~ s in E n d ( N _ ) . Let V be a symmetric neighborhood of 1 in O(N) and V another symmetric 1-neighborhood with

C H A P T E R 5. T H E G E O M E T R Y

160

V C U and s . ( V ) C U for all n E N. Take no E N such that Ko C K V and so(K) C s ( K ) U for n > no. (Note that so converges uniformly on compact sets). Then (cf. B.1.2) s , ( K , ) C s , ( K V ) = s , ( K ) s , ( V ) C s ( K ) U 2. Since U 2 is symmetric, we also conclude that s(K) C s , ( K , ) U 2. Hence s , ( K , ) ~ s(K). D Let B--'-[denote the closure of B ~ in Aft(N_) and r" A the closure S N A = expcmax C B-'~ of S N A in B'-T. Then B'---~, S N B~ and ~.~'cPtAaxe compact semigroups. We describe the structure of cent ~'~'A " T h e o r e m 5.7.6 (The structure of cept ~'A ) Let the notation be as above. Then the following assertions are true:

1) cept= exp(R+Xl) 9 ij A

ooo

9exp(R+Xn), where

Cmax = E n i=1

R+Xi

9

~.) Fo,- F e F~(-~,,.,,) = Fa(con~(~_)) ~e de~,~e eF e E n d ( . _ ) by eF(X) = f o, l X,

if X E g~,a ~ F N A _ , if X E g~,a E F N A _ .

Then the mapping F ~ eF defines an isomorphism of Fa(-Cmax) and the lattice of idempotents E(SCApt) of the compact abelian semigroup ~ AcPt 3) For X ~_ Cm&x we have that lim exp(tX) = eF,

where

F = X • N (-Cmax),

(5.34)

and conversely, eF

=

lim exp(tX)

t--+oo

for every

X e IntFx(Cmax Cl F •

cept = (S N A) . E(S~APt). 4) ~A Proof. 1) is obvious. 2), 3) Let F E Fa(-Cmax). Then there exists an element X E Cmax with F = X • N (-Cmax). The functions t ~ e a(tX) are decreasing for all * x. More precisely, E --Cma =0,

~(x)

if c~ E F


5.7.

161

T H E A F F I N E C L O S U R E OF B~

This shows t h a t lim exp (tX) - e f.

t-+oo

It is clear t h a t e~ - eF and therefore eF e E(SAPt). Let e e E ( S f t ) . Then there exists an element X E Cmax such that e -- limt_+co exp(tX) because Cmax is polyhedral ([154], pp. 11, 26). Thus e = eF for F = X • and with IntF~ (Cmax CI F •

- {Y e Cmax [ Y• N -Cma x - F }

(5.35)

we find t h a t I n t f • (Cmax N F •

= {X e

Cmax

I lim e x p ( t X ) = eF}. t-,co

This proves t h a t F ~ e f is a bijection. Since it is clearly order-preserving, the claim follows. r with si ~. exp(R+Xi). Then either si ~. 4) Let s = S l . . . . . s , E ~'A exp(R+X/) C exp(cmax) or si = limt-~co e x p ( t X i ) ~. E ( S f t ) . Thus s e exp(cmax)E(SAPt), r'l E x a m p l e 5.7.7 In the situation of the SL(2, JR) Example 5.4.23,we identify Aft(N_) with ]R x JR, where the multiplication is

(~, ~)(~', ~') = (~ + ~ ' , ~ ' ) and the action on ]R - n_ is given by (n, 7)" n' = n + 7n'. Then Aff~om(N-) corresponds to the set

{(~, 7) e R x R I Vl~l < 1 =

{(~,7)

9 I-

+ 7~1 < 1}

e R x R IITI < 1, I~1 < 1 - 1 7 1 } .

The embedding of B ~ into Aft(N_) is

(o ~-10) ~+(~,

so the image of B ~ is simply {(n,7) [ 7 > 0,u e R}. Thus we obtain

s n nn = {(n, 7) e R x R I 0 < 7 <

1,

Inl < 1 -

7}

C H A P T E R 5. T H E G E O M E T R Y

162 and

SNA=

{(0,7) e R x R I 0 < 7 < 1}.

The faces of -Cmax * = --Cmax = R - X ~ = -R+c~ are F1 = {0} and F2 = -IR+c~. Therefore we have eFt = (0, 0) and eF2 = (0, 1). Thus ~ AcPt

{(0,7) E R x R I 0_< 7_< 1} m

{(o,o)} u ( s n A)

(s n A)eF, U (s n A)eF, r-!

by direct calculation.

Let a + := {X 6 a l W e A+ : ~(X) > 0} c Cmax be the closure of the positive Weyl chamber. Our next goal is to find the idempotents in S O Bt which occur as limits of elements in exp a +. The results will be useful when AAePt under the action of G. we determine the isotropy group of a point in ,-,+ Suppose that X E a +. We write E x := ( ] R X - Cmax)O Cmax for the face of Cmax generated by X and

Fx =

x ~

n

- - C m a x ----

EX 1 n--Cmax

for the (up to a minus sign) opposite face. For any consider cone(E) = ~ e r . R + a (cf. Remark 2.1.7). In cone(A_) = - c ~ a x. We set A x := E ~ A A , Ax,• Ax,0 := E x~ n A0. Then Remark 2.1.7 shows that because

(5.36) subset E of A we particular, we have := E ~ A A • , and F x = cone(Ax,_)

ax,+ = {~ e ~ + I~ e Fx} = {~ e ~ + : ~(X) =

0}.

An element X ' 6 E x n a + is said to be relatively regular in E x if all roots in A0 which do not vanish on E x , are nonzero on X ~. We note that if X is relatively regular in E x , then A + = A x O A+ = X • O A+ because Ax,+ = X • n A+, and A+,0 = A + O X • follows from relative regularity (recall that X 6 a+). L e m m a 5.7.8 Let X 6 a+ and E x 6 Fa(cmax) the face generated by X . Then the following assertions hold.

1) [A + + ( A + \ E x ~ ) ] A A + C A + \ E ~ c = A + \ A +

and

con~(~ +~) = S~ n con~(a +) e f~(con~(~ +)). 2) There exists a relatively regular element X ' 6 a+ O algint(Ex) with E x = Ex,.

5.7. THE AFFINE CLOSURE OF B I~

163

Pro@ 1) In view of Remark 2.1.7.5), we have that E~x = Fx

-- F x

= cone(Ax,_ )

-

-

cone(Ax,_).

(5.37)

If 7 E A+,o, then the reflection s7 at ker 7 leaves A_ invariant and s.r(X ) = X. Therefore s T ( A x , - ) = A x , - and consequently E x~ is invariant under the subgroup W x of the Weyl group W0 = W(Ao), which is generated by the reflections leaving X fixed. If X E C~ then E x = Cmax, E xx = {0}, and A + = @. So we may assume that X E 0Cmax. Let E = {ao, O~1,..., O~1} be a basis of A + with ao E A+ (cf. L e m m a 5.5.10). Since X E 0Cmax, there exists a root 7 E A+ with 7 ( X ) = 0. Therefore X E a + and 7 = 1 ao + ~-']~i=1niai entail that

o = 7 ( x ) >_

o(X) _> o,

consequently ao E E)~. Note that the coefficient of a0 must be 1 because 7(Y ~ = 1. Hence we may assume that

:= x • n

= {ao, ax,..., a , }.

(5.3s)

1 So c~ = c~o + ~ ] i = 1 nioti ~. A x , + is equivalent to ni -- 0 f o r i > k. Conse-

quently, ao E E)~ and (5.37) imply that

E:~ = It.no @ Ex,o with

Ex,o := E:~ n span{a1, ..., ak } = E ~ n span{a1, ..., cq }.

(5.39)

Next we claim the existence of a set of simple roots which span Ex,o. To see this, we first note t h a t Ex,o is invariant under the finite group W x because W x also fixes s p a n { a l , . . . , a t } = spanAo. We recall t h a t W x is generated by the reflections sa~,..., sa h at the hyperplanes ker cU (cf. [168], 1.1.2.8). Therefore

Ex,o = Ex,o,~ff 9 Ex,o,~x, where

Ex,o,fix = {V E Ex,o [(Vw E W x ) w. Y = Y } and Ex,o,eff = span{w. Y -

Y Jw E W x , Y E Ex,o}.

For Y E Ex,0,fix the relations sa,(Y) = Y imply t h a t ai.LY for i = 1,...,k. Hence Y E span{al, 9.. , ok } • n Ex,o c E x• , o N E x , o = { 0 } .

C H A P T E R 5. T H E G E O M E T R Y

164

In addition, the fact t h a t W x is generated by the reflections sa~, i = 1, ...,k implies t h a t

Ex,o

=

Ex,o,eff = span{sa, (Y) - Y I i = Z, ..., k, Y e Ex,o }

=

span{ar162 CExXo}.

This proves our claim, and from now on we m a y assume t h a t Ex,o = s p a n { a l , . . . , a j } with j < k. Now E x~ = span{a0, ..., a j } and therefore

I l 3i > j " ni > O} . i--1

This implies the first assertion of the lemma. The second assertion is trivial. 2) From E x~ - s p a n { a l , ..., aj } it follows t h a t

lies in a pointed cone. Whence c : - {Y e spanEx I Vfl E A + 9 fl(Y) >_ 0} = ( s p a n E x ) N a + has n o n e m p t y interior in spanEx. But X E a + N a l g i n t ( E x ) . Hence there exists X ' E a l g i n t E x N algint(c). It follows t h a t E x = E x , , X' E a +, and t h a t X' is relatively regular, Lemma

5.7.9 Let X E a + and

e x -- 1-+r lim e x p ( t X ) -- (1, 7) E S 91B~. Then the following assertions hold:

2) Ad(N0~A) ker 7 C ker 7 C n_. Prooj}. 1) is i m m e d i a t e from L e m m a 5.7.8.

rl

165

5.7. T H E A F F I N E C L O S U R E OF B ~ 2) Note that (5.36) implies that ker 7

=

{Y e n_ [ lim ead t X y = O} t--+oo

aEA_,a(X)#O

= = =

g(~+ \ (~+ n x•

g(~+ \ (A+ n E~))~ , _ n g(~+ \ ~ ) ~ .

Thus ker 7 is an ideal in n~ by 1) and therefore invariant under No~. Since it is a sum of root spaces, ker 7 is also invariant under Ad(A). [:] L e m m a 5.7.10 Let X E a + and e x = limt-}oo exp(tX) = (1, 7) E S rl B~. If Aex,Pex" B~ --} B~ are the left and right multiplications with e x in B~, then we have:

V ~:g(~x) n B~ = k ~

~ ~xp (g(~0+ \ a+~))~ e x p ( ~ , + ) , ~ h ~

~_

is identified with the corresponding subset of a. 2) L e t ~ ( X ) : =

[(X •

+)\A + x,o] u

• + n A x,+]\

X•

).

pe-2 (ex ) rl B ~ - exp (g (~(X))) IIexp(A~,+)

~h~

~0+ n ~x,+ = {~ e ~0+ I V~ e Ax,+. (~ I Z) = 0}.

Proof:. 1) The formula

ex(g, ~ ) = (1, 7) (g, ~) = (7(g),7~) shows that Aex(g,5) = e x is equivalent to g E ker7 and 7~ = 7, i.e., 5(kerT) C ker7 and 5(x) E x ker7 for x E ImT. According to Lemma 5.7.9.2), the first condition on 5 is satisfied if 5 E Ad(B~). For ~ = ead Y with Y E no + a the second condition is satisfied by all elements of ead RY if and only if [Y, ImTl C kerT. The set {g E Aut(N_) 9 7g = 7} is a pseudo-algebraic semigroup. Whence 7e adY = 7 implies that 7 etadY -" 7

Vt E R

whenever Spec(adY) C R ([94], Lemma 5.1). This implies in particular that Ae-x1(ex) n No~A is a connected normal subgroup and therefore

A[lx (ex) M N~oA = exp{Y E no + a ] [Y, Im 71 C ker 7).

166

C H A P T E R 5.

THE GEOMETRY

For Y E a this condition means that Y E A xx,_ since

Im ~ = g(ax,+)~ = ~ ( a x , - ) .

(5.40)

If Y e ga with a E A + \ A +, then clearly [Y, Im71 C ker7 (Lemma 5.7.9.1)). This shows the inclusion _D. Conversely, suppose that a E A +x,0 C E~. By Remark 2.1.7 and (5.37) we have that / A x, + = A x3., _ = F xx = E x -

Ex.

Now ( E x - E x ) V I E:~ = {0} implies the existence of fl E A x , - with (a Ifl) # 0. Hence Lemma 5.7.1.2) shows that [IF,Im 7] # {0} for Y E g~ because E[~,ez g~+-a c Im % But then

(ax,+ + A+,o)n A c Ax,+ shows that [IF,Im -y] r ker % 2) First we note that g e x = g ' e x = e x implies that ( g g ' ) e x = e x and g - l ( g e x ) = g - l e x = e x for g , g ' ~. B ~. So p e ~ ( e x ) N B ~ is a subgroup. Moreover, for (n_, ~) E B ~ the condition (n_, 6)(1,-y) - (n_, ~7) = (1, 7) is equivalent to n_ = 1 and (~7 = % Thus p e - l ( e x ) N B ]~ ---- {(1,(~) 9(~[Im-), = idImT}.

(5.41)

This is a pseudo-algebraic subgroup of the group Ad(B~) which consists of real upper triangular matrices. So it is connected by [94], Lemma 5.1, because the exponential function of B ~ is surjective. Therefore it only remains to compute the Lie algebra of this analytic subgroup. First we note that e adRY ]Im 7 - - i d I m 7 is equivalent to [Y, Im 7] = {0}. For Y E a this means that Y E A xx,+. Let Y E ga with a E A +. We have to consider several cases: a) a ( X ) = 0 and a r EXx . Let fl E Ax,+ C E } . Then a + fl r A since otherwise (5.36) implies that f l + a E A+ N X • = Ax,+ C E xx. This shows that [Y, Im 7] = {0}.

b) a E E~. Then we have already seen in the proof of 1) that [IF,Im 71

{0}.

5.7. T H E A F F I N E C L O S U R E OF B ~

167

c) a ( X ) > 0 and (~ r A~x,+. Let ~ E Ax,+ with (a I z) # 0. Then f~(X) = 0 and therefore (f~- a ) ( X ) < 0. So f ~ - a is no root because X G Cmax. Hence (a I ~) < 0 and ~ + a E A+. Now Lemma 5.7.1.2) implies that [Y, g~] # {0}. d) a ( X ) > 0 and a G A~x,+. As in c) we see that f / - a ~ A for all G Ax,+. The reflection sa interchanges the two ends of the astring through ~ and it fixes f~ because ( a l ~ ) = 0. So f~ agrees also with the upper end of this root string and a + f~ ~ A. But this clearly implies that [II, Im V] = {0}El R e m a r k 5 . 7.11 The formula for pex(ex) -1 is relatively complicated. This comes from the fact that in general X is not relatively regular in the face E x . Since every face of Cmax contains relatively regular elements (they form an open dense subset), every idempotent may be reached by such an element. An example for an element which is not relatively regular in general is y0. The face it generates is Cmax, all compact roots vanish on y0 but no compact root vanishes on Cmax Now suppose that X is relatively regular and in a +. Then

x'n

+ = a + =E

na+.

Therefore

(X

nA+)\A + X,O

and

1 (aonax,+)n

X•

because A0 N X • C E x~ = spanAx,+. So the formula for Pe-xl(ex) becomes easier: p e- 1x ( e x ) gl B = exp (g(A + N A:~,+))~ exp(Alx,+). (5.42) El L e m m a 5.7.12 Let L be a connected Lie group and 7 an idempotent endomorphism of L. Then

L ~- ker7 ~ ImT. In particular, ker 7 is connected.

Pro@ It is clear that Im 7 = {g e L I 7(g) = g} and ker 7 are closed subgroups. Obviously, ker 7 is normal in L, so ker 7" Im 7 is a subgroup and the intersection of Im 7 and ker 7 is trivial. If g G L, then g = 7 ( g ) [ 7 ( g ) - l g ] and Hence L ~- ker 7" Im 7 is a semidirect product decomposition.

El

168

CHAPTERS.

THEGEOMETRY

L e m m a 5.7.13 Let L be a connected Lie group and e = (1, 7) be an idempotent in L x End(L). Further, let s = (a, ~). Then L1)-L4) and R 1 ) - R 4 ) are equivalent: L1)

ese = es.

L2) L3) es e eAff(L)e. L4) 6(ker ~) C ker ~/. R1) e s e -

se.

R2) e ( s e ) = (se)e. R3) s e e eAff(n)e.

R4) a E Im "/and 6(Im-y) C Im 'y. Proo~ The equivalence of L1)-L3) and R1)-R3) is trivial. To see that L1) is equivalent to L4), we compute

es = (I, -y)(a, 6) = (-y(a),'y o 6) and

ese

= (-y(a),-yo (~)(i, "y) = (?(a),-yo ~ o ~f).

So es - ese is equivalent to -/~ - -/5-y. On the image of q( this relation is trivial. According to Lemma 5.7.12, it holds if and only if it holds on the kernel of -y, i.e., if and only if (f(ker ~) C ker ~. For the equivalence of R1) and R4), we compute se - (a, 6~/). So se -- ese is equivalent to ~/(a) = a and 6~ = -~6~/. The first condition means that a E Im ~. On the kernel of % the second relation is trivial. In view of Lemma 5.7.12, it holds if and only if it holds on the image of % i.e., if and only if 6(Im V) C Im 7. n A / a c e F of a topological monoid (i.e., semigroup with identity) T is a closed subsemigroup whose complement T \ F is a semigroup ideal. The lattice of faces of T is denoted Fa(T). The group of units in T will be denoted U(T). In particular, if e E T is idempotent, then eTe is a monoid with unit e and U(eTe) is the unit group of this monoid.

L e m m a 5.7.14 Let T be a topological semigroup. For e E E ( T ) we set Te : - {t e T : et E eTe}

5.7.

169

T H E A F F I N E C L O S U R E OF B ~

and F, "= {t e T " et e U ( e T e ) } . Suppose that U ( e T e ) is closed in eTe. Then Te is a closed subsemigroup of T with identity e, the mapping )~e " Te "+ eTe,

t ~ et

is a semigroup homomorphism, and Fe is a face of Te. Proo~ Let t, t' E Te. Then e(tt')e

=

( e t ) ( t ' e ) = ( e t e ) ( t ' e ) = (et)(et'e) = (et)(et') = ett'.

Hence T~ is a closed subsemigroup of T and the mapping A~ is a homomorphism. Now it is clear that F~ := Ag x (U(eTe)), as the inverse image of a face, is a face of T~. n . , = _ . _

L e m m a 5.7.15 Let X E a + and e x = l i m t - . ~ e x p ( t X ) = (1,7) E B~. Then the following assertions hold: 1) B~e x "= {s E B~ " e x s = e x s e x } = B~.

2)

e x is an identity element in the semigroup

exB~.

3) 7 ( a ) = a n I m T . 4) e x S n B~ = { e x s e exB---~ l e x s ( ' ~ ( ~ - ) ) C "~(~-)}.

5) B~ _~ (ker :~ox n B") ,, B~, ,,,here B~x "= Im 7 x exp (g(A+x,o))' exp(Ex~). 6) e x B"-ff -- e x B ~x.

7) U(~xB~) = ~ B ~ a~d U(~xS n B~) = {~x}. Proof:. 1) In view of Lemma 5.7.13, we only have to recall from Lemma 5.7.9 that 5(ker 7 ) C ker 7 V5 E Ad(B~).

2) Lemma 5.7.14 shows that e x B ~ is a semigroup. The other assertion is a consequence of 1). 3) Since e x E S N B~, we have that m

-

.

_

e x . [2_ = 7 ( f l - ) C f l - N ImT.

CHAPTER 5. THE G E O M E T R Y

170 But 7]Im 7 = idIm 7. Hence

~_ NIm7 C 7(~-). 4) Let s E S N B~. Then

e x s . 7('~-) = e x s e x . "~- C "~because e x s e x E S N B~. Thus C holds. If, conversely, s is contained in the right-hand side, then, in view of 2),

~(~-) =

~x(~-)

= ~. 7 ( ~ - ) c 7 ( ~ - ) c

~_.

5) It follows from (5.40) that B~x is a subgroup of B ~. Since B~x N X~-x1 (ex) = { 1 } by Lemma 5.7.10, we conclude that B ~ _~ (ker X~x N B ~) x B~x. 6) The relation exB~x C exB--'i is trivial. But exB~x is closed. Therefore 5) implies that ~ x B ~ = ~ x B ~ = e~B" ~

~~.

7) First we prove that U(exB~x) = exB~x . The inclusion exB~x C

U(exB~x) is trivial. For the converse, we consider the homomorphism r exB~x --3, Aff(Im T) defined by q~(s):= Slim 7. Since s ( k e r T ) = s e x ( k e r T ) = s(1) = { 1} for all s E exB~x, it follows that ~ is a homeomorphism onto a closed subsemigroup of Aff(Im 7) which satisfies r

= id. Therefore

-

r

On the other hand, = Ira7 x Q, where Q is a subgroup of Aut(Im 7) consisting of upper triangular matrices with respect to the root decomposition of the Lie algebra of Im 7. Hence Q is closed in Aut(Im 7) and it suffices to show that U(Q) = Q, where Q is the closure of Q in End(Im 7). But if lim u , = u E U (End(Im

7)) =

Aut(Im 7),

then u E Q. This proves that U(exB~x)= exB~x . Therefore

5.7.

171

T H E A F F I N E C L O S U R E OF B ~

The semigroup e x (S N B~) is compact since S N B~ is. Thus 4) shows that o

is a compact subgroup of the simply connected solvable group Im 7 x Q and therefore trivial. D

N o t e s for Chapter 5 The material in Section 5.1 was first proved in [129]. Part of it can also be found in the work of Ol'shanskii, e.g., [138]. There is extensive literature on the exponential map for symmetric spaces, e.g., in [29, 44, 104]. The semigroup H e x p C was first introduced by Ol'shanskii [137, 138] for the group case. In recent years they have become increasingly important in geometry and analysis, as we will see in the next chapters. References to applications will be given in the notes to those chapters. Further sources are [50, 52, 64, 63, 93, 129, 130] and the work of Ol'shanskii and Paneitz. That .M is globally hyperbolic was first proved by J. Faraut in [25] for the case Gc/G. This was generalized in [129] to arbitrary noncompactly causal symmetric spaces using the causal embedding from Lemma 5.2.8. The proof presented here is an adaptation of that in [25]. A different approach can be found in [114]. The characterization of G ~ exp(Cmffix) as a compression semigroup was noted first by Ol'shanskii. The proof presented here appeared in [58]. The nonlinear convexity theorem was proven by Neeb in [116]. The proof given is taken from [124]. The results on B ~ have been proved in [55].

Chapter 6

The Order Compactification of Noncompactly Causal Symmetric Spaces The order compactification of ordered homogeneous spaces defined in Section 2.4 is a fairly abstract construction. In this chapter we show that for the special case of noncompactly causal symmetric spaces, many features of the order compactification can be made quite explicit. In particular, the orbit structure can be determined completely and described in terms of the restricted root system. The basic idea is to identify gH E .A4 with the compact set g . O C A', where O is the open domain in the flag manifold ~' defined in Section 5.1. Similarly as for the order compactification, this yields a compactification of 54 via the suitable Vietoris topology. The point is that this compactification is essentially the same as the order compactification but easier to treat, since it deals with bounded convex sets in a finite-dimensional linear space rather than translates of a "nonlinear cone."

6.1

Causal Galois Connections

In this section we suppose that G is a connected Lie group and S an extended Lie subsemigroup of G with unit group H. Let A4 - G / H and consider the order < on 54 induced by < s (cf. Section 2.4). In the following, k' denotes a metrizable compact G-space and O C k' an open subset 172

6.1. CAUSAL GALOIS CONNECTIONS

173

with the property

S= {geGIg.

O}

and

S~

{geGlg.Oc

0}.

(6.~)

(cf. Corollary 5.4.21 for an example of this situation.) We endow the set 3r(X) of closed (hence compact) subsets of X with the Vietoris topology (cf. Appendix B). We write 2 G for the set of all subsets of G, and define the mappings F 9Jr(X) --+ 9~(G),

F ~ {g e G i g -a. F C "0}

and F'2 G--+~(X),

A~

n

a.O.

(6.2)

(6.3)

aEA

We call F a causal Galois connection. That this is no misnomer is a consequence of the following lemma. L e m m a 6.1.1 The mappings

~. (7(v), c)-~ (j:(x), c)

and

r . (y(x), c) -+ (7(v), c)

are antitone and define a Galois connection between the above partially ordered sets. Moreover, the following assertions hold: 1) ,I,I"(F)= F ( F ) for every F E 9r(X).

2) r(F) ~ =

{g e G i g -~ . F C 0}.

3) For every subset A C G we have that F(A) = F(-A) and F($A) = ~(A).

4) r(u~e~ A~) = nie~ t(A~). 5) F(lJ/e, f i ) =

nielF(F~).

6) r( ['] f . ) = U r(f.) Io~ eve~u de~e,,i,g ,eq~e,~e f . i, 7(X). hEN

hEN

7) t ( g . A) - g. t(A) and r(g. F) = g. r(F)/or all g e G, A e 2 G, F e 7(x).

s} r(g. o) = ,l,g Io~ all g ~ G. 9) t(~ g) = g.-~ Io,.

au g ~

G.

CHAPTER6.

174

THEORDERCOMPACTIFICATION

Proof:. The fact that F and F define a Galois connection follows from the fact that

A c P(F)

r A c { g e G l F c g.-O}

1) Let g G F ( F ) and s G S. Then g-X. F C O and therefore (gs-X) - 1 . F = s g - x . F C s . "0 C "0, hence $ g - g S -1 C F(F). 2) Let g E F ( F ) ~ Then, since S has dense interior, there exists a n s ~. S ~ with g s ~. F ( F ) . Therefore g - 1 . F = s ( g s ) - 1 . F C s . "0 C O .

Conversely, suppose that g-X. F C O. Then we find a neighborhood U of g in G such that U -1 9F C O because F is compact and O is open. Thus

g e v c r(F). 3) It is clear that F(A) C F(A) - F($A) because a$-I-~

= a . ( 8 - 1 . ~ ) Z) a " O

f o r e v e r y a fi G , s e S.

Let x G F(A) a n d a G A w i t h a -- l i m , ~ o o a , and a , G A. For every n G l~l we find an element f , G ~ with z - a n . fn. We may assume that f : - limn-~oo fn exists in the compact set O. Then u

x=

lira a,, . f,, = a . .[ E a . O .

n-,-],oo

4), 5) These assertions are trivial. 6) Set F := ~ , e N Fn. First we note that F(Fn) C r ( f ) and therefore that the right-hand side of 6) is contained in the left. Let g G F ( F ) and s , ~_ S ~ be a sequence with lim,~oo s , = 1. For every n G N we have that

s , g - 1 . F c s , . "0 c O, i.e., F C g s ~ 1 90 . Note that F , ~ F in the Vietoris topology. Hence we find no G N such that F , o C gs'~ 1 . O , which implies that g s ~ 1 ~. F(F,0). This shows that gs'~ 1 e [,JkeNF(Fk) for every n e N. Now

g = n--l, lira g~e oo

U r(F~). hEN

6.1.

CAUSAL GALOIS CONNECTIONS

175

7) This is immediate from F(g. F ) = {x 6 G l x - l g 9F C "0} = {x 6 G l(g-~z) -~. F C O} = g . r ( f ) and from

A) =

N

aEg.A

. . v=

N g. . v = g.

aEA

8) We deduce from 7) that F(gO) - g . F(O), and therefore it remains to show that S -1 - {g E G I g - 1 . O C O}. First, according to our assumptions about O and X, we have that S -1 C F(O) because g-1 .O C (.9 implies that g-X. ~ C O. We claim that g G S -1 for every g G F(O). Let Sn G S ~ be a sequence with limn-,oo Sn = 1. Then 8 n g - 1 " ' 0 -" Sn "

(g-1 .~) C Sn

"'O C 0

and therefore sng-1 G S ~ Thus g-1 = lim,-~oo sng-1 E -fig = S. 9) In view of 3), we have F($g) = F({g}) = g . g .

[3

C o r o l l a r y 6.1.2 For two elements g,g~ G G we have g <_s g' ~

g' "'0 C g . ' O

and g G ($ g,)O ~

g,. "~ C g . O.

Pro@ These are direct consequences of the fact that

s~

{geGIg.

c O}

and

S-' =r(g).

o

L e m m a 6.1.3 The mapping F:~'(X) --~ ~'(G) is continuous with respect to the Vietoris topologies on .~(,~') and ~'(G). Proofi Suppose that F , ~ F in ~'(X). We split up the proof into two steps. 1) r ( f ) ~ c liminf,_~o~r(F,)" Let g 6 r ( f ) ~ According to Lemma 6.1.1.2), we have g - * . F C O, hence F C g . O. Consequently, we find no 6 N s u c h t h a t F , C g . O f o r n _ no. T h u s g - X - F , C O C O a n d g 6 F(Fn) for n > no. 2) limsup._,oor(F.) c r(f): Let g 6 limsup._,~r(F.) and choose a subsequence F , k and gk 6 F ( F , k ) with gk ~ g. Then F = limk-,oo F , k. Pick f 6 F. If fk 6 F , k with fk ~ f, then fk 6 gk" 0 so f 6 limk-~cr g~= g . O . Thus g 6 r(F), since f was arbitrary. As l i m i n f , - , o o F ( f , ) is closed, we now have

r(F)o c l i m i n f r ( F . ) C limsupF(F.) C r(F). But

r(f)

r(F)

is closed and satisfies r(F) = $r(F), so Lemma 2.4.7 implies = r ( f ) o and hence Lemma S.l.1 shows that r ( F n ) --+ r(f). m

176

CHAPTER6.

THE ORDER COMPACTIFICATION

D e f i n i t i o n 6.1.4 We set .Ado := {g. O Ig e G} c y(x). The map e:G ~ ~ O , g ~ g. El

is called a causal orbit map. Note that Lemma 6.1.1.8) implies that F

o e

= ~

o

Ir,

(6.4)

where rr: G --+ G / H is the quotient map and ~ the order compactification from Lemma 2.4.2. L e m m a 6.1.5 ( F i x e d P o i n t s of t h e Galois C o n n e c t i o n ) ing assertions hold: 1} F o F(A) = A for every A e

The follow-

r(~4~

z) ~ o r(F) = F for every F C X for which there exists a decreasing sequence F,~ = g . . 19 with

F = n--b~ lim F , = n g , . _._, O. hEN Proof:. 1) There exists a compact subset F G A/[~ with A = F(F). The fact that F and F define a Galois connection (Lemma 6.1.1) implies that

r~(A) = r ~ r ( F ) = r ( F ) = A. 2) We use Lemma 6.1.1.6) to see that r ( F ) = U.eNr(F.). According to Lemma 6.1.1.3), this leads to

P r o p o s i t i o n 6.1.6 The mapping F" .h/[~ --r .h/lopt is a quotient morphism of compact G-spaces, where G acts on .h/[~ by (g, F) ~-~ g . F. Moreover, the action of G on .It4~ is continuous. Proof:. We show first that r(r ~ -- .h/[opt. In fact, consider g . O G j~r Then, according to Lemma 6.1.1.8), c ( g . ~ ) = Sg = g s -~ = ~(gH).

6.1. C A U S A L G A L O I S C O N N E C T I O N S

177

This proves the claim, since r : f14 ~ --+ O(G) and ~: f14 --+ O(G) are continuous and {g. O l g e G} is dense in .A4~ just as A4 is dense in ~ c p t . It follows from L e m m a 6.1.1.7) that r is equivariant. L e m m a 6.1.3 together with the above implies that F is a quotient mapping of compact spaces. The action of G on (I)(X) is continuous by L e m m a B.1.2. Thus f14 ~ is G-invariant and the restriction is obviously a continuous action on

Mo. Theorem

n 6.1.7 Let

M~ := (rl~o)-'(M +~

)"

(6.5)

Then the following holds:

I) , ( s ) = M + . o

2) s = {g e G i g .

(M~) ~ c (M~)~

s) s ~ = {g e G i g .

M~ c (M~)~

Proof. 1) This follows 2) and 3)" First we claim that [.M +cpt]~ J = have F e A4~

from r (t(S)) =

~(A4+).

hA cpt note that F(A4~) = ,.,+ by Proposition 6.1.6. W e F([jV~] ~ and IMp] o = r-'([MT'] o). m fact, we AA cpt r F(F) e ,-.+ ~, 1 e r ( F ) ,~ F C -0-

because of L e m m a 2.4.4, and, using L e m m a B.1.2, we see that F e [A4~] ~ implies t h a t there exists a neighborhood U of 1 in G such t h a t U . F C O. On the other hand, F ( F ) E [.A4~pt]~ holds if and only if there exists a neighborhood U of 1 in G such that U . F C O because of Proposition 2.4.4. This shows that

r([M~l ~ c [M ~'~~ + I " The reverse inclusion follows from the continuity of r . Now we have r([fl4~l ~ = [.M~.P'I~ which upon taking the preimage under r also shows t h a t [M~] ~ = r-~([M7'lo). Now we see that g e S is equivalent to

g. [MT'I ~ c 1;~7'1 ~ -: :- g. r ( l M ~ l ~ c r([M~l ~ -: :- r(g. [M~I ~ c r ( l M ~ l ~ cpt

:- g. [.M~I ~ c [ . ~ l ~

Similarly, g - f 1 4 + C [fl47'1 ~ ~ g" f14~ C [fl4~ ~ An application of Proposition 2.4.4 completes the proof of the last two claims. D

C H A P T E R 6. T H E O R D E R C O M P A C T I F I C A T I O N

178

6.2

An Alternative

Realization

of

t

From now on we assume t h a t A4 = G / H is a noncompactly causal s y m m e t ric space such t h a t G is contained in a simply connected complexification Gc. Moreover, the semigroup S = S(Cmax), the flag manifolds A'G/Pmax and XC = Gc/(Pmax)C, and the open domain {9 C A', are the ones from Section 6.1. Recall the causal orbit m a p L: G --r A4 ~ from Definition 6.1.4. Theorem

6.2.1 L(S) -- [ K A . e (S f'l A)] N ~'(V).

Proof:. C" Let E -- limn~OO sn" O fi L(S) C 5r'(O). From [1151, 2.9, we know t h a t G - K A H . Therefore we find elements k , ~. K, an ~. A, and hn E H such t h a t s , = k n a , h n . T h e n s , . O - khan" 0 because H . O - O. According to [66], p. 198, the group K is c o m p a c t since t + ip is a c o m p a c t real form of the complex semisimple Lie algebra gc and G C Gc. Therefore we m a y assume t h a t ko - limn-~OO kn exists in K and we find t h a t ,--.--

n

E = lim k , a , . 0 = ko. lim a , 9O n--boo

B--~OO

because ko x. E - lim,_,oo k o X k , a , 9"0 -- lim,_,oo a , - O. Thus

0 # r ( k o x. E) - k o ~ r ( E ) = lira r ( a . . ~ ) = lim S a . n--bOO

n--+OO

lim n--+OO

O(an)

since r(E) e 0(S) contains S -a. Now we use L e m m a 5.6.5 to find an a e A such t h a t ana -1 ~. S N A for all n _> no. T h e n

E = koa lim ( a - l a , ) 9~ E K A . ~ (S f'l A) N 3:('0). B--~OO

D: Let E = k a . lim,_~OO an 9O C O. Taking s E S ~ we find t h a t lim skaan . 0 C s . 0 C O. n--~OO

Therefore we find an no E N such t h a t s k a a , . O C O for n > no. Thus skaan E S and consequently s . E = lim s k a a , . (9 ~. e(S). n--bOO

T h e fact t h a t 1 E ---5 S now implies t h a t E E e(S).

6.2. A N A L T E R N A T I V E R E A L I Z A T I O N

OF/~/f~PT

179

R e m a r k 6.2.2 Consider the commutative diagram 0

)

N_ - o x

fl_

~

n_

)

A' N_

The vertical maps are G-equivariant homeomorphisms and O, respectively ~2_, is relatively compact in the open set N _ . o x , respectively n_. Therefore j r ( ~ _ ) can be identified with the compact set j r ( ~ ) = {F 6

I ( x ) I F c ~}

(cf. Lemma B.I.1). In particular, we have an action of Affeom(N-) on j=(O) and can consider f14~ as a subset of ~-(~_). [3 Next we consider the set t(S O A) C }'(O). L e m m a 6.2.3 Recall the closure B'-"ffof B ~ in Aft(N_) and the closure "qcpt A of S n A in B~. Then the following assertions hold:

1) S n B~ = {7 6 B-t I 7" 0 c 0 } is a compact semigroup.

~) M ~ = ,(s) = s n B~. g. s) M ~ = c K A . ( ~

9

Pro@ 1) In view of Theorem 5.4.8 and Remark 5.7.4, we only have to show that S n B~ D {7 6 B'--ff ] 7" O C O}, because the other inclusion is clear. Let 7 6 {a 6 B--~] a . O C O} and 7- 6 B ~ with 70 -+ 7. W e choose X 6 C~ and set a(t) = exp(tX). Then ead t x ( ~ _ ) C f l - for every t > 0. Therefore ~ _ C [e -ad tx (~_)]o entails the existence of no 6 N such that 7 , ( ~ - ) C e-~d tX (~_) for all n > no. Then a(t)7, 6 S n S~. We conclude that 7 = lim a(t)-l[a(t)7,] e a(t)-lS n B~. n-+oo

Letting t -+ 0, we find that 7 6 S n B~ because S n B~ is compact. 2) Note that compactness shows

(SNB~).fl_

c S n B~ .~ _ = S n B~ .fl_ c

(SnB~).fl_

so that

,(s) = ( s n B~). ~ _ = S n B~. ~_. The first equality follows from Theorem 6.1.7.

CHAPTER6.

180

THE ORDER COMPACTIFICATION

3) Since S~f t is a compact subsemigroup of S n B~ which contains S N A, Theorem 6.2.1 and Proposition 5.7.5 entail that

e(S) C K A . e(S n A) = K A . (S~f ' . -0).

1:3

Lemma 6.2.3 shows that information on the orbit structure of e(S) = r .h4~ may be obtained from ~'A "~ , which is an orbit of a compact abelian semigroup. The topological structure of e(S) is encoded in the compact semigroup S n B~. For a face f e Fa(cone(A_)), we set

~ F :--'~eF" ~ -

(6.6)

and note that we may view ~ F as a subset, OF C X C ,'~.

(6.7)

6 . 2 . 4 The causal aalois connection F:~'(X) -4 Yr(a) defined in (6.2) induces a homeomorphism ~ -4 ,AA , , +cpt .

Theorem

Proof. We note first that

(6.8)

e(S) C K A . { ~ v l F e Fa(cone(A_))} which is a consequence of Lemma 6.2.3.3) and recall that ~AcPt

= exp(cmax)E(SCf t) = exp(Cmax){eF ] f e Fa(cone(A_))}

from Theorem 5.7.6 Further, we know from Theorem 6.1.7 that ~WL. AAcPt I= F(j~r Since F: A4 ~ -4 f14~pt is a quotient map by Theorem 6.1.6, it only remains to prove that FI-~ is injective. Let E E e(S). Then there exists g E G and F E Fa(cone A_) such that E = g. ~ F C O. Thus

~r(E) = ~ r ( g . ~ v ) = g. ~ r ( ~ v ) = g. ~'F = E because ~ F = limt_+~ exp(tX). O for X E IntF• (Cmax n F • 5.7.6, Lemma 6.1.5). This proves the claim.

6.3

(cf. Theorem E!

T h e Stabilizers for A4~ t

We remain in the situation of Section 6.2. This section is devoted to the study of the stabilizer groups in G of points in ,AA , , +cpt .

6.3. THE STABILIZERS FOR ,/~PT

181

Let X be an element of a +. We determine the connected component of the isotropy group of the element limt~oo(exp tX).'O in A4 ~ We endow the space of vector subspaces of g with the Vietoris topology which coincides with the usual topology coming from the differentiable structure on the Gragmann manifold. P r o p o s i t i o n 6.3.1

The limit IJx

:= limt-,oo

~x = ~ ( x ) +

Proof:.

e~dtXl} exists and

o(a + \ x •

We write g = m + a + ( ~ a e a ga and pr~ "g --+ [},

Y ~

1 [y + r(Y)l

for the projection of g onto ~. Then

aE&+

because pr~(ga) = pr~(g_a). The subspace m = ~ ( , ) is fixed under A and for Y E g=, a E A+ we find that lim e ~a tXpr~(RY)

t--~oo

=

l i m e ~a 'XlR [Y + r(Y)]

t--+oo

= t--+oolimR[et~(x)Y+e-ta(x)r(Y)] _ -We conclude that l i m t ~

et ad X

ypr~(RY), ~ RY,

ifa(X)=O if a(X) > 0.

I~ exists and equals

g~ = s~(x) e o(a + \ x ~ ) . aEA+nX •

D

aEA+\X•

Lemma 6.3.20xN(aWn ~)={0}

a n d g - - I I x W a W n Ii.

Proof:. Let Y = Y1 + Y 2 = Z I + Z 2 E t ~ x n ( a + n " ) with Y1 E Y2 e 0(A + \ X • Z1 e a, and Z2 e n ~ = g(A+) ~. Then YI ~-- Z I "]- Z2 - Y2 e (fl -]- III~ -~- If) N ~ $ ( X ) :

a (~

@

3~(X),

ga.

aE&FIX •

We conclude that Y2 - 0 because the sum a + n ~ + n is direct. Then Y1 E (a + n ~) N t} - {0} and therefore Y - 0. This proves that

CHAPTER6.

182

THEORDERCOMPACTIFICATION

l}x f] (n ~ + a) = {0}. The second assertion follows from dim~ = dim I)x = dim It - (dim a + dim n ~). ci Note that Lemma B.1.5 implies

H x C HE := {9 E

Gig. nF = ~F},

(6.9)

where g x = (exp IJx) because ~ r = limt~oo exp(tX). O. P r o p o s i t i o n 6.3.3 Let l}f be the Lie algebra of HE. Then

~ = ~x + [ ~ n (, + .~)]

and

H~

n

B~ = p,-~ (ex) n B~.

Proo~ The first assertion follows from Lemmas 6.3.2 and B.1.5. Write ex = (1, 7)- We know already that ~ F corresponds to 7 ( ~ - ) C n_ (cf. Lemma 5.7.2 and Theorem 5.4.8) and that

H F n S ~ = { (n_, 6) E B ~ l (n_, 6). 7(~-) = 7(~-) }.

Let (n_, 6) e H E N B ~. Then e x ( , - , 6). 7 ( ~ - ) = ex" 7 ( ~ - ) = 7 ( ~ - ) . Therefore ex(n_,6) E U ( e x ( S N B ~ ) )

= {ex} (Lemma 5.7.15.7)). We

conclude that (n_,6) e Xe-Xx(ex) C ker7 x N~oA (Lemma 5.7.10.1)). Now (a, 6). 1 = a 6 7(I2-) C Im 7 implies that a 6 I m 7 f'l k e r r = {1}. Next we have 6 (7(~_)) C 7 ( ~ - ) , which implies that 6(Im 7) C Im 7 and 6 Ae-Xx(ex) entails that 76hm 7 = 7. Thus 6lira7 = idImT, so (5.41)implies that

HE fl S ~ C pe-2(ex). Conversely, (n_, J)ex = ex entails that ( - - , 6)" 7 ( ~ - ) = ( - - , 6 ) e x . ~ - = e x . ~ - = 7 ( ~ - ) .

t3

C o r o l l a r y 6.3.4 Suppose that X E a+ is relatively regular. Then

• )I1 9 ( D p,(g~,) II 9 ~ = ,- 9 a,~,+ 9 g(a+ \ x l ) ~ g(a: n ax,+ aE A +

Proof. Remark 5.7.11, Proposition 6.3.1, and Proposition 6.3.3.

[3

6.4.

T H E O R B I T S T R U C T U R E OF

6.4

.~CPT

183

T h e O r b i t S t r u c t u r e o f .A4 ~pt

We retain the hypotheses of Section 6.2. In this section we finally determine the orbit structure of the order compactification of .h4. P r o p o s i t i o n 6.4.1 Let w E NKnH(a) and X E

Cmax

with

lim exp ( t X ) . O = ~ F .

t--~oo

Then

lim exp (t A d ( w ) X ) . O -- w . flF.

t-+oo

Proof:. We have t h a t

--

lim exp (t Ad(w) 9X ) .

lim w e x p ( t X ) w - 1 . t--+oo

t--~oo

=

Um w e x p ( t X ) . O = w . f l F .

t--+oo

which proves the assertion. C o r o l l a r y 6.4.2 For every orbit G . "~F there exists X E a + such that lira exp ( t X ) . O E G . ~ F .

t-too

Proof:. Since

Cmax -- W0

9a +

this follows from Proposition 6.4.1.

O

Let X E a+, e x = (1,-y) the corresponding idempotent of S CI B~, and F = F x . We set n x , - := 0 ( A x , - ) = I m 7

and

N x , - := exp(nx,_).

(6.to)

R e m a r k 6.4.3 Let L be a group acting on a space X and y C X be a subset. We define Ny(L)

:=

Zy(L)

:=

(g e L l g. Y = Y}, { g e L I V ~ e y 9g . ~ = ~ } .

Then the ~ubgro.p Zy(L) a gy(L) i~.orm~. In ~ t , let ~ e Y, h e

Ny(L)

and g E Z y ( L ) . But then hgh-'.y

- h . [g. (h - 1 . y)] - h . (h - 1 . y) - y

because h -1 . y E ~ and this implies the claim.

Q

Recall from (6.7) t h a t the flF correspond to compact subsets OF of

OCX.

184

C H A P T E R 6. T H E O R D E R COMPA C T I F I C A T I O N

L e m m a 6.4.4 The following assertions hold:

1) f~F is a compact subset of N x , - ' o x

with dense interior.

2} Set ZF := {g G G IVz E ~ F " g . z = z}. Then ZF

=

{gEGlVz6Nx,_.o"

=

N gemsxg-l" gENx,+

g.z=z}

3} The normalizer o.f ZF contains A, M , and HF. Proof 1) This follows from the fact that ~ F -- [eF" e x p ( ~ - ) ] 9oX a n d thai; eF" exp(12) = exp(12) N N x , - is open in Nx,- (cf. Lemma 5.7.15). 2) The equality {gGGIVYGNx,-'~

g'Y--Y}--

N

gPmaxg-1

gENx,+

is clear, since gPmaxg -1 is the stabilizer of g . o x . It is also clear that ZF contains this subgroup. To see that the converse inclusion holds, let g 6 Z F. Then the mapping

9 " N x , - --~ ,~',

n_ ~-~ n-Xgn_ . o x

is analytic and constant on the open subset exp(12) Iq N x , - . Therefore it is constant because N x , - is connected. We conclude that g.(n_ "ox) = n _ . o x for all n_ G N x , - . 3) Lemma 6.4.3 shows that ZF is a normal subgroup of HF. That M A normalizes ZF follows from 2) and the invarianee of N x , - . o x under M A . To see this invariance, let g E. M A . Then g . o x = o x because M A C Pmax and therefore

g N x , - 9o x - g N x , _ g -I 9o x = N x , - . o x . L e m m a 6.4.5 Let PF be the normalizer of ~F. subgroup of G containing Pmin ~- M A N .

13

Then PF is a parabolic

Proof. Let PF be the Lie algebra of PF. Then, by Lemma 6.4.4.3), I)F + a C PF and M A C PF. The subalgebra generated by prb(g~ ) and a contains ga + g - a for every a G A+X with X relatively regular (cf. Corollary 6.3.4). So m + a q- g(A +) = rrt -[- a -~ n C PF. Therefore Pmin -" M A N group.

C PF and consequently PF is a parabolic subEl

6.4.

THE ORBIT STRUCTURE OF/~CPT

185

, F I• ClCmax L e m m a 6 . 4.6 Let -~2F, "~F' be such that F ~ F' and F• are faces of Cmax generated by relatively regular elements X , X ' in a +. Then G . ~ F ~ G . ~'~F'.

Pro@. Suppose t h a t G . ~ F = G- f~F'. Then there exists g 6 G such that g" ~ f = ~ F , . Therefore HF, = gHFg -1 and gZFg -1 = ZF,. This implies also t h a t gPFg -1 = PF,. But PF and PF, are parabolic subgroups of G containing Pmin. Hence they are equal ([168], p. 46). Moreover, PF is its own normalizer. So g E PF and ZF, -- ZF. Thus A~x,+ = ~F f3 a = ~F, N a = A~x,,+. Then the faces A X,+ • fl Cmax a n d A~X, ,+ l'l Cm&x are equal and F = F ~ follows from Remark 2.1.7. Cl Proposition 6.4.1 and L e m m a 6.4.6 now yield the following. g

m

T h e o r e m 6.4.7 The G-orbits of ~F and ~2F, for F , F ' e Fa(cone(A_)) agree if and only if F and F' are conjugate under the Weyl group Wo. Lemma 6.4.8

1)

G.

"~_ C K A . {'~F

IF e

Fa(cone(A_))}.

2) For all F ~ Fa(cone(A_)) we have G . "~F = K A . "~F. Pro@ 1) In view of Theorem 5.7.6.2) and Theorem 5.7.6.4) this is exactly what was shown in the first part of the proof of Theorem 6.2.1. 2) Using Corollary 6.4.2 and 1), we find G . - ~ F C G . "~_ = K A . {'~F'

IF' e

Fa(cone(A_))}.

If g. f~f E K A . f~f,, then we have G. ~%e = G. f~F, so L e m m a 6.4.6 implies t h a t F = w. F ' for some element w q W0. But W0 9A = A and the action of W0 on A is induced by conjugation with elements from K, so K A . "~F = K A . - ~ w . F ' -" K A w . "Of, = K w ( w - l A w ) " "~F, = K A . ~ f ' . . - - .

. _ _

It follows in particular that g. f~f 6 K A . f~F.

[3

Lemma 6.4.9 Suppose that G . F ( ~ F , ) C G . F ( ~ F ) l'or some F ' , F fi Fa(cone(A_)). Then there exists a Weyl group element w e Wo with w . F~ C F, i.e., w . F ~ is a face of F. Pro@. The hypothesis means that there exists a sequence of elements gn E G such t h a t ~v, = limn-~o, gn " F~F. L e m m a 6.4.8 shows t h a t we can find kn E K and an E A satisfying ~F' = limn-~,o khan " F~F. Without loss of

186

C H A P T E R 6.

THE ORDER COMPA CTIFICATION

generality we may assume that the kn converge to some ko E K , so that ko I 9~ e , = lim,~oo a n . ~ F e A. ~F. This implies that e l ( k [ 1 9"~F') "k~-1 "~e,. On the other hand, we know that r ( k [ ~.~F) = ko ~-r(~F) ~ 0. Therefore Lemma 5.6.5 tells us that there exists an ao E A with k o l ' ~ f , E ao-OF, C ao(S n A)-O. Thus

k: 1" "~F, e e F A S A pt" "0 = AS~pt. ~F = U

A.-OF,,.

F"CF

Pick F " C F with k o 1" ~ F ' E A. ~F,'. Then G. ~ F ' -" G - ~ F " F " e Wo. F ' (cf. Theorem 6.4.7).

entails that 13

L e m m a 6.4.10 Let F e Fa(cone(A+)). Then the following assertions are equivalent. 1) F = F n cone(A_) :for an F e Fa(cone(--A+)). 2) There exists an X E a + such that F = X s O cone(A_). 3) There exists a relatively regular X E a + such that F = X • n cone(A_).

4) F X A Cmax /8 generated by a relatively regular element of a +. Proo~ 1) =~ 2): /~ E Fa(cone(--A+)) means t h a t / ~ = op(E) for some E e Fa(cone(--A+)*) = F a ( - a + ) . If X e algint(S), then 15 = X X n c o n e ( _ A +) and hence F = F n cone(A_) = X • n cone(A_). 2) =~ 3) follows from Lemma 5.7.8.2). 3) =~ 4)- If F = X X A cone(A_), then Cmax n F •

=

Cmax n fl[RX+c~

--

Cmax n (~[RX"~-Cmax] = Cmax n [Rq-X - Cmax],

where a c for cone C in a is the edge of the cone, is generated by X (cf. Section 2.1 for the notation). 4)=~ 1): If F• is generated by X e a +, then F = X X n c o n e ( A _ ) . Let F := X • n cone(- A +) e Fa(cone(- A+)). T h e n / ~ n cone(A_) = F. rn R e m a r k 6.4.11 For all F e Fa(cone(A_)) one can find a 7 e ]IV such

that 7(F) satisfies the conditions from Lemma 0.4.10. If 7(F) and 7 ' ( F ) both satisfy these conditions, then Lemma 6.4.6 shows that 7(F) - 7'(F). rn

6.4.

THE ORBIT STRUCTURE

OF ./t4 c P T

187

The space Fa(cone(A_)) carries a natural partial order ~. It is given by inclusion. The corresponding strict order will be denoted by -~. On the other hand, we introduce an ordering on the set of G-orbits in M opt via G . m -~ G . m'

:r

G-m-~G.m'

G. m # G-m';

G . m C G . m', G . m -~ G . m ' or G.m=G m'.

:r

It is clear that the relation _ on G\./t/I ept is reflexive and transitive. For the antisymmetry, note that G . m ~ G . m ' -g G . m implies that G . m is strictly contained in G . m', which in turn is contained in G . m . This shows that G . m = G . m', in contradiction to the hypothesis. Thus __ is a partial order. Note that G . m -~ G . m ' means that m E G . m ' and the G-orbit of m is not dense in G . m'. L e m m a 6.4.12 Let F' C F be ]aces of Cmax. Then r ( ~ F , ) e G . F ( ~ F ) . Pro@ F C F' implies that eFeF,

= eF, eF

for X ' E Int(F,)- (Cmax CI (F') we can calculate ~F' = eF,~-

= eF,

2.) (of. =

=

lim t --+oo

exp'"teL'

Theorem 5.7.6). Using L e m m a 5.7.2,

lim exp t X ' . eF'~-

t--~oo

lim exp t X ' . ~ F E G . ~ F .

t-+oo

Thus

e G.

O

For an element E E 2r we define the degree d ( E ) := dim kereF , where E E G . ~2F. We note that e F " n_ --+ a_ is a projection and m "= dim n_ - dim ker e F agrees with the topological dimension of the set E ~- f~F ~-- e F" ~ - , which is an m-dimensional compact convex set. Upon cpt identification of j~4+ ~ with dAAePt via F, we have the function d also on M + v r P r o p o s i t i o n 6.4.13 Suppose that F, F' e Fa(cone(A_)) and F ' is contained in F. Then a.

G.

strictly

CHAPTER 6. THE ORDER COMPA CTIFICATION

188

Proof. Recall that -Cma x is a polyhedral cone spanned by the elements of A_. Thus the hypothesis shows that F ' n A_ is strictly contained in F n A_. But then we have d('~F,) = dimkereF, < dimkereF = d('~F).

So ~F, e {y e M~ I d(y) > d('~F) + 1} and hence G . ~F' n j~4~ c {y E

M~ I d(y) > d(~F)+1} o r

AAcP t G r(~F,)n M 7 ' c {, e ,-.+ I d(-) > d (r(~F)) + 1}. 9

(6.11)

But since F(~F) is not contained in the right-hand side of this inclusion, it is also not contained in G. F ( -~- F , ) n ,A.A,c+P t , whence G. r(~v,) # G r(~v) and the assertion follows from Lemma 6.4.12. [3 T h e o r e m 6.4.14 Let ~ "= {F e Fa(cone(A_)) [ 3 F e Fa(cone(-A+)) F n cone(A_) = F}. Then the mapping T:~" --r (McPt\ {O})/G,

F ~ G.'~F

is an order isomorphism. Proof. The surjectivity of T follows from Lemma 6.4.8 and the injectivity from Lemma 6.4.6 and Remark 6.4.11. According to to Proposition 6.4.13 it is order-preserving, and that the inverse is also order-preserving is a consequence of Lemma 6.4.9. o We describe the structure of the lattice ~" in more detail. Note first that ~" is isomorphic to - ~ ' . Since cone(A +) is polyhedral and A+ is generated by a set T = {a0,... ,hi} of linearly independent elements (cf. Lemma 5.5.10), it is clear that the mapping Fa(cone(A+)) ~ 2T,

f ~ F n T

is an order isomorphism. So we have to determine the image of the mapping 2v ~ Y,

D ~ cone(D) n cone(A+).

To each subset D C T there corresponds a subgraph of the Dynkin graph of A. Recall that T contains only one noncompact root a0. Let Do C D correspond to the connected component of a0 in the Dynkin graph. Then (spanD) n A = [(spanDo)n A]0[span(D \ Do) n A] and therefore

~one(D) n ~o.e(A+) = ~o.~(Do) n ~o~e(~+).

9

6.4.

T H E O R B I T S T R U C T U R E OF

.~r

189

Hence it suffices to consider subsets D C T such that c~0 E D and the corresponding subgraph is connected. The resulting lattices have been listed in [55]. E x a m p l e 6.4.15 In the situation of the SL(2,1R) Example 5.7.7 we have ~Ft = { 0 }

and

~F2=~-

and jr _ {F1,F2}. In particular, .A,4"r" \ {0} has two SL(2,]R) orbits.

E!

C o r o l l a r y 6.4.16 G. F ( ~ F ) = UF'CF G. F(~F,). m

Proo~ G . r(~2F) is the disjoint union of G-orbits. For each of these G-orbits G.x we have G.x C G . F(~F), so Lemma6.4.9 implies that G.x --- G.F(~F,) for some face F ' of F. This proves the inclusion C, whereas the reverse inclusion is clear from Corollary 6.4.2. ra

T h e o r e m 6.4.17 For F ~_ Fa(cone(A_)) we have

g'~FeM~

**

g-~rC~-.

Pro@ Recall that Theorem 5.7.6.3) shows that ~ F = limt-,r162 exp(tX).'~for X E IntF x (Cmax f'l F • r If g ' - ~ F ~- A/l ~ and s ~. S ~ then we have s g . ~ F e (A4~ ~ by Theorem 6.1.7 and hence s g e x p ( t X ) . ~ _ ~..~ill~ for large t. We choose sn ~_ S ~ with sn --+ 1 and t , such that t , -+ oo and

s . g e x p ( t . X ) . ~_ e M ~ .

(6.~2)

Now the calculation in the proof of Lemma 6.1.5 shows that (6.12) holds precisely when ~ ( s n g e x p ( t n X ) H ) ~. A/le+pt because of Theorem 6.1.7. But this in turn is equivalent to sng exp t n X E S and hence to sng exp(tnX) 9 f~- C f~-. If we let n tend to c~ we obtain g . ftF C l~_. =~" This time we choose s , ~_ S ~ and t , E R with s , ~ 1, t , ~ oo and s , g e x p ( t , X ) . ~ _ C ~ - . Reading the argument in the first part of the proof backwards, we find s n g e x p ( t n X ) . ~ _ E A4 ~ and upon taking the limit, g. ~ F e A4~. O C o r o l l a r y 6.4.18 Let F ~. Fa(cone(A_)). Then

AA~P*. G . r ( ~ F ) n M +~pt = G. r ( ~ v ) n ,-,+

CHAPTER6.

190

THE ORDER COMPACTIFICATION

Pro@. The inclusion C is obvious. To show the reverse inclusion, let F ' be a face of f and note that F ( ~ F , ) = limt_,r162 exp t X ' . F ( ~ F ) for X ' E IntF,x(CmaxNF'• If x E G 9F ( ~ F , ) N , AACpt , , + , then there exists a g E G such -that x - g. r(f~F,) E ,AZCPt - , + and Theorem 6.4.17 implies that g. ~ F ' C - . Choose s , E S ~ and tn > 0 with sn ~ 1, t . ~ oo and s , g e x p t , X ' . f~F C ~ _ . Then Theorem 6.4.17 shows that s , g e x p t , X ' . F ( ~ F ) E A 4 ~ ' and in the limit we find g. F(f~F,) E ,AACPt - , + 9In other words, -

-

AA cpt G 9F(~F,) n,.,+ c

Finally, Corollary 6.4 9

G.

AA cpt F(~F) n,.,+ .

shows that

AA c p t + n G. r(~F) = U [G. F(~F,) n M~.P'] c G. r(~F) n ,.,+

,A~ cP t

F'CF

For k E No we set

(M~P')k := {x e MTt I d(x)= k}. C o r o l l a r y 6.4.19

1) (A/[+cpt~/ k

"--Ud(~F)=k

(6.13)

(G. F ( ~ F ) n "AACPt " + )"

Z) Each G . F ( ~ F ) with d('~v) : k is open in (M~_Pt)k with respect to the induced topology. Proo~ 1) is obvious. cpt , since 2) It suffices to show that each G 9F ( -~- 2 F ) n , AACpt - , + is closed in ,A/[+ the union is finite 9 Using Corollary 6.4.16 and Corollary 6.4.18, we calculate AAcpt n (~p'),, G r(~F) n,.,+ 9

=

=

-

_-

G. F(~F) n

U [a. r(~F,l n (MT'lk)

F'CF

G . r(~F) n

since d('~F,) < d(-~F) for F ' strictly contained in F.

6.5

(~F')~

T h e S p a c e SL(3, R ) / S O ( 2 ,

AA cpt

[3

1)

We conclude this chapter with a detailed discussion of the space .h4 = S L ( 3 , R ) / S O ( 2 , 1 ) . Thus we let G = SL(3,R), g = s[(3,R). As a Caftan involution g we use 0 9 g -+ g , X ~ - t X , which yields K = S0(3),1~ = s0(3, R) and P=

*c

-TrA

6.5.

T H E S P A C E SL(3,R)/SO(2, i)

191

As a maximal abelian subspace a of p we choose

{('+S 0 O)

a=

0 0

so that

-r+s 0

0 -2s

{(al 0 O)

A = exp a =

0

a2

0

0 a3

0

r, s G R

,

ai > O, a l a2a3 = 1

9

Moreover, we have M

=

Zg(a)

0 1), (Io I ~ )(0 1 ,2)} 0 {,3(,20 Now consider the involution T : g --+ g given by tc

-tA tb

d

(o 1

c ) -d

1~

01) O(X) ( 0

-

and its global counterpart, which is given by the same formula. Then H

=

so(2,

~) d

=

tc

=

so(2,1)=

q

--tc

ESL(3, R) tb

0

b, c E ]~2,

d E R, t A A - ctc = 1,_1 tab -- dc, Ilbll 2 - Idl 2 -

1

tA = - A , b E R 2 } ,

-TrA

The c-dual objects are G~ =

SU(2, i)

/ A~) (t C

E SL(3, C)

A e M(2,C), b, c E C2, d E C, A * A -"Stc = 1

A'b = de, Ilbll ~ - Idl 2 - -X

/

CHAPTER6. THE ORDER COMPACTIFICATION

192

gc =

5~(2,1)

t) + iq

tb - itc - i Tr B K

{(aO

C

{(a

tC

o )~. -TrA

0

(1 0 0)

~}

(10 0)

We write a = RH1 + RH2 with HI:=

0 0

-1 0

0 0

and

//2"=

0 0

1 0

0 -2

and note t h a t X ~ = 1//2 is a cone-generating element in qp. As a system of positive roots we choose A + = {a13, a23, a12 } and note t h a t 2r

al2(rH1 + s n 2 )

=

~23(rH1 + 8//2) al3(rH1 + sn2)

=

- r + 3s

=

r + 3s,

so t h a t A + = {a13, a23 }, The corresponding root vectors are E12

=

E23

=

lO) (o oo) (o ~ 0 0 0 0

0

El3

=

0

0 Therefore we have n+

Ao~ -- {~12}.

0 0

0 0

,

0 0

1 0

,

0 0

0 0

.

{(o o x) 0 0

0 0

U 0

x, y E R

6.5. THE SPACE SL(3,R)/SO(2,1)

{(1o X) o) o) {(1, *)/ {(lo i)}{(oo o)

9193

=

0 1 Y 0 0 1

x,y

no =

0 0 0

zER},

N+

E

R},

0 0 0

1 0 0 0 1

No

=

0

N

=

0 1 0 0 1

and

N ~=

Moreover, ac=

xER},

* 1

0 b 0 0 0 -a- b

a, bG C}

is a Cartan algebra of ~tc = s[(3, C). This leads to (pc)+

(pc)g~

-{(o1 ~)1 ~~2} 1

{(~0

b bGC2}, ~c,+ {(o0 o) 0) } (p~)-= {(o 0 c E C 2 } , O) o ) ) det A-1 A G GL(2, C) , 1

~ec

2

,

t~

and

K~.(pc)+ = {(A0 detA -1

[AGGL(2,C),cGC 2 .

Now we can write down the maximal parabolic,

b )

Pm~x=GNK~.(p~)+ = {(A0 det A- 1 A G GL(2,R), b G R2 / , and identify the corresponding flag manifold as

G/Pm,,x = K/K gl Pm~x = SO(3)/O(2) = RF 2.

194

C H A P T E R 6. T H E O R D E R C O M P A C T I F I C A T I O N

The domain (ft-)c is given by 1

0

z ec 2 }

and the action of Gc = SL(3, C) on (f2_)c can be described by A

b) " z = (tc+ d t z ) ( A + b t z ) -1

d

because (A tc

b)(1 d " tz

0) ( 1 1 = (tc + dtz)(A + btz) -x

0) (A' 1 0

b') d' "

Next we compute the H-orbit of 0 E n_. Since ( H f ' I K ) . 0 = {0}, we have

:)

(atb

to consider only symmetric elements g =

E H. Since H = - H ,

we may also assume that d > 0. Tn~n a = ~ / t + Ilbll2 and 'A = A implies that A = ~/1 + btb, where b E R 2 is arbitrary. In particular, we find that g, o = ' b ( J t + Ilbll2)-a.

Since the spectrum of btb is {0, Ilbll2 } and ~ _ is rotations-invariant, we see that ~ _ -- H . 0 - {z e R2 9Ilzll < 1}. In order to compute the affme semigroup we consider the group (al B ~ = N ~ A ~ N _ x sdir ( N ~ A ) =

0 $

0 a2 $

0 ) ai>0),

(.1.2) -1

which acts on N_ ~- R 2 via al z

a2

0

0

I

x

y

(ala2) -1

0

9( z ' , ~ ' )

=

ala2 \ al

]"

-- z y ' + (a2 - - a l )X, a2

The linear contractions A of 12_ are those with I[AI[ < 1. Using the fact that IIAII2 -IIAA*II, one obtains for example that

II(o b)lL c

= 2

(a2 +

b2+ c2) + ~/](a2 + b2+

) -4a2c2]

"

6.5.

195

T H E S P A C E SL(3, I t ) / S 0 ( 2 , 1)

al

0

0

z

a2

0

0

0

(ala2) -1

Forg=

) G_ N~oA we have

that

g" (x,y) = (axa2) -1 ( ai-10 a~ 1

V

"

So g. f/_ C f2_ if and only if

1 (

2a21a2 (al 2 + a2 2 + Z2) ~- ~/l(ai -2 + a~ "2 + Z2)2 --4(ala2)-21

)

_< 1.

O)

For z = 0 this is equivalent to max{aT 2, a 2 2 } / (a 21a 2) 2 _< 1. So we find for g = exp(rH1 + sH2) =

er+ S

0

0 0

e -r+s 0

0

e-2S

the condition e 2[rl-6s < 1

< :-

It[-- 3s < O.

Because of Cmax

--

= Cmin = =

{~H1 + sH2 I I~l <_ 3,} R+(3H~ + H2) + R+(-3H1 + H2),

Cmax = { r i l l + ~H2 I I~1 _< ,} R+(H1 + H 2 ) + R + ( - H 1 +H2),

this condition is also equivalent to r i l l + sH2 E Cmsx. The cone cone(A_) = --Cmax = --Cmin has four faces:

Fo

=

{0},

F1

=

-R+(H1 + H2),

F~ =

-R+(H1 - n 2 ) ,

and

F2 -----Cmin 9 The faces Fs and F[ are conjugate under the Weyl group which acts by reflections on the line R+H1. Corresponding to these faces we have four idempotents in "A~'cpt" Clearly eFo is the identity, whereas eF2 = 0. We have eF1 -" e x with X - -3H1-

H2 --

( oo) 0

2

0

0

0

2

G a-.

196

CHAPTER

6.

THE ORDER

COMPACTIFICATION

An interesting feature in this example is t h a t the causal Galois connection F: ~'(X) ~ ~'(G) is not injective. To see this, let

at :=

0 0)(,,

0 0

e 2t 0

0 e 2t

and

a 't 9=

0 0)

0

e -4t

0

0

0

e 2t

T h e n a t . (x, y) = (e6tx, y), lim a t . f l - = R x [-1,1],

t-+oo

a~a a~. (~, v) = (~, ~'v), ~o l i m a 't . ft_ = [-1,11 x R.

t-+co !

T h e sequences a . and a . are not b o u n d e d in the order induced on A, hence

r(o,. ~_) = ~(o,) -+ 0 and

r ( . ; . ~_) = ~(.;) -+ 0 in ~-(G). since lira a t . f l - # lira a t 9

t--~oo

t--+oo

we see t h a t F is not injective. We now describe the groups H x before. T h e n A x , + = {a23 } and

I~x

=

and HR.

,

Let X = - 3 H 1 - / - / 2 be as

lim e ad t x r / = pru(g~23 ) q) ga~2 q) fl~3 t-~oo

=

0 0

0 z

z 0

x,y,z~.R

.

This is a solvable Lie algebra of the type R 2 x sdi~ R, where R 2 is the sum of two real root spaces. T h e stabilizer algebra is given by

lit = R X + Dx = R X (B Pr~(ga23) (3 g~,2 (B g~,8. Let us write e x = eft = (1, 7). T h e n

-1

Ira7

=

g-o23,

ker'),

=

g-a18,

=

exp(RX),

pex(ex)

6.5.

T H E S P A C E SL(3, R ) / S O ( 2 , 1 )

and L(ker

~ex) ----0-~13

197

X,dir (g-~2 ~ RX).

The maximal parabolic Lie algebra Pmax of dimension 6 is

Pmax

=

"-

{(Ol1

m ~ a e g(~o) e g(~x+) a21

a22

a23

0

0

- - a l I - - a22

)

aij E R

9

The set I~F1 is 7(1~_) = ] - 1, 1['E32, and its pointwise stabilizer has the Lie algebra N Ad(y)Pmax = R X ~ g~12 ~ g~ls" yEIm 7 The normalizer of its nilradical is the maximal parabolic algebra

Notes for Chapter 6 The material of this chapter has been developed in [55] in order to get a hold of the ideal structure of the groupoid C*-algebra naturally associated to any ordered homogeneous space (cf. [54], [108], and the notes for Chapter 9).

Chapter 7

Holomorphic Representations of Semigroups, and Hardy Spaces In the next three chapters we will ~ve an overview of some of the applications of the theory of semigroups and causal symmetric spaces to harmonic analysis and representation theory. We present here only a broad outline of the theory, i.e., the main definitions and results, but mostly without proofs. We refer to the original literature for more detailed information. In the notes following each chapter the reader will find comments on the history of the subject and detailed references to the original works. In this chapter we deal with highest-weight modules, holomorphic representations of semigroups, the holomorphic discrete series, and Hardy spaces on compactly causal symmetric spaces. The original idea of the theory goes back to the seminal article by Gelfand and Gindikin in 1977 [34], in which they proposed a new approach for studying the Plancherel formula for semisimple Lie group G. Their idea was to consider functions in L2(G) as the sum of boundary values of holomorphic functions defined on domains in Gc. The first deep results in this direction are due to Ol'shanskii [139] and Stanton [159], who realized the holomorphic discrete series of the group G in a Hardy space of a local tube domain containing G in the boundary. The generalization to noncompactly causal symmetric spaces was carried out in [63, 133, 135]. This program was carried out for solvable groups in 198

7.1. HOLOMORPHIC REPRESENTATIONS OF SEMIGRO UPS [64] and for general groups in

7.1

199

[91].

Holomorphic Representations of Semigroups

Let Gc be a complex Lie group with Lie algebra gc aud let g be a real form of gc. We assume t h a t G, the analytic subgroup of Gc with Lie algebra g, is closed in Go. Let C be a regular G-invariant cone in g such t h a t the set S(C) = G e x p i C is a closed semigroup in Gc. Moreover, we assume t h a t the m a p G x C 9 (a,X) ~-+ a exp i X e S(C) is a homeomorphism and even a diffeomorphism when restricted to G x C ~ Finally, we assume t h a t there exists a real automorphism a of Gc whose differential is the complex conjugation of 0c with respect to g, i.e., a ( X + iY) = X - i Y for X, Y E g. All of those hypotheses are satisfied for Hermitian Lie groups and also for some solvable Lie groups; cf. [64]. Let Ir : G --+ U ( V ) be a unitary, strongly continuous representation of G in a Hilbert space V. A vector v E V is called a smooth or C~176 if the map R B t ~-4 t3(t):= Ir(exp t X ) v E V is smooth for all X E g. Here a map f : U -4 V, U C R n open, is called differentiable at the point xo E V if there exists a linear m a p Df(xo) := T=,f : V ~ V such t h a t

f(x) = f(xo) + Df(xo)(X- xo) +

o(II - ~oll).

The function f is of class C I if f is differentiable at every point in V and x ~-~ D r ( x ) E H o m ( V , V ) is continuous. Moreover, f is of class C2 if O f ( x ) is of class C 1. We denote the differential of O f ( x ) by D2f(x). We say t h a t f is of class Ck+l if D k f is of class C 1. In t h a t case we define D k + l f := D ( D k f ) . Finally, we say t h a t f is smooth if f is of class Ck for every k E N. Let V ~176 be the set of smooth vectors. V ~176 is a G-invariant dense subspace of V. We define a representation of g on V ~176 by

r~176

= lim ~ r ( e x p t X ) v - v . t~o

t

We denote this representation simply by lr or use the module notation, 1r(X)v = X . v. We extend this representation to Oc by complex linearity, r ( X + iY) = r ( X ) + Jr(Y), X, Y e g. Let U(g) denote the universal

200

C H A P T E R 7. H O L O M O R P H I C R E P R E S E N T A T I O N S

enveloping algebra of gc. Then lr extends to a representation on U(g), again denoted by lr. The set V ~176 is a topological vector space in a natural way, cf. [168]. Furthermore V~176 is G-invariant and U(g)-invariant. As lr(g) e x p ( t X ) v = exp(t Ad(g)X)lr(g)v, we get

Ir(g)Ir(X)v = Ir(Ad(g)X)r(g)v for all g E G and all X E g. Define Z* = - a ( Z ) , Z E gc. Then a simple calculation shows that for the densely defined operator It(Z), Z 6 gc, we have r ( Z ) * = ~r(Z*). Define

C ( r ) := {X e g lVu e V ~1769 (irr(X)ulu)

< 0},

where ('l') is the inner product on V. Thus C(~r) is the set of elements of g for which r ( i X ) is negative. The elements of C(Tr) are called negative elements for the representation lr. L e m m a 7.1.1 C(lr) is a closed G-invariant convex cone in g.

El

Let C be an invariant cone in g. We denote the set of all unitary representations ~r: G --+ U(V) with C(~r) C C by .A(C). A unitary representation lr is called C-admissible if r E .A(C). Let S be a semigroup with unit and let ~ 9S --+ S be a bijective involutive antihomomorphism, i.e.,

(ab) ~ = b~a~ and

a~ = a

We call ~ an involution on the semigroup S and we call the pair (S, ~) for a semigroup with involution or involutive semigroup. E x a m p l e 7.1.2 The most important example will be the semigroup S ( C ) with the involution (this is a special case of the involution ~ on p. 121 for general symmetric pair). In this case (a exp i X ) ~ = a - ' expi A d ( a ) X e S(C). [] E x a m p l e 7.1.3 ( C o n t r a c t i o n S e m i g r o u p s o n a I-Iilbert S p a c e ) Another example is the semigroup C ( V ) of contractions on a complex Hilbert space V:

C(V) = {T e Horn(V)I IITI[ _< 1} Denote by T* the adjoint of T with respect to the inner product on V. Then ( C ( V ) , ,) is a semigroup with involution. []

7.1. H O L O M O R P H I C R E P R E S E N T A T I O N S OF SEMIGRO UPS

201

D e f i n i t i o n 7.1.4 Let (S, fl) be a topological semigroup with involution, then a semigroup homomorphism p : S --+ C ( V ) is called a contractive representation of (S,~) if p(g~) = p(g)* and p is continuous w.r.t, the weak operator topology of C ( V ) . A contractive representation is called irreducible if there is no closed nontrivial subspace of V invariant under

p(s)

o

D e f i n i t i o n 7.1.5 Let p be a contractive representation of the semigroup S(C) C Gc. Then p is holomorphic if the function p: S(C) ~ --~ Horn(V) is holomorphic. [3 The following lemma shows that, if a unitary representation of the group G extends to a holomorphic representation of S(C), then this extension is unique. L e m m a 7.1.6 If f: S(C) ~ C is continuous and fls(c)o is holomorphic

such that f i g = O, then f = O.

[3

To construct a holomorphic extension p of a representation Ir we have to assume that 7r E jr(C). Then for any X E C, the operator irr(X) generates a self adjoint contraction semigroup which we denote by

T x ( t ) = e 'i~(x) . For s = g exp i X 6 S(C) we define

p(s) : = p(g)Tx(1) T h e o r e m 7.1.7 p is a contractive and holomorphic representation of the

semigroup S(C). In particular, every representation rr E ft.(C) extends uniquely to a holomorphic representation of S(C) which is uniquely determined by lr. D

For the converse of Theorem 7.1.7, we remark the following simple fact. Let (S, ~) be a semigroup with involution and let p be a contractive representation of S. Let

G(S) : =

e S

= ,,8" = 1}

Then G(S) is a closed subgroup of S and lr := PIG(s) is a unitary representation of G(S). Obviously,

G c G(S(C)) Thus every holomorphic representation of representation of G by restriction.

S(C)

defines a unique unitary

202

CHAPTER 7. HOLOMORPHIC REPRESENTATIONS

T h e o r e m 7.1.8 Let p be a holomorphic representation of S(C). PIG E A(C) and the p agrees with the extension of PIG to S(C).

Then r'l

Two representations p and lr of the Ol'shanskii semigroup S(C) are (unitarUy) equivalent if there exists a unitary isomorphism U: V p --+ V~r such that up(,) = v , e s(c) In particular, two contractive representations p and lr of S(C) are equivalent if and only if Pie and 1rig are unitarily equivalent. We call a holomorphic contractive representation p of S(C) admissible if PiG E ~4(C) and write p E jr(C). We denote by S(C) the set of equivalence classes of irreducible holomorphic representations of S(C). L e m m a 7.1.9 Let rr be an irreducible holomorphic representation of S(C).

Then the function :=

is well defined for every s E S(C) ~ holomorphic and positive definite,

Furthermore 0,~ : S(C) ~ --+ C is ra

T h e o r e m 7.1.10 (Neeb) Let rr and p be irreducible holomorphic representations of S(C). Then 7r and p are equivalent if and only if 0,, = Op. [] A nonzero function a: S(C) ---r R + is called an absolute value if for all s,t E S(C) we have a(st) <_ a(s)a(t) and c~(s~) = a(s). Let c~ be an absolute value. A representation p of S(C) is a-bounded if

IIp(,)ll _< for all s E S(C). Note that this depends only on the unitary equivalence class of p. We denote by S(C)(a) the subset in S(C) of a-bounded representations. If lr E S(C), then, by abuse of notation, a(s) := [[r(s)l I is an absolute value of S(C). Let (p,V) and (Tr,W ) be holomorphic representations of S(C). Let V~)W be the Hilbert space tensor product of V and W . Define a representation of S(C) in V~)W by L0 o

:= p ( , )

Then p | 7r E A(C). We denote the representation s ~ id by ~. T h e o r e m 7.1.11 (Neeb, Ol'shanskii) Let (p, V) be a holomorphic representation of the Ol'shanskii semigroup S(C) and let a ( s ) = IIP(')II. Then

203

7.2. H I G H E S T - W E I G H T M O D U L E S there exists a Borel measure I~ on S ( C ) supported on integral of representations

(~)

S'(C)(c~) and

a direct

(o)

such that: 9

9

2) There exists a subset Y C S ( C ) ( ~ ) such that p ( Y ) = 0 and if w E S(C~)(c~) \ N , then (O~, V~) is equivalent to (rrw | lr.., E w and W.., is a ttilbert space.

t t w 6 W ~ ) , where

3) If w E S(C)(c~) then set n(w) := dim W,,,. Then n is a p-measurable function from S ( C ) ( ~ ) to the extended positive axis [0, 0o] which is called the multiplicity function. D

7.2

Highest-Weight Modules

Representations with negative elements can exist only if there exists a nontrivial G-invariant cone in the Lie algebra g. If g is simple, this implies in particular t h a t g is Hermitian. We will thus assume from now on t h a t g is a semisimple Hermitian Lie algebra. Thus G is a semisimple Hermitian Lie group. For simplicity we will assume that G is contained in a simply connected complexification Gc. Then G = G~ and (Gc, G) is a noncompactly causal symmetric pair. Let Z ~ be a central element in t~ defining a complex structure on p. Then X ~ = - i Z ~ is a cone-generating element for (Gc, G) and the corresponding eigenspace decomposition is gc = n+ 9 h ~ 9 n_ as before. A comparision with the standard notation for Hermitian Lie algebras (cf. Example 5.1.10, p. 124) yields

n+=p+,

n_=p-,

and

I}~=l~c.

Thus N+ corresponds to P + = exp p+, N_ corresponds to P - := exp p - , and Go corresponds to K c = G~. Thus

G c P+KcP-

204

CHAPTER 7. HOLOMORPHIC REPRESENTATIONS

(cf. Lemma 5.1.4, Remark 5.1.9, and Example 5.1.10). Let us recall some of the notation introduced in Section 2.6.1. For x = pkq E P + K c P - we write p+(x) = p, p - ( x ) = q and kc(x) -- k. By Lemma 5.1.2, the set G K c P is an open submanifold of the complex flag manifold , ~ = G c / K c P - and G f 3 K c P - = K. This implies that G / K is holomorphically equivalent to an open submanifold D of No. We also have the map x ~ ~(x) = log(x), which maps G / K biholomorphically into an open symmetric domain ~+ C f~+ (cf. Theorem 5.1.8). If Z E p+ and g E G c is such that g exp Z E P + K c P - , then g. Z = z(g exp Z). Moreover, we have the universal automorphic factor j(g, Z):-- kc(g exp Z). For j we find

j(k,Z)

=

j(p,Z)

=

j(ab, Z)

=

k, j(a, b . Z)j(b, Z),

if k E K c , Z E p+, p E P + , and a, b E G are such that the expessions above are defined. A (g, K)-module is a complex vectorspace V such that 1) V is a g-module. 2) V carries a representation of K, and the span of K . v is finitedimensional for every v E V. 3) For v E V and X E t we have X . v = lim ex._,___,vnftX~ t~o

t)

t

4) For Y E g and k E K the following holds for every v E V: k . ( X . v) = ( A d ( k ) X ) . [k. v]. Note that (3) makes sense, as K . v is contained in a finite dimensional vector space and this space contains a unique Hausdorff topology as a topological vector space. The (g,K)-module is called admissible if the multiplicity of every irreducible representation of K in V is finite. If (r, V) is an irreducible unitary representation of G, then the space of K-finite elements in V, denoted by VK, is an admissible (g, K)-module. Let t be a Cartan subalgebra of ~ and g. In this section, let A denote the root system A(gC, tC).

7.2. H I G H E S T - W E I G H T M O D U L E S

205

D e f i n i t i o n 7.2.1 Let V be a (g, K)-module. Then V is a highest-weight module if there exists a nonzero element v E V and a A E t~ such that 1) X . v

= A(X)v for all X E t.

2) There exists a positive system A + in A such t h a t IIc(A+) 9v = 0.

3) v = v ( g ) The element v is called a primitive element of weight A. Let now C E Conea(g) and let (p, V) E jr(C). We assume that - Z ~ E C ~ We assume furthermore that p is irreducible. Then V K is an irreducible admissible (g, K)-module, and =

v

(a)

xet~ where VK(A) = VK(A, i t ) . Let v E VK(A) be nonzero. Let c~ e A(p+, tO) and let X E p + \ {0}. Then X k . v ~.

VK(~ + ko~).

In particular, - i Z ~ (X k. v) = [ - i A ( Z ~ + k]v.

This yields the following lemma. L e m m a 7.2.2 Let the notation be as above. Then the following holds:

1) -i

(z ~ <_ o

2) There exists a A such that p + . VK(A) = {0}.

Furthermore, the following holds. L e m m a 7.2.3 Let W x be the K-module generated by is irreducible and V g -- U ( p - ) W A.

VK()i ).

Then W x

17

Let Ak = A(tc, tC) and let A + be a positive system in AK. Let p be the highest weight of W x with respect to A+ and let v x be a nonzero highest weight vector. Then v x is a primitive element with respect to the positive system A + U A +, where A , = &(PC, t C ) a n d A+ = A(p+, tC). A s - i Z ~ is a positive linear combination of the vectors Ha, c~ E A+, we get the following theorem.

CHAPTER 7. HOLOMORPHIC REPRESENTATIONS

206

T h e o r e m 7.2.4 Let (p, V) E A(C) be irreducible. Then the corresponding

(g,K)-module is a highest-weight module and equals U ( p - ) W x. In particular, every weight of V K i8 O f the form V~

E

~(p+,~)

F~.

F~th~.~o~, (~1~) < 0 lo~ .u ~ e ~+~.

I-I

We will now show how to realize highest-weight modules in a space of holomorphic functions on G / K . We follow here the geometric construction by M. Vavidson and R. Fabec [18]. For a more general approach, see [119]. To explain the method we start with G = SU(1,1). We set according to Example 2.6.16: E = E I = (10

--O) T h u s Z ~

0

0 ,F=E_t=

(o o) 1 0

='iHandp+=CE'tc=CHandp-=CF"

Let

(a ~) E SU(I I) and let 7=

(o b) ESL(2,C). c

d

We identify p+ with C by z E ~-~ z and similarily tc - C by z H ~ z. Then induces an isomorphism ~(g) = ~ of G / K onto the unit disc D = {z E C IIzl < 1}. Furthermore,

az+b cz+d

7"z= and

j(g,z)

=

(cz + d) -1 .

The finite-dimensional holomorphic representations of Kc are the characters x.(exp ziH) = e i n z . In particular,

x.(z ~

i2

=

2 or

-ix.(z ~

n

=

~.

7.2. HIGHEST- W E I G H T MODULES

207

Let (Tr,V) be a unitary highest-weight representation of SU(1, 1) and assume that (r, V) 6_ .d(C). Then n < 0. Let V(n) be the one-dimensional space of xn-isotropic vectors. Then v.

=

v(.kEN

and the spaces V ( m ) and V(k) are orthogonal if m # k. Let a be the conjugation of s[(2, C) with respect to 5u(1, 1). Then a is given by

( 0

]

so that a(E) = F. By Ir(T)* = -lr(a(T)) for all T 6_ $[(2, C) we get 7r(F)* - ~r(-E). Finally, it follows from [F, E ] -

II~(F)%II 2

- H that for v 6_ V(n)"

= =

(Tr(F)kv I~(F)%) (lr((-X)kF)v It,)

L e m m a 7.2.5 Let the notation be as above. Then

7r(-E)k r( f )kv

r ( . + 1)

(-1)kk!r(.- k + 1) V (-,-,)~,,:

where (a)k - a(a $ 1 ) . . . (a + k - 1). As

~(-n)~ I ~=(1-1~12) ", k=.O

(cf. [361) converges if and only if ]z I < 1, it follows that oo

q(~g)V := ~'~-5 k Fkv n! k=0

converges if and only if z X 6_ f~+. Let , o w G be arbitrary. Let a 9 gc ~ gc be the conjugation with respect to g. We use the notation from earlier in this section. Using the usual s[(2, C) reduction, we get the following theorem.

CHAPTER 7. HOLOMORPHIC REPRESENTATIONS

208

T h e o r e m 7.2.6 ( D a v i d s o n - F a b e c ) Let T E p+. Define qT" W x --+ V by the formula oo

q~, := ~ ~(y)~, " •! ~)" k--0

1) If v ~ O, then the series that defines qT converges in the Hilbert space V if and only if T E f~+. 2) Let 7ra be the representation of K on W x. Let

s~(g,z) := ~(j(g, z)). Then

~(g)o = q~ 0s~(g, 0)*-xo forgEGandvEW

x.

E!

It follows that the span of the q z W x with Z E f~+ is dense in V, since V is assumed to be irreducible. Define Q ' f ~ + • f~+ --+ G L ( W x) by

Q(W, Z) = q~vqz . Then the following theorem holds. T h e o r e m 7'.2.7 ( D a v i d s o n - F a b e e ) Let the notation be as above. Then the following hold:

1) Q(w, z) = Sx(exp(-a(w)), T) *-x. 2) Q(W, Z) is holomorphic in the first variable and antiholomorphic in the second variable.

3) (Q(W, Z)ulv) = (qzulqwv) Ior alt u, v ~ Wx. 4) Q is a positive-definite reproducing kernel. 5) Q(9. W,g. Z) = Jx(g,W)Q(W,Z)Jx(g,Z)*.

r~

For Z E 12+ and u E W x, let Fz,u " 12+ -+ W x be the holomorphic function

Fz,~(W) := Q(W, z)u and define

(Fz,u[Fr,w)Q := (Q(T, Z)u, w). Let t t ( ~ + , W x) be the completion of the span of {Fz,u I Z E ~+, u E W ~} with respect to this inner product. Then tt(f~+, W ~) is an Hilbert space

7.3. THE HOLOMORPHIC DISCRETE SERIES

209

consisting of W X-valued holomorphic functions. Define a representation of G in H(ft+, W x) by (p(g)F)(W) := Sx(g -1, W ) - I F ( g -1. W ) . Then p is a unitary representation of G in H(fl+, W x) called the geometric realization of (lr, V). T h e o r e m 7.2.8 ( D a v i d s o n - F a b e c ) The map qzv ~ Fz,v extends to a unitary intertwining operator U between (rr, V) and the geometric realization (p,H(f~+,WX)). It can be defined globally by [Uw](Z) =q*z w,

rn

w E V, Z G ~+.

As the theorem stands, it gives a geometric realization for every unitary highest-weight module. What is missing is a natural analytic construction of the inner product on H(~2+, WX). This is known only for some special representations, e.g., the holomorphic discrete series of the group G. For that, let p = 89~ a e a + a and let p denote the highest weight of the representation of K on W x. Furthermore, let dZ denote the usual Euclidean measure on p+. For f , g G H(f~+,WX), let (Ilg)x := fG

/K

(Q(Z,Z)I(Z)Ig(Z))w~dZ

T h e o r e m 7'.2.9 (I-Iarish-Chandra) Assume that (p + pla) < 0 / o r all cr G A +. Then (flg)x is finite .for f , g G H ( I I + , W x) and there exists a positive constant cx such that

(IIg), = Moreover, (p,H(~2+,Wx)) is unitarily equivalent to a discrete surnand in L2(G). [3

7.3

The Holomorphic

Discrete

Series

In this section we explain the construction of the holomorphic discrete series of a compactly causal symmetric space A4 - G/H. In the next section we will see that those are the admissible representations of the Ol'shanskii semigroup that can be realized as discrete summands in L2(A4). We start with a simple structural fact about SU(1,1).

210

C H A P T E R 7. H O L O M O R P H I C R E P R E S E N T A T I O N S

Define an involution on SU(1,1) by r(a) = a, cf. Section 2.6.16: { Hc:=SL(2,C) r

=

hz=

=

{(~

(coshz sinhz

b

sinhz) coshz

} zEC

a, b e C, a 2

a

= 1

}

and H = SU(1,1) r = 4-{ht I t e R}. Let ap be the maximal abelian subalgebra of qp given by ap = RXo. Let at = exp t Xo =

( cosh ( 89 i shah ( 89

- i sinh ( ~ ) ) cosh ( 89 '

tER "

Define a

---

b

and z --

cosh (t/2) ~/cosh t ' - i sinh (t/2) ~/cosh t 1 ~/cosh t '

3' cosh(t/2) - a7 2 b

i7 sinh(t/2) - b7 2 a

Then a :~-

:=

b)

E Hc,

0) (10) b

a

0

7 -1

z

1

EKe,

E P-

and at = hkp. Now we go back to the general case. Let a be a maximal abelian subalgebra of qk. Then ia is a maximal abelian subspace of q~. Since GC/H C G c / H c is a noncompactly causal symmetric space as in Section 4.1 and 4.2, we find homomorphisms ~oj : SL(2, C) --+ Gc intertwining the above involution on SL(2, C) and the given involution r on Gc. We may assume that our algebra ap is spanned by the elements X~ "= - i ( E ~ - E _ j ) = ~j

T- i ( 0 s"

7.3. THE H O L O M O R P H I C D I S C R E T E SERIES

211

where 7 1 , . . . , 7r are the strongly orthogonal roots. Let Ap := exp ap. Using G = H A p K (cf. [97]) and the fact that Kc normalizes P - , a ( P +) = P and G = G -1 = a(G), we get the following theorem. T h e o r e m 7.3.1 Let the notation be as above. Then r

1) Let a ( t l , . . . ,t,) = e x p ( E j = l tj) E Ap = expap. Then

[1,

a ( t l , . . . , tr) e HC exp ~ E - log(coshtj) Hj j=l

]

P-

2) G C H c K c P - n H c K c P + n P - K c H c n P + K c H c . If x = hkp E H c K c P - , we write

h(x) = h,

kH(x) = k,

and

p~t(x) = p .

Note that h(x) and kH(x) are only well defined modulo Hc 13 Kc. Let 7r be a holomorphic representation of Kc with nonzero (He f'l Kc)fixed vector. Denote the highest weight of 7r by/J~. A simple generalization of Theorem A.2.2 and Lemma A.2.5 to the reductive Lie group (K n H ) e x p iqk gives the following lemma. L e m m a 7.3.2 tt~ E ia and V ~ enKc is one-dimensional. Let Vo be a nonzero (HcnKc)-fixed vector. Define ~

: P - K c H c -+ V,~

by

(I)Ir (X) :'-- 71"(kH(X -1)-l)Oo. We define a map V~ --+ O ( P + K c H c ) v ~ ~(Tr, v) by

v(., where O ( P + K c H c ) denotes the holomorphic functions on P + K c H c . By construction we have the following lemma. L e m m a 7.3.3 Let the notation be as above. Then the following hold: 1) Let p e P+, h E Hc, and x e P + K c H c . ~) For k E KC we have ~o(71",v)(k-lx) v ~ ~(Tr, v) is a Kc-intertwiner.

Then ~p(~r,v)(pxh) =

-" ~o(ff, 71"(]g)v)(x); i.e., the map 0

212

C H A P T E R 7. H O L O M O R P H I C R E P R E S E N T A T I O N S

As ~o(rr, v) is right Hc-invariant, we view it as a function on P +K c Hc / Hc = P +K c " o C .M c .

As G C P + K c H c , we can by restriction view ~o(lr, v) as a function on j~vt. To decide when ~o(Tr,v) is in L2(A4), we write the G-invariant measure on ~4 in polar coordinates using G = K A p H . Let a E A({t, ap). Let P~

=

dim{X e

Ie

(x) = x } ,

q~

=

dim{X e lta I 0 r ( X ) = - X } .

Then there is a positive constant c such that

cf. [32], where A +(o, %) is a positive system in A(0, ap). A = A(Oc, ac) = A(Oc, tc). Then

,a =

Recall that

,c)O,a(p +, ,c)O,a(p-, ,.c).

Let A + be a positive system in A([~c, ac), let A + "= A(p +, ac), and finally, l e t A + = A + U A +. Let 1

p= ~ ~

[dirnc(gc)~]a.

aEA+

By Lemma 7.3.1.1), we get the following theorem. T h e o r e m 7.3.4 Let v E V,~, v # O. Then ~o(r, v ) l ~ E L2(A4) if and only if ( ~ + pl~) < 0 for all a e A +. m T h e o r e m 7.3.5 Assume that (p,~ + pla) < 0 for all ~ e A +. Let E,~ be the closed, G-invariant module in L2(A4) generated by {~(r, v) l v E V~}. Then the following hold. 1) E~r is irreducible. 2) E,~ is a highest-weight modute with a primitive element ~(r, w), where w is a nonzero highest-weight vector for lr. 3) The multiplicity of E,r in L2(A4) is 1.

E1

Denote the representation of G in E~ by p~. The representations (p~, E~) are called holomorphic discrete series of .A/l.

7.3.

THE HOLOMORPHIC DISCRETE SERIES

213

As E~ is a highest-weight module, there exists an G-intertwining operator T,~ 9E,~ + H(n+, V,~). To construct such an operator, define

@,(g,x) := ~r(kc(g))@,(g-Zx), g 6 G, x e P + K c H c . Then g ~ ~I,,(g, x) is right K-invariant and defines a holomorphic function on G / K . T h e o r e m 7.3.6 The map T" E,~ ~ H ( f l + , V , ) , defined by

Tf(z) :=/~

f(x)~=(z,x)dx,

is a nonzero intertwining operator and H o m G ( E , , H(fl+, V , ) ) = C T .

rl

E x a m p l e 7.3.7 Let G be a connected semisimple Lie group and let G1 G x G, H = diag(G). As we have seen, G = G I / H in this case. Furthermore, q - { ( X , - X ) I X 6 g}. The Cartan subspace a is constructed by taking t to be a compact Cartan subalgebra of g, t C [~, and then setting a = { ( X , - X ) I X 6 t}. Let ~ 6 A(gc, tc) and let X• 6 (gc)+a. Then

[(x,-x), for every ( X , - X )

=

6 a. In this way we get a bijective map,

We see also that the root spaces are exactly (gc)a x (gc)-~. In particular, Z ~ - (Z~ ~ and the space p+, where Pl - P x p C gl, is given by p+ = p+ x p - . The bounded realization of G I / K z is Gz/K~ = G / K x G / K , where ~ means the opposite complex structure. So a holomorphic function on G I/K1 is the same as a function f : ~+ x fl+ ~ C that is holomorphic in the first variable and antiholomorphic in the second variable. If 7r1 is an irreducible unitary representation of Kz - K x K, then ~rl is of the form r | ~, where ~r and 6 are irreducible representations of K. In particular, V~ 1 _~ V~ | V6. Assume that Uo is a nonzero diag(K) = (K1 gl H)-invariant element in V~ 1. Define %o" V~ --+ V~r by

< u,

> = (u

vluo),

214

CHAPTER

7. H O L O M O R P H I C

REPRESENTATIONS

where (.[.) is the Kl-invariant inner product on V a t . Then, for k E K,

< ~, ~(p(k)o) >

=

(,, | p(k)oluo)

=

(,,(k-I),,| oluo)

=

(7r1(~,~)(71"(k-1)u| v)lUo)

---~ • 71"(k-1)u, qo(v)~>

= < u,~(~)~(~)> Thus qo 9 V6 --+ V~ is a K-intertwining operator. As both spaces are irreducible, it follows that ~o is an isomorphism. In particular, we have the following lemma. L e m m a 7.3.8 Let rrl be an irreducible representation of K1 with a nonzero diag(K)-fixed vector. Then there exists an irreducible representation 7r of K such that rrl ~_ r | 7rv. r~

Now V~ | V* ~_ Hornc(Va, V~) and the representation is carried over to rrx (k, h ) T = l r ( k ) T r ( h ) -x .

The invariant inner product on Homc(V~, V~) is (TIS) - Tr T S * . In this realization the invariant element uo is (up to constant) the identity id and (Tluo) = T r ( T ) .

Let p be the highest weight of 7r. Then < 7r + p, a > < 0 for every c~ 6 A ( p + , t c ) . Thus ~r corresponds to a unique holomorphic discrete series representation E~ of G. We have (cf. [135]) the following theorem. T h e o r e m 7.3.9 The holomorphic discrete series E,rl is canonically isomorphic to E~r | E~r.

7.4

Classical Hardy

r']

Spaces

In this section we explain the construction of the Hardy space related to a regular cone field on a compactly causal symmetric space. We start with a short overview of the classical theory as it can be found, e.g., in the book by Stein and Weiss [160]. Let C be a regular cone in R n and let f~ = C ~ Let E(f~) := R n + if/. Then =(f~) is an open subset of C n . Let O(f~) be the space of holomorphic functions on E(f~). If f E O(f~) and u e f~, then R n ~ x i-~ fu(x):= f(x + iu) E C

7.4. CLASSICAL H A R D Y SPA CES

215

is well defined. We define the Hardy norm of f with respect to C be

Ilfll~ "= ~,easupIIL, II~,,(R,,) = ,,easuPfR,, If(x + iu)l 2 8x. We can now define the Hardy space

n2(C)

" - -

~l~2(C)

if E

(7.2)

I Ilfll2 <

Define the boundary value map fl 97"/2(C) /3(f) --

by

lim

ngu--+0

--+ L2(][~n)

by

f(.+iu)

Then/3 is an isometry into L 2 (lR'). To describe the image of/3 let

Jr: L 2 (R n) -~ L 2 (R n) be the Fourier transform, i.e.,

1 L

~ ( f ) ( v ) = (2~r)n/2

f ( x ) e -i(=lv) dx,

V f e Cc(Rn).

To simplify the notation we define the function eu, u E C n , by

C n ~ x ~ eu(x) -- e ('lu)

T h e o r e m 7.4.1 Then Im(~) = E.

Let E = { f 6 L2(R") I Supp(~'(f)) C C'} "~ L2(C*). D

Let us sketch the construction of the map E --+ Im(/~). Consider f E E , F = ~ ' ( f ) and let u e f L As S u p p ( f ) C C*, it follows that IRe_u[ < I f I. Hence Fe_u e L2(R"), and we may define g 9E(ft) ~ C by

g(x + iu) = jr-1 (Fe_u)(x). Formally, this is

g(x + iu)

=

(2 i./2 1

By construction, g e H2(f~), and one has to show that/~(g) - f.

216

CHAPTER 7. HOLOMORPHIC REPRESENTATIONS

We define the Cauchy kernel associated with the tube domain E(fl) by

K( x + iu)

=

1 fn . ei(z+iulX) dA (2~r),/2

=

where XA denotes the characteristic function of a set A C R n and x + iu E Z(f~). Then ILK(" + iu)ll ~ = K(2iu) < c~. Thus K(. + iu) E L2(IRn). Furthermore, we get the following theorem. Theorem

7.4.2 Let F E 7-12(C). Then

F(z) = fR~ f ( x ) K ( z - x)dx for all z E E(f~), where f = i~(f).

[3

We define the Poisson kernel by

P(z, y) = IK(z + iy)l 2 K(2iy) , Theorem

x + iy E E(12)

7.4.3 Let f E I-I2(f~). Then

F(x + iy) = fR, P ( x - t, y) f (t) dt, where f = ~(F).

7.5

D

Hardy Spaces

In this section G / H is a compactly causal symmetric space. Let C E ConcH(q) be such that C~ ~t 0. There are two different ways to generalize the tube domain E(fl) from the last section to this setting. First, we may construct a local tube domain in T(J~4)c by G • u iC ~ Second, we may view =([2) as the orbit of 0 E R n under the semigroup R" + i~2. The corresponding construction in this setting is to consider the semigroup S = G exp iD ~ where D E Conea(g). Then E(f~) = S - l o C Gc/Hc. Here we have to assume that G C Gc, where Gc is a complex Lie group with Lie algebra ftc. The inverse is necessary, as we want the semigroup to act on functions on E(C~ We will use the second approach (cf. [64, 63]). In particular, we will assume that G C Gc, where Gc is a complex Lie group with Lie algebra Ilc such that r integrates to an involution on Go. We assume that H = G r.

7.5. H A R D Y S P A C E S

217

Let Hc = G~ and define r := G c / H c . Then M C A4c, and A4c is a complexification of M . Let C E Coneu(q) and D be an extension of iC to a G-invariant closed cone in ig such t h a t D Iq iq = priq D = iC, cf. Section 4.5. Let S ( D ) be the Ol'shanskii semigroup G e x p D . Then S ( D ) ~ = S ( D ~ "= G e x p D ~ ~_ G x D ~ # 0. Define an open complex submanifold E ( C ~ C M c by --7(C~ := S ( D ~

(7.3)

Then E ( C ) " = S ( D ) - l o C E(C~ where the bar denotes the closure in A4c. Furthermore, A4 C a E ( C ) . For simplicity we will write S for S ( C ) , S ~ for S(C~ E for E(C), and E ~ for its interior --(C~ That E and E ~ depend only on C and not on the extension as indicated in the above notation follows from the next lemma, which shows that E ~ locallyis a tube domain. L e m m a 7.5.1 The manifold E ( C ) / s independent o.f the extension D, and

E(C)

~_ G x n - i C ~

As S ~

n

C S ~ it follows that ( S ~

C E ~ In particular,

~-'M c =o, v~ e s ( c ~

(7.4)

Thus, if .f is a function on E~ and s E S ~ we can define a function s- f on by [s ~f](x) -- f ( s - l x ) . Let f E O ( ~ ~ and let s E S. By (7.4) the function s - f i r e is well defined as long as s E S ~ In particular, ]Is-f]] is well defined, where ]1.11 stands for the L2-norm on j~/f. Define the Hardy norm of a holomorphic function f on E ~ by Ilfll2 :- sup II~-fll sES o

We define the Hardy space ~ 2 ( C ) by

n 2 ( c ) .= {f ~ o(=-~

< oo}.

(7.5)

As we are using a G-invariant measure on A4 and G S ~ C S ~ it follows t h a t

[[8"f[[2 ~ Ilfl[2" Define the "boundary value map" ~ " ~'~2(C) --~ L2(.~) by /3(I) = l i m ( s . I ) [ ~ ,

s-+l

where the limit is taken in L2(,A~).

218

C H A P T E R 7. H O L O M O R P H I C R E P R E S E N T A T I O N S

T h e o r e m 7.5.2 Let the notation be as above. Then the following hold:

I) 7"12(C) is a Hilbert space, and the action of G is unitary. 2} The map ~ 97/2(C) -~ Lz(J~4) is an isometric G-intertwing operator. 3) Let P~r be the holomorphic discrete series representation of G in E=. Then p. e.4( o)

where D is some extension of C to a G-invariant cone in 9. If C = Cmin, then the sum is over the full holomorphic discrete series. 0 Assume now that G / K is a tube domain and that 7- = 7-iYo. We also assume that G is contained in the simply connected group Gc. We know from Section 2.6 that

G / H = G . ( E , - E ) = {~ e 81 [@m(~) # 0}. We also have the following lemma. L e m m a 7.5.3 The G-invariant measure on G / K is given by

fG

/H

f(g" ( E , - E ) ) d ~ =

fs1 f(~)l~2(~)l-ad~(~)

where dp is a suitably normalized K • K-invariant measure on Sx.

Q

By the right choice of the cone C - Ck we also know that the semigroup S is just the contraction semigroup of the bounded domain G / K ~_ fl+. In particular, Eo C ~+ x f~+. By construction @,, is holomorphic on f~+ x ~2+. We then have the following theorem. T h e o r e m 7.5.4 Eo _ {~ E [2+ x fl+ I ~m(~) i~ 0}. The classical Hardy space 7t2 can, via the biholomorphic Cayley transform Ca, be viewed as space of holomorphic function on f/+ x fl+. The semigroup acts on this space via

s . f ( z , w) = j ( s -1, z)P"j(s -1, w) p" f ( s - 1 . z, s - 1 . w) which is well defined if - p , is the lowest weight of a holomorphic representation of Gc. In that case we also have a holomorphic square rooth @1 of ~2. The result is Theorem 7.5.5.

,~

219

7.6. THE CAUCHY-SZEGO K E R N E L

Theorem 7.5.5 (Isomorphism of Hardy Spaces) Denote by ~/2(~+ x ~+ ) the classical Hardy space on the bounded symmetric domain ~'1+ x ~+. Assume that g is not isomorphic to sp(2n, R) or $o(2,2k + 1), k , n >_ 1. Then .t

n2(~+ X ~+) ~ f b-.),~11 e n2(C) is an isometric G-isomorphism.

D

For the remaining two cases one has to construct a double covering of =o, .h4, G, and S and define the corresponding Hardy space. The classical Hardy space is then isomorphic to the space of odd functions in that Hardy space. Refer to [136] for the exact statements.

7.6

T h e C a u c h y - S z e g 5 Kernel

For w e E(C ~ the linear form f ~ f ( w ) is continuous. Hence there exists an element g w E n 2 ( C ) such that for f G n 2 ( C ) we have f ( w ) = (f[gw). Let g ( z , w):= gw(z). (7.6) The kernel (z, w) ~-~ g ( z , w) is called the Cauchy-Szeg5 kernel. We note that K(z, w) depends on the cone C used in constructing the Hardy space. By Definition 7.1.4 we have =

for s G S(C) and f , g E 7"/2(C). This gives L e m m a 7.6.1 Let z, w 6 ~(C ~ and let s 6 S(C). Then

g ( s - l z , w) = g ( z , a(s)w).

D

We collect further properties of the Cauchy-Szeg5 kernel together in the following theorem. T h e o r e m 7.6.2 Let 7/2(C) be the Hardy space corresponding to an in-

variant regular cone C C q. Let K(z, w) denote the corresponding CauchySzeg5 kernel. Then the following hold: 1) K ( z , w ) = K ( w , z ) . 2) For fixed z e =-(C~ the map ~(C ~ ~ w ~ K(z, w) G C extends to a smooth map on ~(C). If x G .h/l, then g(z,x) = ~(gz)(x).

CHAPTER 7. HOLOMORPHIC REPRESENTATIONS

220

3) Let K(z) := K(z, o), z ~. =-(C~ Then K is holomorphic and K ( s l o , s2o) = K(s~sx)

for every sa,s2 e S(C~ -1. 4) The map ~-x " Im 13 --+ 7"12(C) is given by I:3

(/3-1f)(z) = f ~ f(m)K(z, m) din.

Let z e E(C~ Then for a suitable u e V , we have ~ , ( z ) ~ 0. Assume t h a t K(z, z) = 0. As

K(z, w) = / ~ K(m, w)K(z, m) din, it follows t h a t

K(z,z)

=

f ~ IK(m,z)l 2 dm > O,

and K(z, z) = 0 if and only if K(m, z) = 0 for every m E ~4. But then f ~u(z) = ] ~ ~pu(m)K(z, m) am = 0, a

contradiction. Thus we obtain L e m m a 7 . 6 . 3 .

L e m m a 7.6.3 Let z ~. E(C~

Then K(z,z) # O.

We can now define the Poisson kernel by

P(z, m) := IK(z, m)l 2 K(z,z) Theorem

(7.7)

7.6.4 Let f be a continuous function on E(C) which is holo-

morphic on E(C~

Then f(z) = f ~ P(z,m)f(m)dm

for every z E E(C~

In particular, /. P(z, m)dm = 1.

7.6.

THE CAUCHY-SZEGO KERNEL

221

Notes for Chapter 7 Holomorphic representations of the semigroup S(D) = G exp iD were introduced by Ol'shanskii [137]. The theory was generalized to the situation we present here in [64]. For the general theory of ho]omorphic representations of Ol'shankii semigroups, refer to the work of K.-H. Neeb, [122] and [121, 123], where the Theorems 7.1.9, 7.1.10, and 7.1.11 were proved (cf. also [119]). There is an extensive literature on highest-weight modules and the classification of highest-weight modules. Our exposition follows closely the work of Davidson and Fabec [18]. More general results were obtained by Neeb in [119]. The classification of unitary highest-weight modules can be found in [22, 71]. For the connection between positive-definite operator-valued kernels and unitary representations, refer to [92]. The articles [38] and [39], where Harish-Chandra constructed the holomorphic discrete series of the group, were the starting point of the analysis on bounded symmetric domains, unitary highest-weight modules, and the discrete series of the group. The analytic continuation was achieved by Wallach in [167]. Most of Section 7.3 is taken from [133] and [135] except for the "only if" part in Theorem 7.3.4, which is from [63]. The construction of ~(~r, v) in [135] was by using the dual representation ~rv. The construction here is taken from [82]. The general theory of the discrete series on j~4 was initiated by the seminal work of M. Flensted-Jensen [32], where he used the Riemannian dual of .M to construct "most" of the discrete series. The complete construction was done by Oshima and Matsuki in [102, 142, 144]. The first construction of what is now called the holomorphic discrete series can be found in [103], where S. Matsumoto used the method of Flensted-Jensen to construct those representations. The material in Section 7.4 is standard and can be found, e.g., in [160]. Most of Sections 7.5 and 7.6 are from [63]. The introduction of the Poisson kernel is new. The part on Cayley-type spaces is taken from [136]. Theorem 7.5.4 was also proved in [15]. Further results on the H-invariant distribution character of the holomorphic discrete series representations can be found in [132]. An overview of the theory of Hardy spaces in the group case can be found in a set of lecture notes by J. Faraut ([27]). In these notes the definition of the Poisson kernel was given and Theorem 7.6.4 was proved for the group case. A shorter overview can also be found in [30].

Chapter 8

Spherical Functions on Ordered Spaces In this chapter we describe the theory of spherical functions and the spherical Laplace transform on noncompactly causal symmetric spaces as developed in [28] and [131]. The theory is motivated by the classical theory which we explain in the first section. The second motivation is the HarishChandra-Helgason theory of spherical functions on Riemannian symmetric spaces (cf. [40, 41, 451).

8.1

T h e Classical Laplace Transform

Before we talk about the Laplace transform on ordered symmetric spaces, let us briefly review the classical situation. Let ~/I = IRn and let C be a closed, regular cone in IRn, e.g., the light cone. As explained in Example 2.2.3, ]Rn becomes an ordered space by defining

x>_y,~z-y~C. Let M < is closed in M x M . A causal kernel or Volterra kernel is a map K" M x M --r C such that K is continuous on M < and zero outside M <. Let V (.M) be the vector space of causal kernels. For F, G E V (.M), define

F#G(x,y)

:=

[

,y] 222

F(x,z)G(z,y)dz

(S.1)

8.1. THE CLASSICAL LAPLACE TRANSFORM

223

With this product V(~I) becomes an algebra. The kernel K is invariant if, for all x, y, z E J4,

K ( x + z, y + z) = K(x, y). An invariant causal kernel corresponds to a function of one variable supported on C via F(x) = g ( 0 , x), g ( x , y) = f ( y - x). The above product is then given by the usual convolution of functions, F # G - F 9G. Associated to the Volterra algebra is the Volterra integral equation of the second kind,

A(x, y) = B(x, y) + f

,y]

g ( x , z)A(z, y) dz,

where B and K axe given. If A and B are invariant kernels such that a and b are the corresponding functions, this reads

= T h e o r e m 8.1.1 ( M . Riesz)

+f

dz.

,y]

The Volterra equation has a unique solution

given by A=B+R#B, where R is the resolvent R = ~ = 1 K ( " ) , with K (n+l) = K ( n ) # K . series defining R converges uniformly on bounded sets and R E V(r

The 1:3

For a, b and K = k invariant, the Volterra equation is

a=bWk.a. Recall the exponential functions e - x from p. 215 and assume that f e - x is bounded for A E C + ilR n. Define the Laplace transform of f by Z~(f)()~) := f e - ' X l " f ( x ) , x

for

)~ E C + i R " .

If I has compact support, t h e n / ~ ( I ) is defined for every )~ in IRn. Write )~ = u + iy, u E C. Then s + iy) = (2rr)n/2Jr(feu)(y). Hence

f(x)e-(ul z) =

1 n fR n ~,f(u + iy)ei(ulz) dx (27r)

or

f(x)-

1 fR s (27r)n

+ iy)e (u+iylx) dx.

Furthermore, the Laplace transform has the following two properties.

CHAPTER 8. SPHERICAL FUNCTIONS

224

1) If p(D) is a differential operator with constant coefficients, then

L:(p(D)f)(A)-

p(A)E~(f)(A) .

2) The Laplace transform is a homomorphism: .C,,(f 9g) = L(f)L(g).

From 2) one sees that the Laplace transform transforms the Volterra equation into

~(~) = ~(b)+ L(k)L(~), which gives

~:(~) =

8.2

Spherical

~(b) t - z(k)"

Functions

In order to generalize the notion of the Laplace transform to ordered symmetric spaces, we need to find the functions corresponding to the exponential function eu. These will be the spherical functions. As one already sees in the case of Riemannian symmetric spaces (cf. [45]), there are different ways to define a spherical function. 1) The differential equation: The spherical functions are the normalized, ~(o) = 1, eigenfunctions of the commutative algebra ~ G / K ) of invariant differential operators on G/K. 2) The integral equation: Spherical functions satisfy the integral equation

f

~(xky) dk

~(~)~(U).

3) The algebraic property: Denote the algebra of compactly supported, K-bi-invariant function on G by C~(G//K). Let ~ be a K-biinvariant function on G. Then the map

CT(G//K) ~ f ~ fa f(x)~o(x) dx E C. is a homomorphism if and only if ~o is a spherical function.

8.2. S P H E R I C A L F U N C T I O N S

225

4) The integral representation: For )~ E ab and p half the sum over the positive roots, we define

(8.2) where eK(x) ---- e <~-","K(=)> (cf. (5.18) on p. 136), with H replaced by K. Then ~oK is spherical function, and every spherical function has an integral representation of this form for a suitable A. Furthermore, ~K = ~ /( if and only if there is a w in the Weyl group of A such t h a t A=wp. We remark here t h a t there is no hope in general of using 4) to define a function on j~4 if we replace K by H. This is due to the fact t h a t G ~: H A N and H is not compact, so the integral does not converge for arbitrary x. One of the reasons for the fact that all those different definitions give the same class of functions in the Riemannian case is t h a t D ( G / K ) contains an elliptic differential operator. Thus every joint eigendistribution is automatically an analytic eigenfunction. As this does not hold for the non-Riemannian symmetric spaces, the different definitions may lead to different classes of functions, distributions, or hyperfunctions. Let .M -- G / H be an irreducible, noncompactly causal symmetric space with G C Go. We recall some basic structure theory. Let a be maximal abelian in p contained in qp and A -- A 0 U A+ U A_ be the set of roots of a in g. Choose a positive system A+ in A such that A+ -- A+ O A+. As usual, we set 1 p= ~ ~ [dim0~la. aEA+

Let C = C(A+) be the positive open Weyl chamber in a corresponding to A+ (cf. p. 116). Fix a cone-generating element X ~ E a such t h a t A+ = {a e A I c~(X ~ - 1} and A0 -- {c~ e A I a ( X ~ = 0}. Let C ---- C m a x ( X ~ (cf. (4.19) on p. 98). Finally, let S - S(C) - H exp C. Then S is a closed semigroup and S C H A N (cf. Theorem 5.4.7). In particular, the function

e,x(s) = e l l ( s ) : = e <~-p'aH(')> ,

:k e a~,

(8.3)

is well defined on S. D e f i n i t i o n 8.2.1 A spherical function is a H-bi-invariant function ~o defined on the interior of S such t h a t for all s, t E S ~ the function h

CHAPTER 8. SPHERICAL FUNCTIONS

226 is integrable over H and

fH ~(Sht ) dh

=

~(s)~(t).

We will often, without further comment, identify the H-bi-invariant function on S or S ~ with the H-invariant function on ~4+ and j~4~_, respectively. Thus we may view spherical functions as functions on ~4. Let us fix a Haar measure on G and other groups before we go on. We normalize the Haar measures on A and a* such t h a t they are dual to each other, i.e., the Fourier inversion formula for the abelian group A holds without constants:

Further, we normalize the measures that

dn and dn ~ = r(dn)

on N and N ~ such

fN eK2o(n~)dn~= 1. I

We normalize the Haar measure on No and No~ in the same way by using p0 = 89~ e ~ o + dim g~ a instead of p. We choose dn+ on N+ such t h a t

dn = dn+ dno . dn_ = 7(dn+). f ECc(N_),

Let

We choose the measure

dX

on n_ such t h a t for all

L _ f(n_)dn_ = f,_ f ( e x p X ) d X . Then we relate the measures on G and H by

fa f (x) dZ = fH fA fN f (han)a2p dh da dn for every f E Ca(G) with S u p p ( f ) C

HAN.

8.2.2 LetC = {A e ac I Vc~ E A+ : Re(A+plc~) < 0}. Let A E s and let s E S ~ Then H 9 h ~ e x (sh) E C is integrable and

Theorem

is a spherical ]unction.

0

227

8.2. S P H E R I C A L F U N C T I O N S

The idea of the proof is to use the integral formulas in [128] to rewrite the integral defining ~ox as an integral over K:

JK

NHAN

As the semigroup S ~ acts by compressions on H / ( H N K ) , it follows that H A N N f f ~ k ~-+ e~(sk) is actually bounded. A simple s[2 reduction shows that the exstension of K M H A N B k ~ e - x (k) by zero outside K N H A N is continuous if A E s The proof actually shows that ~px is a well-defined spherical function on the set

s

/KnHAN e-Re)~(k)dk

< c~ } .

(8.4)

which in general is bigger than s For g e Go, the decomposition g - h(g)a(g)n(g) is just the usual Iwasawa decomposition. Let 1 aEA+

Then p = p0 + p+. Let ~ denote the spherical function on the Riemannian symmetric space G o / ( K N H):

~O~(x)= /KNH e)~+o+(xk) dk" Denote the G0-component of g E H G o N by go(x). L e m m a 8.2.3 Let s E S ~ and )~ E s

=[

JH nK\H

Then

dh,

where dh denotes a suitable normalized invariant measure on H / ( H N K ) . In particular, ~w~ = r for every w E Wo. ID

Let D(M) be the algebra of invariant differential operators on 2~4. To )~ E a~ there corresponds a homomorphism X), " D(j~[) ~ C.: In short, this homomorphism can be constructed by choosing an element u in the universal enveloping algebra U(g) that corresponds to D by right differentiation. Then project u to U(a) _~ S(a*) along [~U(g) ~ U(g)n and evaluate at ~ - p . L e m m a 8.2.4 Let )~ E s Let A(C) := expC C A. analytic and D~x = xx(D)~x on H A ( C ) H .

Then ~A]HA(C)H i8 []

228

8.3

C H A P T E R 8. SPHERICAL FUNCTIONS

The Asymptotics

In this section we describe the asymptotic behavior of ~x on A(C). For that we need three c-functions. Define cf~(A) = ~a e-x (exp X ) d X , and

co(~) = f~0 ~-~('~)d.~,

~(x) = ~.(x)~(a)

The function c0(A) is the usual Harish-Chandra c-function for the symmetric space .Ado = G o / ( K N H ) and has thus a well-known product formula co(A) = l"IaezXo+ca(Aa), cf. [35] or [45]. On the other hand, the function cn(A) is known only for some special cases, cf. [26]. The integral defining cf~(A) converges exactly for A E s The integral defining Co(A) converges for A E ab such that Re(Alc~) > 0 for every a E A0+. One should note that one can replace A and p by AJ,n[so,so] and p0, respectively, in all calculations involving co (A). Lemma

8.3.1 Let A E r be such that

Re(Ala )

> 0 / o r every ~ E A+.

Then

C(~) :

fNmf'IHANe-~(ri~)d r i l l .

0

Let us introduce the notation a

oo

for the fact that a E A(C) and for all a E A+ we have lima ~' = c~. m

Rewriting the integral defining ~,~ as an integral over N fq H A N , we get the following theorem. T h e o r e m 8.3.2 Let A E r be such that Re(AIc~) > 0 for every ~ E A+o . Then lira aP-X~x(a) = c(A). a A (_.}r) oo

Furthermore, lim et
=

cry(A)~+p+(a). o

8.3.

229

THE ASYMPTOTICS

E x a m p l e 8.3.3 ( T h e H y p e r b o l o i d s ) Let G = SOo(1,n) and let H = SOo(1, n - 1), n > 2, cf. Section 1.5. Let

{ (cosh, 0 inh,) }

a=

at=

0 sinh t

I,-1 0

tGR

0 cosh t

.

Let X ~ = El,n+l 4- En+l,1. Then a = R X ~ We choose the positive root such that a ( X ~ = 1 and identify a~: with C by z ~ -zcr. Then p =-(n1)/2. The spherical function ~ox is given by ~o~(at) =

(cosht + sinhtcoshO)-~-('-i)/2(sinhO)"-2dO

Let Q~ be the usual Legendre function of the second kind and let 2F1 be the hypergeometric function. From [231 we get Theorem 8.3.4 T h e o r e m 8.3.4 The integral defining ~ox(at) converges for t > 0 and ReX < - ( n - 3)/2. Furthermore,

r(~- ~) ~o~,(at)

=

~_~

7 . F ( ) , + a~_X)Csinht)~/_ x Qx_ 89(cosht)

2._2F/n1) r(~-~) ~,)2 F(X+I) 92F1

._,

(2c~

($+

"2~ X + " - x2 A + l , 2 ' 2 '

'

"

1 (cosh t) 2 )

where 7n is a constant depending only on n.

In particular, for n = 2, 1

-xt

i

~ox(at) = X sinh t e

"

In this case A+ = A+. Thus c(X) = cn (A) and

~(~)

2n_2F { n - 1 2

~ r(x-n~_._.~3) F(A 4- 1)

2._,o Furthermore, 1 cn(A) ~oA(a,) = 2F1

( X + 2~2-~ ' ~

'

2

X+l,

1) cosh 2 t

which extends to a meromorphic function on a~: holomorphic for Re A > - 1 . El

CHAPTER 8. SPHERICAL FUNCTIONS

230

E x a m p l e 8.3.5 ( T h e G r o u p C a s e ) Let G be a connected semisimple Lie group such that Gc/G is ordered. Let a be a maximal abelian subalgebra of q = ig contained in p (9 it. Note that ia is a compact Cartan subalgebra of g. The Weyl group W0 is given as NK(I1)/ZK(II), w h i c h is the Weyl group of a in K. Let ~(w) = det w, w E W. T h e o r e m 8.3.6 The spherical function ~ox is given by ~x(exp X)

~I1~ca+ < ~,,x > ILia+ ~i~h < ~,x >

for X ~ C ~ a. We get for suitable constants 7, 70, and 71 such that 7o71 - 7, 70

co(~) = I ] ~ o + < ~ , ~ > , 71

cn(~) = [I~e~+ < ~,a >' and

7

~(,x)= lI,,e:+ < ,x,~ > Furthermore, c(X)-l~pA(expX) = ~WEWo e'(w)e-<wA'X> I'I~e-a+ sinh < ~ , X > for X E C ~ fla and as a function of A this function extends to a holomorphic function on ac. 0

8.4

Expansion Formula for t h e S p h e r i c a l F u n c t i o n s

Let us recall the case of spherical functions on the Riemannian symmetric space G/K, cf. [40, 41, 45], before we talk about the causal symmetric spaces. Let 7~x(x) = fK e x ( x k ) d k as before. Let or(A) = fN, e_x(n~) dn~ be the usual Harish-Chandra e-function for the Riemannian symmetric space G/K, which is isomorphic to the r-dual space Gr/K r = M r. Notice that we can view A as a subset of M r. If D E D(:ctr), then the radial part A m r ( D ) of D is a differential operator on A(C), such that for every K-bi-invariant function F on G we have

(DF)]AfC) = A~,.(D)(FIAfC)).

8.4. E X P A N S I O N F O R M U L A

231

In particular, this holds for the spherical functions ~ K~.

A,~'(D)(~f~IA(c) ) = xx(D)(~f"IA(c) ) .

Let A = NA+ and construct the function r~,/~ E A recursively by Fo

=

1, -

=

2(pl )]

2 E aEA+

m~ E

+ p - 2kala) - ( a l A ) ) .

k>0 m

Define for a E A(C):

~eA\{0}EFg(A)a~) "

r

From [40, 45] we see T h e o r e m 8.4.1 Let W be the Weyl group of A. Then there exists an open dense set U C a b such that for A E U, {~,x I s E W } is a basis of the space o/functions on A(C) satisfying the differential equation A~,(D)~ = xx(D)~,

VD E D ( ~ r ) .

r-!

For an H-bi-invariant function on HA(C)H, define a Kr-bi-invariant function f7 on K " A ( C ) K r C G" by f'r(klak2) = I(a). Denote the natural isomorphism D(M) ~_ D ( ~ r) by 7. Then (Dr) "r = D'Yf "t. This, together with Theorem 8.4.1 and the asymptotics for the spherical function ~ox (cf. [130]), gives Theorem 8.4.2 T h e o r e m 8.4.2 Let A E s f3 U and a E A(C). Then O wEW0

As a corollary of this we get the following. C o r o l l a r y 8.4.3 The .functions g • A(C) ~ (A,a)

1

1

extend to a b • A(C) as meromorphic functions in A.

CHAPTER 8. SPHERICAL FUNCTIONS

232

We will denote the H-bi-invariant extension hlah2 ~ [1/cn(A)]~ox(a) by the same symbol. Similarly we denote the H-bi-invariant function on HA(C)H (or HA(C)o) that extends q)x by the same symbol, (I)x. Then the product formula for c shows that

c(~) = ~+(~)co(~), where c+(A) is the part of the product coming from the roots in A+. As A+ is W0-invariant, it follows that c+(A) is W0-invariant. Thus

c"(~,) ~(~)

c+(:~) ~,~(~,)'

and c+(A)/cn()O is Wo-invariant. Let

wEW

be the spherical function on M r for the parameter A. Theorem

8.4.4 Let the notation be as above. Then

~

~w.~ Wr ~ - ~

= wE

8.5

The

~

=

W

wEWo\W

Spherical

Laplace

cn(w~) ~o~.

[]

Transform

A causal kernel or Voltevra kernel on M is a function on M x M which is continuous on {(x, y) I x < y} and zero outside this set. We compose two such kernels F and G via the formula

F#G(x,y)

= /.M F(x, m)G(m, y) dm =

f

I

,y]

F(x,m)G(m,y)dm.

This definition makes sense because j~4 is globally hyperbolic (cf. Theorem 5.3.5). With respect to this multiplication, the space of Volterra kernels V(.AA) becomes an algebra, called the Volterra algebra of .h4. A Volterra kernel is said to be invariant if

F(g~,g~) = E(~, ~) Vg e G.

8.5. THE SPHERICAL LAPLACE TRANSFORM

233

The space V(.M)~ of all invariant Volterra kernels is a commutative subalgebra of V ( M ) , cf. [25], Th6or~me 1. An invariant kernel is determined by the function f(m)=f(o,m), meM+, which is continuous on M + and H-invariant. On the other hand, for f a continuous H-invariant function on M + , we can define an invariant Volterra kernel F by

f ( a .o, b. o) = ](a-lb 9o). Under this identification the product # corresponds to the "convolution"

f # g ( m ) -- f f(x . o)g(x - 1 . m) d~. JG/n So the algebra V(.M) ~ becomes the algebra of continuous H-invariant functions on S . o with the above "convolution" product. The spherical Laplace transform of an invariant Volterra kernel F is defined by

- J ~ F(o, m)ex(m) din.

s

Here, by abuse of notation, we view the H-invariant function e), as a function on M + . The corresponding formula for the H-invariant function on S . x o is s

f ~ f(x)ex(x) dx.

Let T)(f) be the set of A for which the integral converges absolutely. Using Fubini's theorem, we get Lemma 8.5.1. L e m m a 8.5.1 Let l,g 6 V(.M) ~ be invariant causal kernels. Then T)(f)N T)(g) C V( f #g). For A 6 V( f ) N V(B) we have =

Let M = ZHnK(A). In "polar coordinates" on M ,

H I M x A(C) 9 (hM, a) ~ ha. o E .M+, we have, for f 6 Cc( S / H ) ,

fS/H

(c)

C H A P T E R 8. SPHERICAL FUNCTIONS

234

where c is some positive constant depending only on the normalization of the measures, and ~(a) =

l'I

(sinh < a, log a > ) t o o .

oE~, +

8.5.2 Let c > 0 be the constant defined above. Let f : S / H --+ C be continuous and H.invariant. If )~ E ~9(f), then ~x exists and

Theorem

f ~,f(~k) = c ] f(a)~o~(a)d(a) da. JA (C) To invert the Laplace transform, we define the normalized spherical function ~x by

1 ~(~) .= --~~(~)

and the normalized Laplace transform by

~(f)(~) :-- fA (c) f ( a ) ~ ( a ) ~ ( a ) d a . Then

f (a)c~wx (a)J(a) da. wEW.

as cn (,~) is W0-invariant. Note that the unknown function c~ (A) disappears in this equation. Let )~ E ia*. Then

~(-~) = co(~) and

~o(-X)~o(X) = ~ o ( - w ~ ) ~ o ( w ~ ) . Thus

and

~0(~) ~(x)

co(-~) ~0(-~a)

_

(8.7)

c~(~) co(w~)

~(~) ~(x) = l,

so )~ ~ co(w)~)/co()~) has no poles on ia*. Let C ~ ( H A ( C ) H / / H ) be the space of H-bi-invariant functions with compact support in H \ H A ( C ) H / H . 8.5.3 Let f E C~(HA(C)H//H). Then s B A ~ L~(f)(A) E C extends to a meromorphic function on a~ with no poles on ia*. D

Theorem

8.6. THE ABEL TRANSFORM

235

From the functional equation for ~Px we get

Z

cr(w)~)s

)(w)~) =

Cl,Sr(f'r)(.X),

wEWo\W

where C1 is a positive constant independent of f and A, and 9r denotes the spherical Fourier transform on f14 r" 'T'(f~')('X)-/G"

/,(~)vi(~) d~.

Let Ex(hlah2) :- ~ ( a ) be the H-bi-invariant function on G with the same restriction to A(C) as 7~. By the inversion formula for the spherical Fourier transform, [45], we have for some constant depending on the normalization of measures and w - [WI,

vf(a)

=

1 ~o .T(f.y)()~ ) -w

.

= L W wE

_ L =

_--

1

~

wE

~

~.~ ~o ~(/)(~):(~):(_~) :(~) E_x (a) d)~ W

*

W

*

~a

W wE

E-x(a)

:~~1

zi(/)(~):(_,~------~E_~(~)e~

~WA ~ . s W

l__woSo . s-

:(-w~)e~

cr(_)~ ) d)~. E_~(a)

This proves the inversion formula at least for f E CT(HA(C)H//H )" T h e o r e m 8.5.4 Let Ex := Ex/c'(A). Let f e CT(HA(C)H//H ). Then

there exists a positive constant c that depends only on the normalization of the measures involved such that for every a E A(C), cf(a) = lWo ~ia" ~ ( f ) ( ~ ) g - A ( a )d~.

8.6

The Abel Transform

As in the Riemannian case, we can define the Abel transform and relate that to the spherical Laplace transform. The main difference is that in this case .A(f) does not have compact support even ff the support of f is

C H A P T E R 8. SPHERICAL FUNCTIONS

236

compact. On the contrary, by the nonlinear convexity theorem, the support of A ( f ) can be described as a "cone" with a base constructed out of the support of f. Let f be an H-invariant function on ~/1+. We define the Abel transform, A.f: A --+ C of f, by

f .A(f)(a) -- a t' fin f(cm) dn whenever the integral exists. Using the nonlinear convexity theorem, p. 151, we prove the following lemma. L e m m a 8.6.1 Let f be a continous H-invariant function on .A~+ (extended by zero outside .A4+) such that n ~ f(an) is integrable on N for all a E A. Let L C Cmnx be the convex hull of log(Supp(flsna)). Then log (Supp(Af)) C L + Cmin.

[2

We rewrite now the Integral over A~f+ C M as an integral over A N , cf. [128], to get Theorem 8.6.2. T h e o r e m 8.6.2 Let f be an H-invariant function s i n Then a

and A E 2)(I).

m

a~'.Af(a) da

/~(f)(,~)-- /

/~A(Jt.f) (--A),

d e x P Cmax

where

ff-,n

is the Euclidean Laplace transform on A with respect to the cone

Cmax.

l-I

The Abel transform can be split up further according to the semidirect product decomposition N = N+No. Set

A+f(9o) = a n+ [ _

f(gon+)dn+

d lv +

for go E Go. Then obviously A + ( f ) is Ko-bi-invariant and

.Af(a) --a " ~

+(f)(o.o)dno .

Denote by .A0 the Abel transform with respect to the Riemannian symmetric space G o / K N H. Then we have

.Af (a) -- .Ao(,A+f ) (a) for all continuous, H-invariant functions f : S / H --} C such that the above integrals make sense and all a E A. As it is well known how to invert the transform A0, at least for "good" functions, the inversion of the Abel transform associated to the ordered space reduces to inverting A+ f.

8.7. RELATION TO REPRESENTATION THEORY

237

Theorem 8.6.3 If f : M + --r C is continuous, H-invariant, and such that the Abel transform exists, then its Abel transform is invariant under Wo, i.e., A f ( w a ) = .hi(a)

8.7

Relation

va e A,,I, ~ Wo.

to Representation

Theory

The spherical functions are related to the representation theory on both G and the dual group G c. Here we explain the relation to the representation theory of G. Let M be the centralizer of A in K and let P be the minimal parabolic subgroup M A N . For A 6. o~, let (rr(~),H(A)) be the principal series representation induced from the character Xx " man ~ a x of P. The space of smooth vectors in H(A) is given by

C~176 = {I' e r176176l Vman e M A N , Vg 6. G 9f(gman) = a-(X+P)j'(g)} and

I~(X)(g)/](*) = l(g-'*) 9 The bilinear form Coo(A) x Coo(-A) 9 (f,g) ~ < f , g > ' = f g f(k)g(k)dk defines an invariant pairing Coo (A) x Coo( - A). Extend ex to be zero outside H A N . Using the above pairing, we find that ex E C-oo(A) H for )~ E - s where C-oo (A) is the continuous dual of Coo(,k). Furthermore,

< f, ex > - / K f ( k ) e , x ( k ) d k ' - / H f ( h ) d h for f E C~176 and A E -~'. The linear form f ~-~< .f, ex > has a meromorphic continuation to all of ab as an H-invariant element in C-oo(A), cf. [2, 3, 128, 140, 141, 145]. Let / E C~~ Then r ~ ~ 1 7 6 E C~176 Hence

/ ~ o ~ ( / ) = < ~;~176

>

is well defined. Definition 8.7.1 A distribution 0 on G is called H-spherical if 1) O is H-bi-invariant.

C H A P T E R 8. S P H E R I C A L F U N C T I O N S

238

2) There exists a character X: D(Ad) such t h a t D(O) = x(D)O, Theorem

8.7.20x

every D E D(.M).

rn

is an H.spherical function and DOx = xx(D)6)x for m

The relation between the spherical distribution O x and the spherical function ~o~ is given by Theorem 8.7.3. Theorem

8.7.3

If A

E

s and

OA(f)

=

Supp(/) C

f

JG

(S~ -1, then

f(x)cp-,x(x-1)dx.

0

Notes for Chapter 8 The material in Section 8.1 is standard, but usually F is defined as the set { (x, y) E .hd I z ~_ y}. In that case the product for invariant kernels becomes F # G - G.F. Spherical functions on symmetric spaces of the form Gc/G were introduced by J. Faraut in [24], where they were used to diagonalize certain integral equations with symmetry and causality conditions. Most of the material in Section 8.2, Section 8.3, and Section 8.5 is from [28]. The proof of Theorem 8.3.6 in [63] used the relation to the principal series in Section 8.7 and the formula of the H-spherical character due to P. Delorme [19]. Examples 8.3.3 and 8.3.5 are from [28]. Lemma 8.2.4 was proved in [131]. A more explicit formula for the spherical function and the cf~(A) function for Cayley-type spaces was obtained by J. Faraut in [26]. The inversion formula for the Laplace transform was proved in [28] by using the explicit formula for the spherical functions. In the same article, an inversion formula was proved for the rank 1 spaces by using the Abel transform. The general inversion formula presented here was proved in [131]. The first main results on spherical functions on Riemannian symmetric spaces are in [40, 41]. The theory was further developed by S. Helgason, cf. [43, 45]. The isomorphism "r : D(Ad) ~ D(,M r) was first constructed by M. Flensted-Jensen in [32]. A Laplace transform associated with the Legendre functions of the second kind was introduced by [17]. In [164] this transform was related to harmonic analysis of the unit disc. A more general Laplace-Jacobi transform associated with the Jacobi functions of the second kind was studied by M. Mizony in [105, 106]. There is by now an extensive literature on the function e~ and its generalizations. Its importance in harmonic analysis on .hd comes from the generalized Poisson transformation, i.e., the embedding of generalized principal series representations into spaces of eigenfunctions on .M. A further application is the construction of the spherical distributions e~. We refer to [2, 3], [12, 13], [128], [140, 141, 145] and [161], to mention just a few.

Chapter 9

The Wiener-Hopf Algebra The classical Wiener-Hopf equation is an equation of the form

[x + w(f)]r = ~, where rl E L2(R+), ~ is an unknown function on R +, and

[w(I)r (~) =

I ( ~ - t)r

~0~176

at,

where f E LI(R). Equations of this type can be studied using C*-algebra techniques because the C*-algebra )4;a+ generated by these operators has a sufficiently tractable structure. It contains the ideal K: of compact operators on L2(R +) and the quotient )4PR+/K: is isomorphic to Co(R). There is a natural generalization of these Wiener-Hopf operators to operators acting on square integrable functions defined on the positive domain in an ordered homogeneous space (cf. [54, 55, 108]). In this section we consider Wiener-Hopf operators on the positive domain M + of a noncompactly causal symmetric space M = G/H. We recall the ordering _<s that has been used in the definition of the order compactification M opt and the corresponding positive domain ~4+ C jg4. Next we consider an invariant measure # ~ on M and the corresponding unitary action of G on L 2 (r as well as the integrated representation Ir~ of the group algebra LI(G) on L2(M), i.e., [lr~(f)r

(x) = f6 f(g)r

x) d#G(g)

r e L2(M), f e

LI(G).

The Wiener-Hopf algebra ) ~ + is defined to be the C*-algebra generated by the compressions of the operators ~r,~(f) to the subspace Lz(j~4+). Our 239

240

C H A P T E R 9. T H E W I E N E R - H O P F A L G E B R A

intention in this chapter is to describe the structure of the Wiener-Hopf algebra in terms of composition series which are determined by the G-orbit structure of .A4ept. Further, we explain how the Wiener-Hopf algebra can be obtained as the homomorphic image of the C*-algebra of a groupoid. We start by showing the connection of the order compactification as described in Section 2.4 with the functional analytic compactification using the weak-star topology in L~176 and characteristic functions, which is commonly used in the context of Wiener-Hopf algebras. L e m m a 9.1.1 The mapping

9 "~r

--+ L~176

A ~ XA,

where XA is the characteristic function of A, is a continuous injection. Proof. 1) 9 is injective: Let A , B E ~'r with XA = XB in L~176 i.e., almost everywhere. Let a e Int(A). If x r B, then there exists a neighborhood U of a in A such that U I"1 B = 0. Therefore p(U) < ~(A \ B) = 0, a contradiction. We conclude that Int(A) C B. In view of Lemma 2.4.7, this implies that A C B. The inclusion B C A follows by symmetry. 2) ,I, is continuous: Let A , --r A in ~',(G). We have Co show that lira XA. = Xa almost everywhere. We proceed in two steps: a) XInt(A) _< liminfxA." Let a E Int(A). Then there exists b e Int(A) n Int(T a). We choose a neighborhood U of b in T a which is contained in A. Then there exists nv E N such that An N U ~ 0 for all n > nv. Let bn E A n N U. T h e n b n E Ta and thereforea ES bn C$ An = An. This shows that liminfxA. (a) = 1. b) limsupxA. < XA: Let a E G with limsupxA.(a) = 1. We have to show that XA(a) ---- 1, i.e., a E A. To see this, let U be an arbitrary neighborhood of a in G. Then the condition limsupxAn (a) -- 1 implies for every no E l~l the existence of n > no with a E An. It follows in particular that An N U ~ @. Hence a E limsupAn = A. D L e m m a 9.1.2 Let B := {f E L~176 [

g.f:=fOAg-~

for

Ilflloo _< 1}

and set

g E G, f E B,

where Xg is left multiplication by g in G. Then the mapping G • B -+ B, (g, f) ~ g . f is continuous, i.e., G acts continuously on the compact space B.

241 Pro@ Let go E G, f0 E B and h E L x. We have to show that the function (g, J') ~-+ (g. f, h) is continuous at (g, j'). For g E G and f E B we have (g " f , h) - (go " fo, h)

= (g" f - g o " l , h ) + (go" l - g o " fo,h) - (I, g-1 . h - go 1 9h) -~- ( f -" ~o,go 1. h)

and both summands tend to 0 for g --+ go and f --+ 2'o because go 1 . h and [[g-1. h - g o 1 . hill ---r O.

Lemma

9.1.3

E

L1 m

1) The mapping 9 ".T$(G) ~ B,

A ~ XA

is a G-equivariant continuous and monotone mapping of compact Gpospaces. 2) The mapping M cpt ~ L~176

A ~ XA

is a homeomorphism onto its image. Pro@. 1) follows from Lemma B.1.2, Proposition 9.1.1, and Lemma 9.1.2. 2) is a direct consequence of i) and the compactness of .A4 cpt D This lemma shows that there is no essential difference between the compactifications in i ~176 (G) and in jr(G). Recall from Theorem 2.4.6 that A4 is open in .Adept. Thus it is reasonable to extend p ~ to A4 cpt via p ~ (A4cPt\A4) - 0. In particular, we can identify A4+ and .h4+pt as measure spaces. We w r i t e / J ~ + for PM]~+- If now

p" L 2 (.h4,/z.~) --+ L 2 (f14+,/z.~a+ ) and

j" L 2 (.A4+,/z/~+ ) --+ L 2 (.h4,/~Aa ) are given by restriction and extension by 0, respectively, the Wiener-Hopf operator W.~a+ ( f ) " L2(.M+,/~/v[+ ) --~ L2(.A4+,/zAa+ ) associated to the symbol f E L I(G) is given by W ~ + (I) := P 0 ~ru(f) 0 j, i.e.,

[w~+ (s)r (~) = /

J~ (.)

I(g)c(g-'. ~) d~,~(g),

C H A P T E R 9. T H E W I E N E R - H O P F A L G E B R A

242

where pG is a suitable Haar measure on G (cf. Lemma 2.4.2). If we multiply two such Wiener-Hopf operators we obtain for ft, f2 E Co(G)

[ w ~ + (/1) o w:~+ (12)r (~)

Thus the product of two such Wiener-Hopf operators is again some sort of Wiener-Hopf operator with a two-variable symbol. To construct a domain on which these symbols are functions, we need the notion of a groupoid. A locally compact groupoid is a locally compact topological space ~ together with a pair of continuous mappings satisfying the following axioms. The domain of the first mapping is a subset ~2 C {~x ~ called the set of compos. able pairs, and the image if (x, y) E {~2 is denoted xy. The second mapping is an involution on ~ denoted by x ~ x -1. The axioms are as follows.

1) (z, y), (y,z)

e ~2 ~

(T,~l,z), (x, ~]z) E ~2 and ( x y ) z -- x(yz);

2) ( x , x - 1 ) , ( x - l , x ) E ~2 for all x E G;

3) (~, u), (z,:) e ~2 ~

~-'(~v) = u and (z~)~-' = ~.

The maps d" G-+ G, z ~ z - i x ,

r " ~i ~

(~,

x ~ xx -1

are called the domain and range maps, respectively. They have a common image G0 called the unit space of ~. Note that a pair (x, y) belongs to G2 if and only if d(x) = r(y), hence {~2 is closed in ~ • g. Also, since u E GO if and only if (u, u) E {~2 and u 2 - u, it follows that G0 is closed in ~. Let u E G~ Then r - l ( u ) N d - l ( u ) is called the isotropy group in u. It carries the structure of a locally compact group. We say that two points u, v in G0 lie in the same orbit if d-X(u)f] r - l ( v ) r 0. Note that this defines a partition of go into orbits because d(x -1) = r(x) for x E G. Finally, we note that r(xy) = r(x) and that d(xy) = d(y) for x, y E {~. We consider the right action of G on ./t4 ept defined by A . g := g-XA. Our groupoid will be the reduction of the transformation group r ept x G McPt to ,A.4cPt . , + (cf. [108] 2.2.4 and 2.2.5). This amounts to the following. We set AAcpt xGlgex}. g-{(x,g) e,.,+ The domain of the multiplication is

;~ = { ((=,g), (~, h)) e ; • ; I = .g = ~},

243 the multiplication map G2 ~ G is

((~,g), (~, h)) ~ (~,g)(~,h)= (~,gh), and the involution G -~ G is defined by

(~, g) ~, (~, g)-x

:=

(~. g, g-s).

The domain and range maps for this groupoid are given by

d(x,g) = (x .g, 1)

and

r(x,g) = (x, 1). cpt

Thus the unit space GO of ~ is equal to j~r x {1}. We note that it is instructive to visualize the elements (x,g) of the groupoid as arrows from d(x,g) to r(x,g). Then the involution corresponds to inversion of arrows and the composable elements are the arrows which fit together. Moreover, multiplication is composition of arrows. We endow ~ with the AA cpt locally compact topology inherited from ,.,+ • G. Next we define a convolution product on the set Cc(~) of all compactly supported complex valued functions on ~ by

F. 9F~(~.g)

= fG F.(~.g~)F~(~ "g ~ . " - ' ) X ~ ; . ' ( ~ "g~) d . o ( . ) -

f.~(~)Fx(x,a)F2(a- 1 .

x , a - l g ) dpG(a)

which is obviously related to the product formula of our Wiener-Hopf operators. We define a map Cc(G) --~ Ce(~) via

i ~ ],

](~,g):= f(g).

Note that the compactness of d AAcPt implies that ] G Co(g) Then Co(g) is w t..~,. a ,-algebra with respect to the involution 9

F*(x,g) = f ( ( x , g ) - X ) A G ( g ) -1 = f ( g - 1 . x,g-1)AG(g) -1 (cf. [54], III.6). We define norms on Cc(~) via (cf. [108], 2.7) Ilfll0 :=

sup liE(x,.)111 opt zEA4+

and

Ilflll := max{llFII0,11F*l[0}.

Now we write LI(~) for the normed ,-algebra obtained form Cc(~) by completion with respect to the norm ]l" lll (cf. [150], p. 51). We also recall from [109], 2.11, that Co(g) and therefore Ll(g) admits a two sided approximate identity. Thus we obtain a universal enveloping C*.algebra C*(G) as the subalgebra of C*(L'(G),) generated by nx(g). The following result is proven in [54].

244

C H A P T E R 9. T H E W I E N E R - H O P F A L G E B R A

T h e o r e m 9.1.4 For F e Cc({7), the prescription [W~+ (F)r (x) = f~

(=)

F(x,g)r

x)~/8(g -1, X) d~G(g),

where s ( g - l , x ) is a cocycle depending on the measure chosen on .All, defines a norm-contractive ,-representation 9c o ( g )

and the extension

W.,~+ " C*({~)--~ B(L2(.M+))

is a C*-representation with image YY~+.

The preceding theorem shows that it is reasonable to try to describe the ideal structure of the algebra YY~+ of Wiener-Hopf operators via the C*algebra C*(G) of the groupoid. But it should be noted here that it is an open problem to determine the kernel of the representation W ~ + . Whenever a subset ,A,#cPtt t.~= . , + of q/vA.#cPt is invariant in the groupoid sense, i.e., satisfies /tAopt t ~ d(x,g) 6 .,,-,+ r(x,g) 6 .M+cpt t , we set If ,-,+ A,~cpt ! is locally compact in #,-.+AAcpt, then {~~,, is again a locally compact groupoid and one can talk about its groupoid C*-algebra. N o w one can show (cf. [55]) the following. cpt . Then we have T h e o r e m 9.1.5 Let U be an open invariant subset of.All+ a short exact sequence of C*-algebras,

o

c*(a)

o.

For f E Cc(~) the ~t'~ap~u is given by restriction to r - l ( j ~ p t \ V). The cpt . 1-1 sequence splits if U is open and closed in .A/l+ Recall that there are only finitely many different G-orbits in A4 opt which are in addition locally closed, i.e., each orbit is open in its closure, and we also know that all other orbits in the closure of a given orbit have lower dimension.

245 Let r denote the set of all orbits of dimension (dimr opt : - .M+opt f3 .Adt (cf. Section 6.4). Let and (.A,4+)t

- k in .M opt

z,, :=

for k 6 No and consider this set as an ideal of C*(~) according to Theorem 9.1.5. Then one finds (cf. [55]) the following theorem. T h e o r e m 9.1.6 Let xt,i, i = 1,... ,it be a set of representatives for the

different invariant subset of (.h4e+Pt)k and Ht,i the stabilizer of xk,i in G. Then the C*-algebra C*(G) has a composition series Io C *.* C k i m M

with

= C*

(~)

ih

I k / l k - 1 ~-- ~

(C*(Hi,k) | K:(L2(tM;Ptlk,pk)) )

i=1

/or k >_ 1 and Io ~- C*(H) | K:(L2([MTtl0,p0)), ,,here

de ote, the C*- lgeb

of comp a operator,

on the Hilbert space L2( [jk4+ cpt]k,pt) and I.tt is a positive Radon measure cpt [M+

The main result of [54] is the following theorem, which contains all the information one has on the first ideal in the composition series of the WienerHopf algebra. T h e o r e m 9.1.7 Let IC (L2(M+))

be the ideal of compact operators on L2(.A4+). Then IC (L2(.A,4+)) C W.M+. More precisely, this ideal is the image of C* (G.~+) under the Wiener-Hop] representation.

Notes for Chapter 9 The groupoid approach to C*-algebras generated by Wiener-Hopf operators goes back to Muhly and Renault [108] (cf. also [126, 109]). They also gave definitions of Wiener-Hopf operators for arbitrary ordered homogeneous spaces, but treated only the case of vector spaces ordered by polyhedral or homogeneous cones in detail. First attempts to also deal with nonabelian symmetry groups are due to Nica [127]. A more systematic approach is given in [54] (cf. also [53]). The case of noncompactly causal symmetric spaces was developed in [55].

Appendix A

Reductive Lie Groups In this appendix, on the one hand we collect the basic notation for semisimple and reductive Lie theory used throughout the book. On the other hand we quote, and in part prove, various results which are purely group theoretical but not readily available in the textbooks on Lie groups.

A.1

Notation

Let K be either R or C and b a finite-dimensional abelian Lie algebra over the field K. Given a finite-dimensional b-module V, we denote the corresponding representation of b on V by lr. If A E K and T E End(V), we define V(A,T) := {v E V lTv = Av}. (A.1) For X E b we set

v(~, x):= v(~, ~(x)).

(A.2)

For a E b* we define V ( a , b) by V ( a , b) := {v E V [VX E b 9X . v = a(X)v } = N

V(a(X),X).

(A.3)

XE~

Furthermore, V * := V(0, b).

(A.4)

If the role of b is obvious, we abbreviate V ( a , b) by Va. Define

zx(v, b):= {,~ e b* \ {o} I v,, # {o}} 246

(A.5)

A.1. N O T A T I O N

247

and

v ( r ) := ( ~ v , , ~ # r c a ( v , b).

(A.6)

aEF

The elements of A(V, b) are called weights, as is the zero functional if

v~ # {0}. If V is a real vector space, we set V c := V| of V c relative to V by u + iv := u -

iv,

We define the conjugation

u, v E V .

(A.7)

Sometimes we also denote this involution by a. We denote Lie groups by capital letters G, H, K, etc., and the associated Lie algebras by the corresponding German lowercase letter. For complexifications and dual spaces we use the subscript c, respectively the superscript *. If ~p" L -+ K is a homomorphism of Lie groups, we use ~ for the corresponding homomorphism of Lie algebras and complexified Lie algebras. In particular, we have ~(exPL(X)) = expg(99(X)) for all X E [. We call a real Lie group semisimple if its Lie algebra is semisimple. A Lie algebra is called reductive if it is the direct sum of a semisimple and an abelian Lie algebra. In contrast to the Lie algebra case, there is no generally agreed-on definition for real reductive groups. Therefore we make explicitly clear that we call a real Lie group reductive if its Lie algebra is reductive. Let G be a semisimple connected Lie group with Cartan involution 0. Let K = G o be the corresponding group of 0-fixed points in G. The Lie algebra of K is given by

= g(z, o ) = { x e g I o ( x ) = x } . Let p = g ( - 1 , 0). Then we have the Cartan decomposition g = ~ @ p.

(A.8)

We denote the Killing form of g by B , i.e. , B ( X , Y) = Tr(ad X o ad Y). For X, Y E g set

( X I Y ) := (X I Y)o :=

-B(X,O(Y)).

(A.9)

Then (. ] ")0 is an inner product on g. We will denote the corresponding norm by I" l0 or simply by [" ]. With respect to this inner product, the transpose ad(X) T of ad(X) is given by - a d ( O ( X ) ) for all X E 9. In particular, ad(X) is symmetric if X E p.

248

A P P E N D I X A. P ~ D U C T I V E LIE GROUPS

Let a be a maximal abelian subalgebra of p and m be the centralizer of

a in t~, i.e., m =

(A.10)

As { a d X [ X E a} is a commutative family of symmetric endomorphisms of g, we get

g=m~a@

~

aEA(S,,)

9~.

The elements of A = A(g, a) are called restricted roots. For X E a such t h a t a ( X ) ~ 0 for all a E A, one can define a set of positive restriced roots to be A + := {a e A l a ( X ) > 0}. (A.11) Set aEA+

Then n is a nilpotent Lie algebra and we have the following Iwasawa decomposition of g:

g=t@a~n.

(A.12)

Let N = exp n and A = exp a. Then N and A are closed subgroups of G and we have the Iwasawa decomposition of G: K • A • N 9 ( k , a , n ) ~ kan E G.

(A.13)

This m a p is an analytic diffeomorphism. For a subset L of G and a subset b of g, we denote by ZL(b) the centralizer of b in L: ZL(b) = {b e L I V X e b : Ad(b)X = X}.

(A.14)

Similarly, we define the normalizer of b in L by Nr.(b) = {b e L I V X e b : Ad(b)X e b}. We fix the notation M = ZK(a), M* = NK(a) and W := W(a) := M * / M . The group W is called the Weyl group of A. We will use the following facts t h a t hold for any connected semisimple Lie group (cf. [44]). 1) If a and b, are maximal abelian in p, then there exists a k E K such that Ad(k)b = ,.

2) If k E M* and a E A, then k . a = c~ o Ad(k -1) is again a root.

A.2. F I N I T E - D I M E N S I O N A L R E P R E S E N T A T I O N S

249

3) Let A + be a set of positive roots in A. For k E M * we have k E M if a n d only if k - A + = A+. 4) If A C A is a set of positive roots, t h e n t h e r e exists a k E M * such that k. A = A+. Let Brain "= M A N . T h e n Pmin is a group. T h e groups c o n j u g a t e to Brain are called minimal parabolic subgroups of G. A s u b g r o u p of G containing a m i n i m a l p a r a b o l i c s u b g r o u p is called a parabolic subgroup. If )~ E a~ we define a c h a r a c t e r a ~ a x on A by a "x "= exp A(X), a = exp X. By restriction, (. I')0 defines an inner p r o d u c t on a a n d t h e n also on a* by duality. Choose a C a r t a n s u b a l g e b r a t containing a. T h e n t is 0-stable a n d t = tk ~ a, where tk = t N [~ C m. T h e elements of A ( g c , tC) are called roots . Similar to the case of restricted roots, one can choose a s y s t e m of positive roots. This can be done in such a way t h a t A + (g, a) = {hi, l a e

x+(gc, tc), A.2

# o}.

Finite-Dimensional

Representations

D e f i n i t i o n A . 2 . 1 Let IK be R or C a n d V a K-vector space. Given a c o n n e c t e d real Lie group G a n d a s u b g r o u p L C G, a r e p r e s e n t a t i o n lr of G on V is called L-spherical if t h e r e exists a nonzero L-fixed vector v E V g e n e r a t i n g V as an L-module. We define VL =

e V IT

e L : lr(a)v = v

}.

Similarly, if [ is a s u b a l g e b r a of t h e real Lie algebra g a n d rr is a representation of it on V , t h e n 7r is called [-spherical if t h e r e is a nonzero v E V such t h a t 7r([)v = {0}. n If G is semisimple a n d L = K , t h e n we call lr spherical if it is K-spherical. For t h e following t h e o r e m , see [45], p. 535. A . 2 . 2 Let G be a connected semisimple Lie group and rr an irreducible complex representation of G in the finite-dimensional Hilbert space V .

Theorem

1) r ( K ) has a nonzero fixed vector if and only if ~r(M) leaves the highestweight vector of 7r fixed.

A P P E N D I X A. R E D U C T I V E LIE GROUPS

250

2) Let )~ be a linear form on it~ (ga, where tk = t N t . Then )~ is the highest weight of an irreducible finite-dimensional spherical representation lr of G if and only if

~1,, = o

,,d

W e A+(g.,)

9 (al~) (~1~) e Z+.

(A.18)

Here (. [.) is the inner product on (irk + a)* abtained from the Killing form by duality. 0

L e m m a A . 2 . 3 Let G be a connected semisimple Lie group with Caftan involution 0 and (Tr, V) be a finite.dimensional (complez or real) representation of G. Then there exists an inner product on V such that lr(x)* = 7r(0(x)-l). In particular, 7r(x) is unitary for all x e K and symmetric for x E exp p.

Proof:. If V is real, we replace V by Vc. Since the real part of an inner product on V c defines an inner product on V by restriction, we may assume that V is complex. The corresponding representation of gc is again denoted by 7r. Consider the compact real form u = t (B ip of gc and let U be a simply connected Lie group with Lie algebra u. Then U is compact. As U is simply connected, there exists a representation Try of U whose derived representation is rrl,. Let (. I')0 be an inner product on V. Since V is compact, we can define a new inner product by

(~ Io) =

fu (~u(,)wI ~u(-)o)o d , .

Relative to this inner product, Try is unitary. From the definition of u, it now follows easily that for any x E G we have

~(~)" = ~ (o(~)-~),

(A.19)

and the lemma follows. L e m m a A . 2 . 4 Let G be a group and V an irreducible finite-dimensional real G-module. Then the following statements are equivalent:

1) The complezified module V c / s irreducible. 2) V carries no complex structure making it a complex G-module. 3) The commutant of G in End(V) ts R - I d v .

A.2. FINITE-DIMENSIONAL R E P R E S E N T A T I O N S

251

Proo~ Suppose that V c is not irreducible. Then we can find an irreducible complex submodule W of Vc. If W MV ~t {0}, then W N V is an invariant nonzero subspace of V. As V is irreducible, it follows that W N V - V. But then W = V c and V c would be irreducible. Thus we have W N V = {0}. Now we have W f'l W = {0} because otherwise there would be a w E W , w ~t 0, such that ~ E W . Then one of the vectors ~ + w, i ( ~ - w) is nonzero and

~+w,i(~-w) ~ v n w , which contradicts V N W = {0}. This implies that the R-linear map

+

)ev

is injective and G-equivariant. On the other hand, the image is nonzero. Therefore the map is an R-linear G-isomorphism W _~ V. Thus V is a complex G-module. Conversely, if I is a complex structure on V for which V is a complex G-module, then the complex linear extension of I to V c is an intertwiner on V c which is not a multiple of the identity. Thus Schur's lemma implies that V c is not irreducible. Clearly, 3) implies 2). Conversely, if 1) holds, the commutant of G in E n d c ( V c ) is C. Idvc. Since E n d R ( V ) N C . Idvc = R - I d v , we obtain 3). o L e m m a A . 2 . 5 Let G be a connected semisimple Lie group and V a finite-

dimensional irreducible real G-module. 1) d i m V g _< 1. 2) If dim V g = 1, then V c /s irreducible. Proof. 1) We apply Theorem A.2.2 to Vc. Let A E a* be a maximal weight of V c and let v be a nonzero weight vector of weight A. Multiplying by a suitable scalar, we may assume that ~ ~ - v . Then (man). v = aXv,

m E M, a E A, n ~. N.

In particular, P. v = R+v.

(A.20)

It follows that (man).V = aXV. Thus u "= v+V is in V and lr(man)u = aXu. Let now OK E V K \ {0}. As G = K A N and (lr(kan)u l o g ) = aX(u l OK), we find that (u I VK) ~ O. If dim V g > 1, we can choose a K-fixed vector WK which is orthogonal to VK. But then (u [ W K ) V K - (u [VK)WK E v K \ {0}, whereas

(" I(" I ','K)"K -- (" I " K ) ' K ) = O,

A P P E N D I X A. R E D U C T I V E LIE GROUPS

252

which is impossible by the above argument. Thus dim V g < 1. 2) This is an immediate consequence of L e m m a A.2.4 ,since a complex G-module has even real dimension and V K is a complex subspace of V if V carries a complex structure commuting with G. 13 L e m m a A . 2 . 6 Let G be a connected semisimple Lie group acting irreducibly on V . Assume that V is spherical so that V c is irreducible. Let X E a* be the highest weight of V c , u be a highest-weight vector, and

VK E V K be such that (u I OK) > O. Further, let P - M A N be the minimal parabolic subgroup of G with A+ ({t, a) = A(n, a). Then u E P . R+VK. Proof. Write vg = E v, with v, E ( V c ) , \ {0}. As different weight spaces are orthogonal and dim V x - 1, it follows that X occurs in the above sum and t h a t vx = cu with c > 0. We can write p - X - ~ n ~ a with na > 0 and c~ E A+(gC, tC). But as v g is M-fixed, it follows that all the v~'s are M-fixed. Thus aitk - 0. Now choose H E a such that c~(H) > 0 for all c~ E A+ (9, a). Let at := exp t H E P. Then lim

e -tx(H)at

" VK

t .--+o o

=

lim t -.+ o o

cu + Z

which proves the claim.

exp(-t Z

n~a(H))v,]

~- C~

r-]

R e m a r k A . 2 . 7 Let L be a connected reductive Lie group with Lie algebra [ and G "= L ~ the commutator subgroup of L. Then G is a semisimple connected Lie group, but not necessarily closed in L. However, if L is linear, then G is closed (cf. [66l). a L e m m a A . 2 . 8 Let L be a connected reductive Lie group and rr a representation on the finite-dimensional real vector space V . Suppose that the restriction of rr to the commutator subgroup of L is irreducible and spherical. Then the center of L acts by real multiples of I d v .

Proof. Since the restriction of lr to G := L ~ is spherical, L e m m a A.2.5 implies t h a t V c is an irreducible G-module. Therefore the commutatant of 7rig is R . I d v by L e m m a A.2.4. Let Z := Z(L) be the center of L. Then rr(Z) of L is contained in the commutatant of 7rig and this implies the claim. 13

A.3

Hermitian Groups

A Herrnitian group is a real connected simple Lie group G for which the corresponding Riemannian symmetric space G / K is a complex bounded symmetric domain. In terms of group theory, this is equivalent to

253

A.3. HERMITIAN GROUPS

~(t) ~ {0},

(A.21)

G/K C P+/(KcP-) C Gc/(KcP-).

(A.22)

where 3(t) is the center of t and g = t % p is a fixed but arbitrary Cartan decomposition. W e call a Lie algebra Hermitian ifit is the Lie algebra of a Hermitian group. For the results in this section, we refer to [44, 83, 84, 155]. If g is Hermitian, then g and t have the same rank, i.e.,there exists a Cartan subalgebra t of g contained in t. Moreover, ~(t) is one dimensional and every ad Zlp, Z E 3(t) \ {0} is regular (see [44],Chapter 6, Theorem 6.1 and Proposition 6.2). Further, there exists an element Z ~ E ~ (t) with eigenvalues 0, i,-i such that the zero-eigenspace is t, and adp Z ~ is a complex structure on p. The :Ei-eigenspace p• of ad Z~ is an abelian algebra. Let g be Hermitian and Gc a simply connected complex Lie group with Lie algebra gc- Then the analytic subgroup G of Gc with Lie algebra g is Hermitian and closed in Go. Let P • be the analytic subgroup of Gc with Lie algebra p• The group P • is simply connected, and exp : p• ~ P • is an isomorphism of Lie groups. Denote the inverse of exp Ip+ by log 9 P + --+ p+. Then G C P+KcP-, where K c is the analytic subgroup of Gc with Lie algebra to, and we have the following embeddings of complex manifolds:

Since P+ ~ p ~ p / K c P - is injective and holomorphic, we get an embedding

G / K ~ P + / ( K c P - ) ~- P+ ~-~ p+.

(A.23)

We denote this composed map by m ~-+ ~(m) and call it the Harish-Chandra embedding. Then ~(G/K) is a bounded symmetric domain in p+. Furthermore, the map

P+ x Kc • P - ~ (p, k, q) ~ pkq e Gc is a diffeomorphism onto an open dense submanifold of Gc and G C P + K c P - . If g e G, then g = p+(g)kc(g)p-(g) uniquely with p+(g) e P+,

k(g) e Kc and p - ( g ) e P - . Let g be a Hermitian algebra and G C Gc a Hermitian group with Lie algebra g and simply connected Gc. We denote the complex conjugation of gc w.r.t, g and the corresponding complex conjugation of Gc by a and note that G = G~ (cf. Theorem 1.1.11). Let t be a Cartan subalgebra of g containing Z ~ Then t C ~. Let A = A(gc, tC), Ak = A(tC, tC) and A , = A(pc, tC). The elements of Ak are called compact roots, whereas the elements of An are called noncompact roots. Choose a basis Zz, Z2,..., Zt

A P P E N D I X A. RED UCTIVE LIE GROUPS

254

of t N [[~,[~]. The lexicographic ordering of it* with respect to this basis is now defined by A > p ff the first nonzero number in the sequence ( A p)(iZ~ (A - p)(iZ1), (A - p ) ( i Z 2 ) , . . . , (A - #)(iZt) is positive. Denote as usual by a superscript + the corresponding set of positive roots. Note that the ordering is chosen such that A+ = A(p+, tC). For a E A we choose Ea E (gc)a such that E - a = a(Ea). The normalization of the Ea can be chosen such that the element Ha = [Ea, E - a ] E it satisfies a ( H a ) = 2 (cf. [441, p. 387). Two roots a, ~ E A are called strongly orthogonal if a 4-fl tg A. Note that strongly orthogonal roots are in fact orthogonal w.r.t, the inner product on it* induced by the Killing form. We recall the standard construction of a maximal system of strongly orthogonal roots: Let r be the rank of D = G / K , i.e., the dimension of a maximal abelian subalgebra of p. Let r r := A(p+,tC) and 7r be the highest root in Ft. If we have defined F . D F~-I D ... D Fk ~ q) and 7j E r j , j = k , . . . r , we define Fk-1 to be the set of all 7 in r k \ {7k} that are strongly orthogonal to 7k. If r k _ l is not empty (or, equivalently, k > 1), we let 7k-1 be the highest root in Fk-1. Set F := { 7 1 , . . . , ")'r}. Then F is a maximal set of strongly orthogonal roots in A(p+, tC). We get a maximal set r := { 7 1 , . . . , 7,} of strongly orthogonal roots. Let E• := E+~j and Hj := H~#. Further, we set

Xj

:=

- i ( E j - E_j) E p,

Xo :-- ~

Xj ,

(A.24)

j--1

1~

:-

Yo:=-~1 ~L~I~ '

Ej+E_jEp,

j--1

Eo

:=

~E/ep

+

(A.25) (A.26)

j=l

and

zo:=

j--1

Furthermore, we let P

a

:=

(A.27) j=l

7EP

r

ah

c pn

:= j--I

+ (,c)-.,), 7Er

(A.28)

A.3. H E R M I T I A N G R O U P S

255

and

r

t-:=i aHj. j=l The strong orthogonality of the roots 7i, 7j, i ~ j implies that the inner automorphisms, Cj E Aut(gc), j = 1 , . . . , r, defined by Cj "= Ad(cj),

~ri cj = exp - ~ X j ,

with

commute. They are called partial Cayley transforms. Their product C := C1 o . . . o Cr E Aut(gc) is called the Cayley transform. It satisfies C = Ad(c)

7ri c = exp-~Xo,

with

(A.29)

so in particular we have c 8 = 1. E x a m p l e A.3.1 For g = su(1, 1) we get HI =

(1 O) EI_(O 1) 0

-1

'

0

0

, E-1 -

(0 O) 1 0

and z~

i (1

o-

01)

'

x~

1 (0

i

-i) o

'

and

Yo = 1 ( 0 1 ) 5 1 o

9

This gives

1 (11

1) 1

Define ~j : ~[(2, C) --+ gc by qaj : HI ~-r Hi,

E1 ~ Ej ,

and

E_ 1 b.~ E_j.

Then [Im ~j, Im ~k] = 0 for j ~ k and ~j o a = a o ~j. Thus, in particular, ~j (su(1, 1)) C g. As SL(2, C) is simply connected, 9j integrates to a homomorphism ~j : SL(2, C) -r Gc such that ~j(SU(1,1)) C G. The use of the ~j allows to reduce many problems to calculations in SL(2, C). We give an application of this principle. L e m m a A.3.2 The elements Zo, Xo, and Yo span a three dimensional subalgebra of g isomorphic to eu(1,1). Set ~ := exp(I./21zo), ~h := exp ([irr/2)Yo), C := Ad(6), and Ch := Ad(ch). Then we have

A P P E N D I X A. RED UCTIVE LIE GROUPS

256

0 C(H~)= - ~ , C(Xj)= X~, C(~) = H~. e) (3(H~)= H~, (3(X~)= ~, (~(~)=-X~. 3) Ch(Hj) - X j , C h ( X j ) - - H i , C h ( I / ~ ) - I/~.

P r o p o s i t i o n A.3.3 a and ah are maximal abelian subalgebras of p. Furthermore, C h ( t - ) -- ia

and

C ( t - ) - iah.

El

The following theorem of C. C. Moore [110] describes the set of restricted roots A(g, ah). A similar statement is also true for a instead of ah. T h e o r e m A.3.4 (Moore) Define aj = 7j o C -z. Then the set of roots of a in g is given either by 1 A(g, a) = +{c~j, ~(ai + c~k) [ 1 < i , j , k < r;i < k} or by 1 1 zx(g, ,) = • ~,~, ,~ , ~(,~ .4- ,~ ) I z <_ i, j, k <_ ,.; i < k}. Taking + gives a positive system of roots. The dimensions of the root spaces for the roots -l-l(oli-4-Otk) all agree. O

T h e o r e m A.3.5 ( K o r ~ a y i - W o f f ) lent:

The following properties are equiva-

1) The Cayley transform C E Aut(gc) has order 4. 2) Z ~ = Zo. 3) adXo has only the eigenvalues 0 and • 4) The restricted root system is reduced. 1/these conditions are satisfied, 12 = Ad(Ch(L))(iEo) is an open convex cone in g(1,Xo), where L := KcN(cGc-~). Moreover, the Cayley transform induces a biholomorphic map Ch o ch : r

~ g(1, Xo) + if~

defined by Z ~ C h l ( C h 9Z) (cf. [155], p. 137).

[21

If the conditions of Theorem A.3.5 hold, the Riemannian symmetric space G / K is called a tube domain.

Appendix B

The Vietoris Topology In this appendix we describe some topological properties of the set of closed subsets of a locally compact space. This material is needed to study compactifications of homogeneous ordered spaces. Let ( g , d) be a compact metric space. We write C(K) for the set of compact subsets of K and Co(K) for the set of nonempty compact subsets. For A E Co(K) and b E K we set

d(A, b) = d(b, A):= min{d(a, b) l a e A} and for A, B E C0(K) we define the Hausdorff distance:

d(A,B) :-- max { max{d(a,B) { a e A},max{d(b,A) { b e B}}.

(B.2)

This metric defines a compact topology, called Vietoris topology, on Co(K) ([11], c a . II, w Ex. 15). We set

d(A, 0 ) - d(@,A):-- oo.

(B.3)

For two open subsets U, V C K we set

K(U, V) := {F e C(K) ] F C U, F N V ~ 0}.

(B.4)

Then the sets K(U, V) form a subbase for the Vietoris topology ([14], p.

162). Let X be a locally compact space which is metrizable and a-compact. We write ~'(X) for the set of closed subsets of X and C(X) for the set of compact subsets. To get a compact topology on ~'(X), we consider the one-point compactification X ~ := X U {w) and identify X with the 257

A P P E N D I X B. THE VIETORIS T O P O L O G Y

258

corresponding subset of X ~. Note that our assumption on X implies that X ~ is metrizable. We define the mapping ~ 9.~'(X) -4 Co(X'),

F ~ F U {w}.

(B.5)

Then/~ is one-to-one and we identify 9v(X), via/~, with the closed subset Im/~ = { g e d0(X ~) I w E g } . As a closed subspace of d0(X~), the space ~'(X) is a compact metrizable topological space. For a sequence (An)heN in ~'(X), we define liminfA, "= t x e x I v.-, e

>

1 , ~ . d ( x , A , ) < -~}

(B.6)

and limsupAn "= {x fi X IVm fi N, Vno fi N, 3n >_ no " d(x, An) < ~ } .

(B.7)

We note that these sets are always closed. L e m m a B . I . 1 The following assertions hold:

1) If U C X , then {F E ~'(x)I F n U # $} is open; and if A C X is closed, then { f e ~'(X) I f C A } / s closed. 2) Let A , be a sequence in Y ( X ) . Then An converges to A e Y ( X ) if and only if A = limsupAn = liminfAn. In this case A consists of the set of limit points of sequences (an) with an E An. 3) If A C X is closed, then { f e 9v(X) I A C F} is closed. ~) If An is a sequence of connected sets, An -r A r 0, and .for every n e N the set Um>n Am is not relatively compact, then every connected component of 74 is noncompact. 5) The relation C is a closed subset of Jr(X) • Jr(X). Proof. 1) The first assertion follows from the observation that {F e Jr(X) [ F n U r $} = Jr(X) n {F e d ( X ~) I F N U ~t 0} because U is first one with 2) =~" Let intersects the

open in X ~. The second assertion follows by applying the U := X \ A. x E A. Then there exist numbers nm E N such that A , (1/m)-baU around a if n > nm. W.l.o.g. we may assume that

259 the sequence nm is increasing. For k = nm + 1, ..., nm+l we choose ak (5 Ak with distance less than 1/m from a. Then a = limk-,oo ak and therefore a (5 liminfAn. If, conversely, a (5 limsupAn, then there exists a subsequence nm (5 N with nm > m and elements am (5 A n . with d(a, am) < 1/m. We conclude t h a t a = lim am (5 lim Ann., = A. This proves t h a t limsupA. C A C liminfAn. (r We use the compactness of bY(X). Let An,,, be a convergent subsequence of An. Then the first part implies t h a t lim Ann -- liminfAnn. Moreover, we have t h a t liminfAn C liminfAnn and limsupAnn C limsupAn. Hence A - liminfA, C liminfAnn C limsupAnn C l i m s u p A , - A implies t h a t lim An.. - A. Finally, the arbitrariness of the subsequence of An entails t h a t An ~ A. 3) This is an immediate consequence of 2). 4) Let a E A be arbitrary and C(a) the connected component of a in A. We have to show t h a t C(a) is noncompact. Suppose this is false. Then C(a) is a compact subset of X and there exists a relatively compact open neighborhood V of C(a) in X such t h a t V 13 A is closed ([11], Ch. II, w Cor.]) and therefore compact. Let ~ "= min{d(a',b) I a' e V 13 A,b e x \ v } > 0 and e := min{~/4,d(w, Y ) / 3 } . Since a e l i m A . , there exists nv E N such t h a t d(A,, a) < e for all n > nv. Moreover, we may assume t h a t d(An, A U { w } ) < e. Let an e An with d(an, V13A) < E. Then d(an,w) > e because d(w, V N A ) > 3e and therefore d(an, A ) < e. Let b e A such t h a t d(a,, b) < e. Then d(b, V 13A) < 3e < 5. Hence b (5 V 13 A and this entails t h a t d(a,, V 13 A) < e. Thus

{~ e A, I d(~, A n v) e]~, 2~[} = ~. Since An is connected, this implies t h a t An C V. This contradicts the assumption t h a t U m > , Am is not relatively compact. 5) Let (An, Bn) --4-(A,B) with An C Bn. Then 2) shows t h a t A C B. [::1 We recall t h a t a Hausdorff space Y endowed with a closed partial order < is called a pospace. Thus we see t h a t (Jr(X), C) is a compact pospace.

260

A P P E N D I X B. THE V I E T O R I S T O P O L O G Y

Proposition B.1.2

L e t p 9G x X

-4 X be a continuous action of a locally

compact group G on X . Then ~ ( p ) " G x Jr(X) --} Jr(X),

(g, F) ~ g(F)

defines a continuous action of G on Jr(X). Proof. Let (g,F) = lim(gn, Fn) with gn E G and F, E ~ ( X ) . We have to show that gn(Fn) "} g(F). To see this, we have to show that each convergent subsequence converges to g(F). So we may assume that gn(Fn) is convergent. Let f E F. Then there exists a sequence fn 6 Fn with fn "} f. Hence gn " In -'} g" f implies that g. f E lim gn (Fn). This proves that g(F) C limgn(Fn). If, conversely, f~ = limgn" fn E limgn(Fn), then

.f, = g;-~. (g,. f,) ~ g-1. f, entails that f - l i m g ,

f~ E l i m g ( F , ) - g(lim F , ) - g(F).

C]

P r o p o s i t i o n B.1.3 Suppose that the locally compact group G acts continuously on X . Further suppose that 0 C X is open. 1) The set S := S(O) = {g E G i g . 0 C O} is a closed subsemigroup of G. 2) If X is a homogeneous G-space, then the interior of S is given by S~

={geGIg.-~cO},

where O is the closure of O in X . Proofl 1) According to Lemma B.I.1 the set {F e J r ( x ) I X \ O C F} is closed. Therefore Proposition B.1.2 implies that also S = {g e G I X \ O C g ( X \ O)} is closed. Since S obviously is a semigroup, this implies the claim. 2) Let g E S ~ and U be a neighborhood of 1 in G such that Ug C S. Then g . O = g . Oc ( U g ) . O C S . O C O. . . . . .

Conversely, if g. 0 C O, then there exists a neighborhood U of I in G such that (Ug). O C O. In particular, we find Ug C S, whence g E S ~ n P r o p o s i t i o n B.1.4 Let G be a Lie group. Set Jr(G)H := {F e ~'(G) I Vh E H " F h = F}. Then the mapping rr* " Jr(G/H) -~ Jr(G)H, is a homeomorphism.

F ~ Ir-X(f)

261

Proof. T h a t ~r* is a bijection follows from the fact t h a t the quotient m a p p i n g Ir : G ~ G / H is continuous. So it remains to show t h a t lr* is continuous. Let Fn ~ F in .T'(G/H) and set En := ~ r - l ( F , ) and E := ~ r - l ( F ) . We m a y assume t h a t E , -+ E ~. Then we have to show t h a t E = E ' . Let e' E E ' and e , e E , with en --+ e'. Then ~r(en) -~ 7r(e') E lim F , = F . Hence e ~ E E and therefore E ~ C E. If, conversely, e E E, then r ( e ) E F and there exists a sequence fr, E Fn with fn -+ ~r(e). Using a local cross section a : U -+ G, where V is a neighborhood of 7r(e) and a (lr(e)) = e, we find t h a t a(fn) --4 e ~. l i m E ~ = E'. El B . 1 . 5 Let G be a Lie group acting on a locally compact space y and X E g. For p E Y let gv be the Lie algebra of the group G p = {g E G ] g . p = p}. /f q = l i m t _ , ~ e x p ( t X ) . p then Lemma

limsuPt~ooead tXgp C gq.

Proof:. Suppose Z E limsupn_~cceadnXgv. Then Z = limk-~oo eadnkXyk for suitable sequences Yk E gv and (nk)keN. But then exp Z . q

-" = =

k-,oolim(exp(nkX) exp(Yk) e x p ( - n k X ) ) l i r n e x p ( n k X ) . p lim e x p ( n k X ) e x p ( Y k ) e x p ( - n k X ) e x p ( n k X ) . p

k--+c~

lim e x p ( n k X ) exp(Yk) . p

k--+cx~

lim exp (n kX ) . p = q

k--+cx~

which proves the asertion.

[]

Notation Chapter

1

H, stabilizer of a base point: 2 r, a nontrivial involution on G and g: 1 G ~ = {a e G I rCa) = a}: 2 g(1, r), the +l-eigenspace of r: 2 [} _. gr, the Lie algebra of H: 2 q, the ( - 1)-eigenspace of v: 2 a, the conjugation X + i Y ~ X - i Y relative to g: 3 ~=roa: 6 ~ ( - z , ~), ( - z ) - ~ i g ~ p ~ e of ~: 2 adq(X) - ad(X)lq, restriction of a d X , X e ~, to q: 2 g~, left translation by a: 4 .M, the symmetric space G / H : 4 j~4, universal covering space of M : 5 bk, l}p, qk, qp: 7 (~, the analytic subgroup for g in the simply connected complexification: 6 (~r the same with g replaced by go: 6 H, the v-fixed group in (~: 6 =

r

6

./Qc = G / H : 6

r x - A d ( e x p r X ) - e~radX: 8 7~x = Ad (exp([~/2]X)) = e['~/2l *dx" 8 gC _ {~~ iq, the c-dual Lie algebra: 6 t c, the maximal compactly embedded algebra {Jk (9 iqp in gc: 8 pc=tjp~iqkcgC: 8 qC = iq, the - 1 eigenspace of rio," 8 r a = r o 8, the associated involution: 9 b a = g ( + l , r ~) = {Jk 9 qp: 9 q~ = g ( - 1 , r ~) = qk 9 bp: 9 262

NOTATION

263

M a, a symmetric space locally isomorphic to G/Ha: 9 or = rigr the Cartan involution on the Riemannian dual Lie algebra gr. 9 g" = IJk (9 ilJp ~ iqk ~) qp, the Riemannian dual Lie algebra: 9 G ~, a Lie group with Lie algebra g~: 9 t r=bk~i~pCgr:

9

K ~ = exp t ~, the maximal almost compact subgroup of Gr: 9 A4 ~, the Riemannian dual space G~/K": 9 qHnK={XeqIVkEHOK : A d ( k ) X = X } : 12 qHonK, the same as above with H replaced by Ho: 12 a,(a) = { Y e [ I V X e a : [Y, X] = 0}, the centralizer of a in [: 14 ~([) = ~,([), the center of [: 14 yo, a central element in [~p such that ad(Y ~ has spectrum 0, 1 and - 1 and = g(0, yo): 21 Qp,q, the bilinear form xlyl + . . . + xpyp - Xp+lyp+l - ... - xnyn: 24 Q• = {x e R" I Q~:,(x,x) = • 24

ip 'q ._ ( lp0

O ) , where Ik is the (k x k)-identity matrix: 25 --Iq

M(l x m, IK), the K-vector space of (k x m)-matrices: 25 Z(G), the center of G: 14 q+ = g ( + l , Y ~ q- = q ( - 1 , Y ~ Cayley type spaces: 13

the irreducible components of q for

Chapter

2

V c = C n - C , the largest vector space contained in the closed cone C: 29 < C > : = C - C, the vector space generated by the cone C: 29 C* = {u E V lVv e C, v # 0 9 (ulv) > 0}, dual cone: 29 v 9 -- {, e V lW

e v

9

= o}. 3o

C ~ - int(C), the interior of C: 30 algint(C), the interior of C in < C >, the algebraic interior of C: 30 cone(S), the cone generated by S: 30 Cone(V), the set of regular, closed convex cones in V: 30 - c/(~), the closure of f~: 30 Fa(C), the set of faces of the cone C: 31 op(F) - F • n C, the face of C* opposite to the face F of C: 32 p w ( c ) , the orthogonal projection of the cone C into V C W : 32 I vTM (C) - C n V, the intersection of C with V: 32 Aut(C) = {a E GL(V) ] a(C) = C}, the automorphism group of C: 33 ConeG(V)" G-invariant cones in Cone(V)" 33,38 Ev,NW,L ( C ) = cony (L. C), the minimal L-invariant extension of C: 34 M(m, K) = M ( m x m,K): 34

NOTATION

264

H(m,K) = { X ~. M ( m , K ) I X " = X } , space of Hermitian matrices over 34 H + (m, K), the cone of positive definite matrices in H(m, K): 34 cony(L), the convex hull of the set L: 36 u g -- f g ( k . u)dk e c o n v ( K , u), a K-invariant vector in C obtained as the center of gravity of a K-orbit: 36 Cmin, a minimal invariant cone: 39 Cmax, a maximal invariant cone, Cmax = Cmin. 39 O, the basepoint o = e H if .A4 = G/H: 40 ~s, strict causality relation via connecting by causal curves: 41 -~, the closure of the relation ___s: 41 T A = {y E Y I ::la E A " a < y}" 41 .l,A = {y e Y ] 3a e A " y <_ a}: 41 [m,n]<={zE.Mlm
d e m e n t s of

Y,(G/H) with

o

noncompact connected upper sets:

48 aA, the boundary of A: 49 Ck = C+ - C_ C q, a cone such that C~ FI t~ r 0 : 5 3 Cp = C+ + C_ C q, a cone with C~ FI p ~ ~}: 53 X• (K gl H)-invariants in q+: 52 12, a Cayley transform commuting with ra: 56,255 p+(g),kc(g),p-(g), the projections of g ~. P + K c P - onto its components: 56 fl+, the bounded realization of G / K : 56 j(g, Z), the Kc projection of g exp Z: 56 g . Z, the P + component of g exp Z: 56 S, the Shilov boundary of G / K : 56 r = log(v) e P+" 56 E = ~(c), a base point in S: 56 S~ = S x S: 57 pn, half the sum of positive noncompact roots: 58

265

NOTATION

rrm, irreducible representation of Gc with lowest weight - m p n " 9 m(Z) = (lrm(c -2 exp Z ) u o I~o): 59 9

w)

:=

- w):

59

59

Chapter 3 X ~ a cone-generating element in q: 77 Ck = ~zo, an isomorphism (g, r, 0) _~ ({t, ra, 0): 78 Cv = ~~ ~ , a n isomorphism (g, r, 0) "~ (it, r a, 0) r: 78 r = qOiyo, a n isomorphism (g, r, 0) "~ (it ~, T, r ~): 78 =

e

~ = o} =

79

A• = {a E A l a ( X ~ = 4-1}: 80 n+ = ~ ( x o ) = + l g~: 80 no = ~ . e a o + g . C go" 80

Chapter

4

X ~ - A/IA[2 e a: 92 Ya e ita, such that IYal2 -- T~[r, Y-a -- v(Ya). Thus [Ya, Y-a] -- Xa: 92 y a = 89 + Y-a) e I~v: 93 Z a -- ~(Y-al _ Ya) E qk: 93 X • = l ( x a 4-Z a) E q: 93 ~Oa, a r-equivariant homomorphism s[(2, C) -+ [tc: 93 s a = I m T a : 94 A = A(gc, ~)" the roots of the Caftan subalgebra q: in ge: 95 /~• = A((pe) +, t~), the set of noncompact roots: 95 A0 := A(~=, t~), the set of compact roots: 95 = { 7 1 , . . . , 7rr }, a maximal set of strongly orthogonal noncompact roots" 96 a~, maximal abelian subalgebra of I]v: 96 Cmin(X 0) --- Cmin -- conv [Ad(Ho)(IR+X~ a minimal Ad(Ho)-invariant cone containing the cone-generating element X~ 98 Cm~x(X ~ = Cm~x{X E q lVY E Cmin : B ( X , Y ) ~_ O} = Cmin(X~ *, the maximal cone containing X~ 98 Cmin(X 0) ~- Cmin = E a E A + R0+ X a -- E a E A + R0-I"O~, the minimal Weyl group invariant regular cone in a: 98 Cmax(X 0) ----Cmax -'- {X Ell I Vo~ E A+ : o~(X) _~ 0} -- train, the corresponding maximal cone in a: 98 5min = ~ e s lR+ &' the minimal W-invariant cone in a Cartan subalgebra

NOTATION

266

containing a: 102 {X E it, r I W e zx+ 9~ ( x ) > o} - {x e it ~ I ~ e ~ + - ~ ( x ) > o} ~* n." 102 = Cmi Cmi., a minimal Ge-invariant cone in i{te: 103 Cmax = C~in, a maximal Ge-invariant cone in ig c" 103 ,~,

~

Cmax --

Wo = W ( A o ) =

NK,(te)/ZK,:(te):

115

C~(A+) = {X E a l v a E A + " a ( X ) > 0}, open Weyl chamber: 116 Wo(~) = {w e Wo I~ o w = ,o o r}: l i t Wo - {w e r162 I wl. - i d } : 117

Chapter

5

Pmax = HaN+, a maximal parabolic subgroup in G: 121 I~.G._+ G, g ~ r(g) -1" 121 A~ = exp ah, where ah C lip is maximal abelian: 121 0 = (Gr)o "ox C X = G/Pmax: 122 X = G/Pmax, the real flag manifold" 122 n . n_ ~ X, n(X) = (exp X). Pm~, real Harish-Chandra embedding: 123 ~'~- -- t ~ - - l ( o ) C tt_, real bounded domain, isomorphic to H / ( H n K): 123 fl+, the bounded realization of H / ( H n K) inside n+" 124 Xc = Gc/(Pmax)C, the complex flag manifold: 125 (n• ~- G*/Ke, the complexification of H / ( H n K)" 125 S(C) = H exp C, the closed semigroup in G with tangent cone ~ @ C: 129 S(O), the compression semigroup {g E G Ig.OC O}:133 S(L, Q), the compression semigroup {g E G [ gLQ C LQ} = S(LQ/Q): 134 O(g) = d~ Ag(Cm.x + ~), Vg e N~AH: 135 P(t,h) " N ~AH ~ N~AH, g ~ tgh -1" 135 Ih, the inner automorphism g vr hgh-l: 135 an, the causal Iwasawa projection, g E H exp(an(g))N: 136 W(a) = NK(a)/ZK(a), the Weyl group of a in G: 138 W,.(a) = {s e W(a) I s(a n Ij) = a n Ij}: 138 Wo(a) = NKntl(a)/ZKnlt(a), the Weyl group of a in H a. 138 SA = S(G ~, Pm~x) ~ n Aq: 141 /X = A(gc, t~), the set of roots of the Cartan subalgebra t~" 143 = ~ e X + ( g c ) ~ : 143 = iff: 143 = exp(~)" 143 = exp fi): 143 L x v V : 144 Ira" ~',(G) ~ jr, (B), f ~ f n B~. 153

267

NOTATION

riB," ~ ' , ( a ) --+ I , (B), F ~ F n B~- 154 Aft(N_) = N_ ~ End(N_) = N_ x End(a_): 159 Aff~om(Y-) = {(n_, 7) e Aft(N-) I n - 7 ( f l - ) C fl-}" 159 B~, the closure of B ~ in Aft(N_): 160 A~x, Pex" B~ ~ B~, left and right multiplication with e x in B~" 165 SApt = S r'l A = exp Cmax C B-'~, the closure of ~'AeCPtin B"-~: 160 0, i f X 6 g ~ , a ~ F A A _ e f " X b-~ X , if X 6 g ~ , a 6 F n A _ " 160 E x = (RX - Cmax) n Cmax, the face of Cm,x generated by X: 162 F x = X-L n --Cm~ * x = E } n -Cm, * x " 162 A x = E x ~ A A : 162 W x , the subgroup of W0 generated by the reflections fixing X 9 163 Ex" 163 Ex,o "- E:~ gl span{a1, ...,ak } = Ex,o,eff ~ Ex,o,flx: 163 Ex,0,fix = {Y E Ex,o I(Vw e W x ) w. Y = Y}" 163 Ex,o,etr = span{w. Y - Y ] w e W x , Y e Ex,0}" 163 U(T), the group of units in a monoid T: 168 Chapter

6

9 ~ ( X ) ~ ~ ( G ) , F ~ {g e G [ g - 1 . F c 0 } , causal Galois connection: 173 I'" 2 a -+ ~'(X), A ~ naeA a . O, dual map of r: 173 P

M ~ = {g. O IN ~. G} C .T(X), causal orbit: 176 ~: G ~ A4 ~ g ~ g. O, causal orbit map: 176 ~ f -- e f " ~ - , projection of fie" 180 ~x = lim,-,oo e ~d'xl} = 3~(X) + g(A+ \ X• 181 = {g e a = stabilizer of a projection ~F: 182 OF = 0X + [0F Cl (a + n~)], the Lie algebra of HF: 182 n x , - = g ( A x , - ) , image of the idempotent ex: 183 N x , - = exp(nx,_)" 183 Ny(L) = {g e L I g . Y = Y}: 183 Zy(L) = {g e n IVy e Y 9g . y = y}: 183 d(E) = dimker e l , the degree of E E G. flF: 187 T: jr _r ( M c p t \ {0})/G, f ~ G. ~F, classifying map for the G-orbits of A4~Pt: 188

NOTATION

268

Chapter 7 V ~176 the space of smooth vectors: 199 C(Tr), the cone of negative elements: 200 A(C), the set of C-admissible representations: 200 C(V), the space of contractions of V: 200 S(C), holomorphic dual of S: 202 O~, the character of ~r: 202 VK, the K-finite elements in V: 204 r := ~(kH(x-1)-l)t~o: 211 ~(r, v)(x)= (v]@=(~)), the generating function for E= C L 2 ( ~ ) 9211 E~, the holomorphic discrete series: 212 @,~(g,x) - r(kc(g))~,~(g-lx), g e G, x e P+KcHc" 213 Y, the classical Fourier transform: 215 eu, the function x ~-~ e (xlu}" 215 ~(C),~~ the o-orbits of the Ol'shanskii semigroups S(C) and S(C~ 9 217 ~2(C), the Hardy space corresponding to the cone C: 215,217 fl(f), the boundary value map: 217 K(z, w)" The Cauchy-Szeg5 kernel: 219 P(z, m), the Poisson kernel: 220

Chapter 8 M< - {(x,y) E M ] x < y}, the graph of the order <: 222 ]}(j$4), the Volterra algebra: 222,232 ]}(r #, the algebra of invariant Volterra kernels: 233, 233 f #G(x, y) = f[z,y] F(x, z)G(z, y)dz, the product of two Volterra kernels: 222,232 D(~4), the algebra of invariant differential operators on ~ 4 : 2 2 4 C~(G//K), the algebra of K-bi-invariant functions on G with compact support in K \ G / K : 224 eg(x) = e , e),(x) = e f = e '~-p'aH(~)>" 225 ~ (x) = f K e~ (kx) dk, the spherical function on G/K: 225 P = 89~ E A + (dimg~) ~: 225 ~ ( s ) = fH e~(sh)dh, the spherical function with parameter )~: 226 P0 = 89~ a e ~ + dimga a: 226 ~c = {)~ e ac I Va e A+ 9 Re(A + pla) < 0}, domain of definition for parameters for spherical functions: 226

NOTATION C ' = {A

269

E a~ IfKnHANe-ReA(k)dk < oo}. S a m e

as above: 227

P+ = ~1 ~ ~ z x + m~a: 227 D(fl4), the algebra of G-invariant differential operators on .A4" 227 A(O) := exp0 C A" 227 c~(A) = fn e-A (exp X) dX, the causal c-function: 228 c0(A) = f-~o e-A(n~) dn~' classical c-function for Ha/(H N K): 228 c(A) - ca(A)c0(A): 228 a A(~) oo, convergence to oo in a Weyl chamber: 228 cr(A) = fN~ e-x(rill) dnll, the classical c-function: 230 A ~ r ( D ) : the radial part of D E D(fl4r): 230 F~" 231 9x(a) = a A-p ~ ~ ^ F~(A)a-~. 231 c+ (A): 232 L:F(A) - f ~ F(o, m)ex(m) din, the Laplace transform of a Volterra kernel: 233 O~(HA(O)H//H), the space of H-biinvariant functions on HA(O)H that have compact support in H\HA(O)H/H: 234 Z)(f), domain of definition of the Laplace transform of f: 233 @x(x) - [1/c(A)]pA(x), the normalized spherical function: 234 /~(/)(A) - [I/c(A)]L:(/)(A), the normalized Laplace transform: 234 f~, the Kr-bi-invariant extension: 231 ' e(--~p~x(x) 9235 1 /~x "- ~r(x)E~" 235

.A(f)(a) = a p fg f(an)dn, the Abel transform: 236 A+I(go) = ap+ fN+ I(gon+)dn+" 236 Chapter

9

W(f), classical Wiener-Hopf operator with symbol f: 239 VV~+, C*-algebra generated by all Wiener-Hopf operators: 239 W~+ (f): Wiener-Hopf operator with symbol f: 241 G, a groupoid: 242 G2, composable pairs in ~: 242 d(x), domain map for {7:242 r(x), range map for G: 242 G~ unit space of G: 242 LI(G), closure of 0e(G): 243 C*(G), C*-algebra generated by L1({7) w.r.t, suitable norm: 243

NOTATION

270 W ~ + , Wiener-Hopf representation of C*(G): 244 Appendixes

V()~, T) = {v E V ITv = ~ v}, eigenspace of T for the eigenvalue )~: 246 V()~, X) = V(A, 7r(X)), for a representation r: 246 V ( a , b) = Va, simultaneous eigenspaces: 246 V b = V(0, b): 246 A(V, b), the set of weights: 246 V(F) = ( ~ V~, the sum of weight spaces: 247 O, the Cartan involution: 247 t~ = g0, the maximal compactly embedded subalgebra: 247 p = g(-1,O): 247 a, a maximal abelian subalgebra in p: 248 m = t~~ 248 A = A(g, a): 248 n = g(A+), the sum of positive root spaces; Iwasawa n: 248 W(a) = W, the Weyl group of (g,a): 248 V L = {v e V i V a ~. L : a . v = v}, L-fixed vectors in V: 249 Z ~ a central element in {~defining a complex structure on p: 253 Ak, the set of compact roots: 253 A , , the set of noncompact roots: 253 E~, root vectors for a E A , (suitably normalized): 254 H~, co-roots for a: 254 F, maximal set of strongly orthogonal positive noncompact roots: 254 E-t-j, normalized root vectors for 7j E F: 254 Hi, corresponding co-roots: 254 Xj = - i ( E j - E_j): 254 Y~ = Ej + E_j: 254 Xo 9254 Yo - 89 254 Eo: 254 Zo: 254 t - , the subspace of t generated by Hi: 254 Cj, partial Cayley transform: 255 C(K), the space of compact subsets of K: 257 C0(K), the space of nonempty compact subset of K: 257 d(A, b), the Hausdorf distance: 257 d(A,B): 257 K(U, V), a subbasis for the Vietoris topology: 257

= 89

NOTATION ~-(X), the space of closed subsets of X: 257 X ~, the one-point compactification of X: 257 ~, the one-point compactification map: 258 liminf, the limes inferior for sets: 258 limsup, the limes superior for sets: 258 Y:(G) H, the set of H-fixed points of ~'(G) under translation: 260

271

Bibliography [I]

Ban, E., van den: A convexity theorem for semisimple symmetric spaces. Pac. J. Math. 124 (1986), 21-55.

[2]

Ban, E., van den: The principal series for reductive symmetric spaces I, H-fixed distributions vectors. Ann. Sci. Ecole Norm. Sup. 21 (198S), 359-412.

[3]

Ban, E., van den: The principal series for a reductive symmetric space. II. Eisenstein integrals. J. Funct. Anal. 109 (1992), 331-441.

[4]

Beem, J. K., P. E. Ehrlich,: Dekker, New York, 1981.

[5]

Berger, M.: Les espaces sym~triques non compacts. Ann. Sci. Ecole Norm. Sup 74 (1957), 85-177.

[61

Bertram, W.: On the causal group of some symmetric spaces. Preprint, 1996.

[7]

Betten, F.: Kausale Kompaktifizierung kompakt-kausaler Riiume. Dissertation, GSttingen, 1996.

Is]

Bourbaki, N.: Groupes et alg~bres de Lie, Ch. 4, 5 et 6. Masson, Paris, 1981.

[9] [~0]

Bourbaki, N.: Groupes et alg~bres de Lie, Ch. 9. Masson, Paris, 1982.

[11] [12]

Global Lorentzian Geometry. Marcel

Bourbaki, N.: Groupes et algdbres de Lie, Ch. 7 et 8. Masson, Paris, 1990. Bourbaki, N.: Topologie g~ndrale. Hermann, Paris, 1991. Brylinski, J.-L., P. Delorme" Vecteurs distributions H-invariants pour les s6ries principales g6n~ralis6es d'espaces symetriques reductifs et prolongement meromorphe d'int6grales d'Eisenstein. Invent. Math. 109 (1992), 619-664. 272

BIBLIOGRAPHY

273

[13] Carmona,

J., P. Delorme: Base m6romorphe de vecteurs distributions H-invariants pour les s6ries principales g6n6ralis6es d'espaces sym6triques r6ductifs: Equation fonctionneUe. J. Funct. Anal. 122 (1994), 152-221.

[14] Carruth,

J. N., J. A. Hildebrand, R. J. Koch: The Theory oJ Topological Semigroups, Vol. II. Marcel Dekker, New York, 1986.

[15] Chadli,

M.: Domaine complexe associ6 ~ un espace sym6trique de type Cayley. C.R. Acad. Sci. Paris 321 (1995), 1157-1161.

[16] Clerc, J.-.L.:

Laplace transforms and unitary highest weight modules. J. Lie Theory, 5 (1995), 225-240.

[17] Cronstr6m,

C., W. H. Klink: Generalized O(1,2) expansion of multiparticle amplitudes. Ann. Phys. 69 (1972), 218-278.

[18] Davidson,

M. G., R. C. Fabec: Geometric realizations for highest weight representations. Contemp. Math., in press.

[19] Delorme,

P.- Coefficients g6n6ralis6 de s6ries principales sph6riques et distributions sph6riques sur Gc/GR. Invent. Math. 105 (1991), 305-346.

[20] Dijk, G. van: Orbits on real affine symmetric spaces. Proc. Koninklijke Akad. van Wetenschappen 86 (1983), 51-66. [21] Doi, H.: A classification of certain symmetric Lie algebras. Hiroshima Math. J. 11 (1981), 173-180. [22] Enright, T. J., R. Howe, N. Wallach: A classification of unitary highest weight modules. Proceedings: Representation Theory of Reductive Groups. Progr. Math. 40 (1983), 97-143. [23] Erdelyi, A., et al.: Higher Transcendental Functions, I. McGraw-Hill, New York, 1953. [24] Faraut, J.: Alg~bres de Volterra et transformation de Laplace sph~rique sur certains espaces symetriques ordonnes. Syrup. Math. 29 (1986),183-196. [25] Faraut, J.: Espace symetriques ordonn~s et alghbres de Volterra. J. Math. Soc. Jpn. 43 (1991), 133-147. [26] Faraut, J.: Functions sph~riques sur un espace sym~trique ordonn~ du type Cayley. Preprint, 1994.

274

BIBLIOGRAPHY

[27] Faraut, J.: Hardy spaces in non-commutative harmonic analysis. Lectures, Summer Schools, Tuczno, 1994. Preprint, Paris, 1994. [28] Faraut, J., J. Hilgert, G. Olafsson: Spherical functions on ordered symmetric spaces. Ann. Inst. Fourier 44 (1995), 927-966. [29] Faraut, J., A. Kors Press, Oxford, 1994.

Analysis on Symmetric Cones. Clarendon

[30] Faraut, J., G. Olafsson: Causal semisimple symmetric spaces; The geometry and harmonic Analysis. In: Semigroups in Algebra, Geometry and Analysis, De Gruyter, 1995. [31] Faraut, J., G. A. Viano: Volterra algebra and the Bethe-Salpeter equation. J. Math. Phys. 27 (1986), 840-848. [32] Flensted-Jensen, M.: Discrete series for semisimple symmetric spaces. Ann. Math. 111 (1980), 253-311. [33] Flensted-Jensen, M." Analysis on non-Riemannian symmetric spaces. CBMS Reg. Conf. 61 (1986). [34] Gel'fand, I. M., S. G. Gindikin: Complex manifolds whose skeletons are semisimple real Lie groups, and analytic discrete series of representations. Func. Anal Appl. 7 (1977), 19-27. [35] Gindikin, S.G., F.I. Karpelevich: Plancherel measure of Riemannian symmetric spaces of non-positive curvature. Dokl. Akad. Nauk USSR 145 (1962), 252-255. [36] Gradshteyn, I., I. Ryzhik: Tables of Integrals, Series and Products. Academic Press, 1980. [37] Giinter, P.: Huygen's Principle for Linear Partial Differential Operators of Second Order. Birkh~iuser, Boston, 1987. [38] Harish-Chandra: Representations of semisimple Lie groups, IV. Amer. J. Math. 77 (1955), 743-777. [39] Harish-Chandra: Representations of semisimple Lie groups V, VI. Amer. J. Math. 78 (1956), 1-41, 564-628. [40] Harish-Chandra: Spherical functions on a semisimple Lie group, I. Amer. J. Math. 80 (1958), 241-310. [41] Harish-Chandra: Spherical functions on a semisimple Lie group, II, Amer. J. Math. 80 (1958), 553-613.

BIBLIOGRAPHY

275

[42] Hawking, S. W., G. F. R. Ellis: The Large Scale Structure of SpaceTime. Cambridge University Press, Cambridge, 1973.

[43]

Helgason, S.: A duality for symmetric spaces with application to group representations. Adv. Math. 5 (1970), 1-154.

[44]

Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York, 1978.

[45]

Helgason, S.: Groups and Geometric Analysis. Integral Geometry, lnvariant Differential Operators and Spherical Functions. Academic Press, New York, 1984.

[46]

Helgason, S.: Geometric Analysis on Symmetric Spaces. AMS Math. Surveys and Monographs 39, Providence, R. I., 1994.

[47]

Hilgert, J.: Subsemigroups of Lie groups. Habilitationsschrift, Darmstadt, 1987.

[48] Hilgert, J.: Invariant cones in symmetric spaces of hermitian type. Notes, 1989.

[49]

Hilgert, J." A convexity theorem for boundaries of ordered symmetric spaces. Can. J. Math. 46 (1994), 746-757.

[50] Hilgert, J., K. H. Hofmann, J. D. Lawson: Lie Groups, Convex Cones and Semigroups. Oxford University Press, 1989. [51] Hilgert, J., K.-H. Neeb: Braunschweig, 1991.

Lie-Gruppen und Lie-Algebren. Vieweg,

[52] Hilgert, J., K.-H. Neeb: Lie Semigroups and Their Applications. Lect. Notes Math. 1552, Springer, 1993. [53] Hilgert, J., K.-H. Neeb: A general setting for Wiener-Hopf operators. In: 75 Years of Radon Transform, S. Gindikin and P. Michor, Eds., Wien, 1994. [54] Hilgert, J., K.-H. Neeb: Wiener Hopf operators on ordered homogeneous spaces I. J. Funct. Anal. 132 (1995), 86-116. [55] Hilgert, J., K.-H. Neeb: Groupoid C*-algebras of order compactified symmetric spaces. Jpn. J. Math. 21 (1995), 117-188.

[56l

Hilgert, J., K.-H. Neeb: Compression semigroups of open orbits on complex manifolds. Arkil Mat. 33 (1995), 293-322.

276

BIBLIOGRAPHY

[57] Hilgert, J., K.-H. Neeb: Compression semigroups of open orbits on real flag manifolds. Monat. Math. 119 (1995), 187-214. [58] Hilgert, J., K.-H. Neeb: Maximality of compression semigroups. Semigroup Forum 50 (1995), 205-222. [59] Hilgert, J., K.-H. Neeb, B. Orsted: The geometry of nilpotent coadjoint orbits of convex type in Hermitian Lie algebras. J. Lie Theory 4 (1994), 185-235. [60] Hilgert, J., K.-H. Neeb, B. Orsted: Conal Heisenberg algebras and associated Hilbert spaces. J. reine und angew. Math., in press. [61] Hilgert, J., K.-H. Neeb, B. Orsted: Unitary highest weight representations via the orbit method I" The scalar case. Proceedings of "Representations of Lie Groups, Lie Algebras and Their Quantum Analogues," Enschede, Netherlands, 1994, in press. [62] Hilgert, J., K.-H. Neeb, W. Plank: Symplectic convexity theorems. Comp. Math. 94 (1994), 129-180. [63] Hilgert, J., G. Olafsson, B. Orsted: Hardy spaces on affine symmetric spaces. J. reine und angew. Math. 415 (1991), 189-218. [64] Hilgert, J., G. Olafsson: Analytic extensions of representations, the solvable case. Jpn. J. Math. 18 (1992) 213-290. [65] Hirzebruch, U.: Uber Jordan-Algebren und beschr~inkte symmetrische Rs von Rang 1. Math. Z. 90 (1965), 339-354. [66] Hochschild, G.: The Structure of Lie Groups. Holden Day, San Francisco, 1965. [67] Hofmann, K.H., J. D. Lawson: Foundations of Lie semigroups. Lect. Notes Math. 998 (1983), 128-201. [68] Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Springer, New York-Heidelberg-Berlin, 1972. [69] Jaffee, H.: Real forms of Hermitian symmetric spaces. Bull. Amer. Math. Soc. 81 (1975), 456-458. [70] Jalfee, H.: Anti-holomorphic automorphisms of the exceptional symmetric domains. J. Diff. Geom. 13 (1978), 79-86. [71] Jakobsen, H.P.: Hermitean symmetric spaces and their unitary highest weight modules. J. Funct. Anal. 52 (1983) 385-412.

BIBLIOGRAPHY

277

[72] Jakobsen, H.P., M. Vergne: Restriction and expansions of holomorphic representations, J. Funct. Anal. 34 (1979), 29-53. [73] Jorgensen, P.E.T.: Analytic continuation of local representations of Lie groups. Pac. J. Math 125 (1986), 397-408. [74] Jorgensen, P.E.T.: Analytic continuation of local representations of symmetric spaces. J. Funct. Anal. 70 (1987), 304-322. [75] Kaneyuki, S.: On classification of parahermitian symmetric spaces. Tokyo J. Math. 8 (1985), 473-482. [76] Kaneyuki, S.: On orbit structure of compactifications of parahermitian symmetric spaces. Jpn. J. Math. 13 (1987), 333-370. [77] Kaneyuki, S., H. Asano: Graded Lie algebras and generalized Jordan triple systems. Nagoya Math. J. 112 (1988), 81-115. [78] Kaneyuki, S., M. Kozai: Paracomplex structures and affine symmetric spaces. Tokyo J. Math. 8 (1985), 81-98. [79] Knapp, A.W.: Representation Theory of Semisimple Lie Groups. Princeton University Press, Princeton, N. J., 1986. [80] Kobayashi, S., T. Nagano: On filtered Lie algebras and geometric structures I, J. Math. Mech., 13 (1964), 875-908. [81] Kobayashi, S., K. Nomizu: Foundation of Differential Geometry I, II. Wiley (Interscience), New York, 1963 and 1969. [82] Kockmann, J.: Vertauschungsoperatoren zur holomorphen diskreten Reihe spezieller symmetrischer l~ume. Dissertation, Gfttingen, 1994. [83] Korhnyi, A., J.A. Wolf: Realization of Hermitian symmetric spaces as generalized half-planes. Ann. Math. 81 (1965), 265-288. [84] Korhnyi, A., J. A. Wolf: Generalized Cayley transformations of bounded symmetric domains. Amer. J. Math. 87 (1965), 899-939. [85] Kostant, B., S. Sahi: Jordan algebras and Capelli identities. Invent. Math. 122 (1993), 657-665. [86] Koufany, Kh.: Semi-groupes de Lie associ6 h une alg~bre de Jordan euclidienne. Thesis, Nancy, 1993. [87] Koufany, Kh.: R6alisation des espaces sym6triques de type Cayley. C.R. Acad. Sci. Paris (1994), 425-428.

278

BIBLIOGRAPHY

[88] Koufany, Kh., B. Orsted: Function spaces on the Ol'shanskii semigroup and the Gelfand-Gindikin program. Preprint, 1995. [89] Koufany, Kh., B. Orsted: l~space de Hardy sur le semi-groupe metaplectique. Preprint, 1995. [90] Koufany, Kh., B. Orsted: Hardy spaces on the two sheeted covering semigroups. Preprint, 1995. [91] KrStz, B.: Plancherel-Formel fiir Hardy-RKume. Diplomarbeit, Darmstadt, 1995. [92] Kunze, R.A.: Positive definite operator-valued kernels and unitary representations. Proceedings o.f the Conference on Functional Analysis at Irvine, California. Thompson Book Company 1966, 235-247. [93] Lawson, J.D.: Ordered manifolds, invariant cone fields, and semigroups, Forum Math. 1 (1989), 273-308. [94] Lawson, J.D.: Polar and Olshanskii decompositions. J. reine und angew. Math. 448 (1994), 191-219. [95] Libermann, P.: Sur les structures presque paracomplexes, C.R. Acad. Sci. Paris 234 (1952), 2517-2519. [96] Libermann, P.: Sur le probleme d'equivalence de certaines structures infinitesimales. Ann. Math. Put. Appl. 36 (1954), 27-120. [97] Loos, O.: ~ymmetric 8paces, I: General Theory. W.A. Benjamin, Inc., New York, 1969. [98] Makarevic, B.O." Open symmetric orbits of reductive groups in symmetric R-spaces. USSR Sbornik 20 (1973), 406-418. [99] Matsuki, T.: The orbits of affine symmetric spaces under the action of minimal parabolic subgroups. J. Math. Soc. Jpn. 31 (1979), 331-357. [100] Matsuki, T." Orbits on affine symmetric spaces under the action of parabolic subgroups. Hiroshima Math. J. 12 (1982), 307-320. [101] Matsuki, T.: Closure relations for orbits on affine symmetric spaces under the action of minimal parabolic subgroups. Hirsoshima Math. J. t8 (1988), 59-67. [102] Matsuki, T.: A description of discrete series for semisimple symmetric spaces II. Adv. Studies Pure Math. 14 (1988), 531-540.

BIBLIOGRAPHY

279

[103] Matsumoto, S.: Discrete series for an affine symmetric space. Hiroshima Math. J. 11 (1981) 53-79. [104] Mittenhuber, D., K.-H. Neeb: On the exponential function of an ordered manifold with affine connection. Preprint. [105] Mizony, M.: Une transformation de Laplace-Jacobi, SIAM J. Math. 116 (1982), 987-1003. [106] Mizony, M.: Algbbres de noyaux sur des 6spaces symetriques de SL(2,1R) et SL(3,R) et fonctions de Jacobi de premiere er deuxieme esp6ces. Preprint. [107] Molchanov, V.F.: Holomorphic discrete series for hyperboloids of Hermitian type. Preprint, 1995. [108] Muhly, P., J.Renault: C*-algebras of multivariable Wiener Hopf operators. Trans. Amer. Math. 5'oc. 274 (1982), 1-44. [109] Muhly, P., J. Renault, D. Williams: Equivalence and isomorphisms of groupoid C*-algebras. J. Operator Theory 17 (1987), 3-22. [110] Moore, C.C.: Compactification of symmetric spaces II. Amer. J. Math. 86 (1964), 358-378. [111] Nagano, T.: Transformation groups on compact symmetric spaces. Trans. Amer. Math. Soc. 118 (1965), 428-453. [112] Neeb, K.-H.: Globality in semisimple Lie groups, Ann. Inst. Fourier 40 (1991), 493-536. [113] Neeb, K.-H.: Semigroups in the universal covering group of SL(2). Semigroup Forum 43 (1991), 33-43. [114] Neeb, K.-H.: Conal orders on homogeneous spaces. Invent. Math. 134 (1991), 467-496. [115] Neeb, K.-H.: Monotone functions on symmetric spaces. Math. Annalen 291 (1991), 261-273. [116] Neeb, K.-H.: A convexity theorem for semisimple symmetric spaces. Pac. J. Math. 162 (1994), 305-349. [117] Neeb, K.-H.: Holomorphic representation theory II. Acta Math. 173 (1994), 103-133.

280

BIBLIOGRAPHY

[118] Neeb, K.-H.: The classification of Lie algebras with invariant cones. J. Lie Theory 4 (1994), 139-183. [119] Neeb, K.-H.: Realization of general unitary highest weight representations. To appear in Forum Math. [120] Neeb, K.-H.: Holomorphic Representation Theory I. Math. Annalen

(1995), 155-181. [121] Neeb, K.-H.: Representations of involutive semigroups, Semigroup Forum 48 (1996), 197-218. [122] Neeb, K.-H.: Invariant orders on Lie groups and coverings of ordered homogeneous spaces. Submitted. [123] Neeb, K.-H.: Coherent states, holomorphic representations, and highest weight representations. Pac. J. Math., in press. [124] Neeb, K.-H.: A general non-linear convexity theorem. Submitted. [125] Neidhardt, C." Konvexit~it und projective Einbettungen in der Riemannschen Geometrie. Dissertation, Erlangen, 1996 [126] Nica, A.: Some remarks on the groupoid approach to Wiener Hopf operators. J. Operator Theory 18 (1987), 163-198. [127] Nica, A.: Wiener-Hopf operators on the positive semigroup of a Heisenberg group. In: Operator Theory: Advances and Applications 43. Birkh~iuser, Basel, 1990. [128] Olafsson, G." Fourier and Poisson transformation associated to a semisimple symmetric space, Invent. math. 90 (1987), 605-629. [129] Olafsson, G.: Causal Symmetric Spaces. Math. Gotting. 15 (1990), pp 91. [130] Olafsson, G.: Symmetric spaces of Hermitian type. Diff. Geom. and Appl. 1 (1991), 195-233. [131] Olafsson, G." Spherical function on a causal symmetric space. In preparation. [132] Olafsson, G.: Fatou-type lemma for symmetric spaces and applications to the character formula. In preparation. [133] Olafsson, G., B. Orsted: The holomorphic discrete series for affine symmetric spaces, I. J. Funct. Anal. 81 (1988), 126-159.

BIBLIOGRAPHY

281

[1341 Olafsson, G., B. Orsted: Is there an orbit method for affine symmetric spaces? In: M. Duflo, N.V. Pedersen, M. Vergne, Eds.: The Orbit Method in Representation Theory. Birkh/iuser~ Boston 1990. [135] Olafsson, G., B. Orsted: The holomorphic discrete series of an affine symmetric space and representations with reproducing kernels, Trans. Amer. Math. Soc. 326 (1991), 385-405.

[ 361

Olafsson, G., B. Orsted: Causal compactification and Hardy spaces. Preprint, 1996.

[ 371

Ol'shanskii, G.I." Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series. Func. Anal. Appl. 15 (1982), 275-285.

[138] Ol'shanskii, G.I.: Convex cones in symmetric Lie algebra, Lie semigroups and invariant causal (order) structures on pseudo-Riemannian symmetric spaces. Soy. Math. Dokl. 26 (1982), 97-101. [139] Ol'shanskii, G.I.: Complex Lie semigroups, Hardy spaces and the Gelfand-Gindikin program. Diff. Geom. Appl. 1 (1991), 235-246. [1401 Oshima, T." Poisson transformation on affine symmetric spaces. Proc. Jpn. Acad. Set. A 55 (1979), 323-327. [141] Oshima, T.: Fourier analysis on semisimple symmetric spaces. In: H. Carmona, M. Vergne, Eds. Noncommutative Harmonic Analysis and Lie Groups. Proceedings Luminy, 1980. Lect. Notes Math. 880 (1981), 35 7-369. [142] Oshima, T.: Discrete series for semisimple symmetric spaces. Proc. Int. Congr. Math., Warsaw, 1983. [143] Oshima, T., T. Matsuki: Orbits on aftine symmetric spaces under the action of the isotropic subgroups. J. Math. Soc. Jpn. 32, (1980), 399-414. [144] Oshima, T., T. Matsuki: A description of discrete series for semisimple symmetric spaces. Adv. Stud. Pure Math. 4 (1984), 331-390. [145] Oshima, T., J. Sekiguchi: Eigenspaces of invariant differential operators on an affine symmetric space. Invent. Math. 57' (1980), 1-81. [146] Oshima, T., J. Sekiguchi: The restricted root system of a semisimple symmetric pair. Adv. Stud. Pure Math. 4 (1984), 433-497.

282

[ 471

BIBLIOGRAPHY

Paneitz, S.: Invariant convex cones and causality in semisimple Lie algebras and groups. J. Funct. Anal. 43 (1981), 313-359.

[148] Paneitz, S.: Determination of invariant convex cones in simple Lie algebras. Arkiv Mat. 21 (1984), 217-228. [149] Penrose, R.: Techniques oJ Differential Topology in Relativity. Regional Conference Series in Applied Math. SIAM 7, 1972. [150] Renault, J.: A groupoid approach to C*-algebra. Lect. Notes Math. 793, Springer, Berlin, 1980. [151] Renault, J.: Representation des produits crois~s d'alg~bres de groupoides. J. Operator Theory 18(1987), 67-97. [152] Rossi, H., M. Vergne: Analytic continuation of the holomorphic discrete series of a semi-simple Lie group. Acta Math. 136 (1976), 1-59. [153] Rudin, W.: Functional Analysis. McGraw-Hill, New York, 1973. [154] Ruppert, W.A.F.: A geometric approach to the Bohr compactification of cones. Math. Z. 199 (1988), 209-232. [155] Satake, I.: Algebraic Structures of Symmetric Domains. Iwanarni Shouten, Tokyo, and Princeton University Press, Princeton, N. J., 1980. [156] Schlichtkrull, H.: Hyperfunctions and Harmonic Analysis on Symmetric Spaces. Prog. Math. 49. Birkh~iuser, Boston 1984. [157] Segal, I.E.: Mathematical Cosmology and Extragalactic Astronomy. Academic Press, New York, 1976. [158] Spindler K.: Invariante Kegel in Liealgebren. Mitteilungen aus dem Math. Sere. Giessen, 188 (1988), pp 159. [159] Stanton, R.J.: Analytic extension of the holomorphic discrete series, Amer. J. Math. 108 (1986), 1411-1424. [160] Stein, E.M., G. Weiss: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, NJ, 1971. [161] Thorleifsson, H.: Die verallgemeinerte Poisson Transformation fiir reduktive symmetrische R~ume. Habilitationsschrift, GSttingen, 1994. [162] Varadarajan, V.S.: Lie Groups, Lie Algebras and Their Representations. Prentice-Hall, Englewood Cliffs, N.J., 1974.

BIBLIOGRAPHY

283

[163] Varadarajan, V.S." Harmonic Analysis on Real Reductive Groups. Lect. Notes Math. 576, Springer Verlag, Berlin, 1977. [164] Viano, G.A.: On the harmonic analysis of the elastic scattering amplitude of two spinless particles at fixed momentum transfer, Ann. Inst. H. Poincard 32 (1980), 109-123.

[165] Vinberg, E.B.: Homogeneous cones. Soy. Math. Dokl. 1 (1961), 787790. [166] Vinberg, E.B.: Invariant convex cones and ordering in Lie groups, Func. Anal. Appl. 15 (1982), 1-10. [167] Wallach, N.: The analytic continuation of the discrete series, I, II. Trans. Amer. Math. Soc. 251 (1979), 1-17, 19-37. [168] Warner, G.: Harmonic Analysis on Semisimple Lie Groups L Springer Verlag, Berlin, 1972. [169] Wolf, J.A.: The action of a real semisimple Lie group on a complex flag manifold, I: Orbit structure and holomorphic arc components. Bull. Amer. Math. Soc. 75 (1969), 1121-1237. [170] Wolf, J.A.: The fine structure of hermitian symmetric spaces. In: W. Boothby and G. Weiss, Eds., Symmetric Spaces, Marcel Dekker, New York, 1972.

Index Abel Transform 236 Composable pair Absolute value 202 Cone Absolutely continuous 41 convex a-dual 9 closed Algebraic interior 30 dual Associated dual 9 extension of Associated to 2 forward light cone A-subspace 22 generating Automorphism group 33 generating element Boundary value map 215 homogeneous Bounded symmetric domain 253 interior of Cartan decomposition 247 intersection Cartan involution 247 invariant Cauchy-Szeg5 kernel 216,219 minimal Causal maximal compactification map 46 open curve 41 pointed Galois connection 173 polyhedral Iwasawa projection 136 positive kernel 222 projection of maps 40 proper orbit map 176 regular orientation 41,42 self-dual semigroup 43 tangent space 42 Conjugation structure 40,72 Convex hull Cayley-type 76 Degree Cayley transform 255 Domain map partial 255 Dominant C-dual 7 Edge Centralizer 248 Essentially connected Chamber 116 Extension Compactly causal 76 Face 284

242 29, 30 29 30 29 34 30,36,51 30 77 33 30 34 33, 36,37 39,50,98 50,98 3O 3O 31 46 34 30, 37 3O 3O 44 247 36 187 242 147 29 79,79 34 31, 168

Index

(g, K)-module 204 admissible 204 highest-weight module 205 Globally hyperbolic 52 Globally ordered 42 Group of units 44 Groupoid 242 Hardy norm 215,217 Hardy space 215,217 Harish-Chandra embedding 253 Hausdorff distance 257 Holomorphic discrete series 212 Homogeneous vector bundle 144 Involution associated 9 of a semigroup 200 Infinitesimally causal 73,74 Integral 147 Interval 42 Isotropy group 242 Iwasawa decomposition 248 Isomorphic 4 Killing form 247 Laplace transform 223 spherical 233 normalized 234 Lexicographic ordering 254 Lie group Hermitian 252 semisimple 247 Light cone 30,37,51 Minimal parabolic subgroup 249,252 Minkowski space 41 Module highest-weight 205 42 Monotone map 200 Negative elements 76 Noncompactly causal 248 Normalizer 242 Orbit in a groupoid Order 46 compactification

285 preserving map 42 Poisson kernel 216,220 Positive definite 34 Pospace 259 Preceding 41 Primitive element 205 q-compatible 138 q-maximal 138 Quasi-order 41 Reductive pair 2 Relatively regular element 162 Range map 242 Representation a-bounded 202 C-admissible 200 contractive 201 holomorphic 201 irreducible 201 q-regular 23 r 23 L-spherical 249 spherical 249 Root 249 compact 253 noncompact 253 positive restricted 248 restricted 248 semisimple 247 Semigroup affine 159 affine compression 159 compression 134 real maximal Ol'shanskii 134 SL(2) 13,93,122,126,143,152, 16 i, 189,206,210,255 Shilov boundary 56 Span 29 Spherical distribution 237 Spherical function 224,225 normalized 234 Spherical representation 249 L-spherical 249

286 [-spherical 249 Strongly orthogonal 52,95,254 subsymmetric pair 4 Symmetric pair 2 effective 5 isomorphic 4 irreducible 4 reductive 2 Symmetric space 1 associated dual 9 causal 76 of Cayley type 76 c-dual 7 compactly causal 76 noncompactly causal 76 non-Riemannian semisimple 7 Riemannian dual 9 semisimple 6 universal, associated to 5 Tube domain 52,256 Unit group 168 Unit space 242 Universal enveloping C*-algebra 243 Vietoris topology 257 Volterra kernel 222,232 algebra 232 invariant 223,232 Weight 247 dominant 147 integral 147 Weyl group 248 Wiener-Hopf algebra 239 operator 241, 241 symbol 241

INDEX

Perspectives in Mathematics Vol. 1 Vol. 2 Vol. 3

J.S. Milne, Arithmetic Duality Theorems A. Borel et al., Algebraic D-Modules Ehud de Shalit, lwasawa Theory of Elliptic Curves with

Vol. 4

M. Rapoport, N. Schappacher, P. Schneider, editors,

Complex Multiplication

Vol. 5 Vol. 6 Vol. 7 Vol. 8 Vol. 9 Vol. 10 Vol. 11 Vol. 12 Vol. 13 Vol. 14 Vol. 15 Vol. 16 Vol. 17 Vol. 18

Beilinson's Conjectures on Special Values of L-Functions Paul Giinther, Huygens' Principle and Hyperbolic Equations Stephen Gelbart and Freydoon Shahidi, Analytic Properties of Automorphic L-Functions Mark Ronan, Lectures on Buildings K. Shiohama, editor, Geometry of Manifolds E Reese Harvey, Spinors and Calibrations L. Clozel and J.S. Milne, editors, Automorphic Forms, Shimura Varieties, and L-functions, Volume I L. Clozel and J.S. Milne, editors, Automorphic Forms, Shimura Varieties, and L-functions, Volume H Gabor Toth, Harmonic Maps and Minimal Immersions Through Representation Theory James W. Cogdell and Ilya Piatetski-Shapiro, The Arithmetic and Spectral Analysis of Poincarg Series D. Gieseker, H. Knorrer, and E. Trubowitz, The Geometry of Algebraic Fermi Curves Valentino Cristante and William Messing, editors, Barsotti Symposium in Algebraic Geometry Gerrit Heckman and Henrik Schlichtkrull, Harmonic Analysis and Special Functions on Symmetric Spaces Henrik Schlichtkrull and Bent Orsted, editors, Algebraic and Analytic Methods in Representation Theory Joachim Hilgert and Gestur 01afsson, Causal Symmetric Spaces: Geometry and Harmonic Analysis

This Page Intentionally Left Blank