Causal Symmetric Spaces (eBook)

Cover 1
Contents 5
Preface 8
Introduction 10
Chapter 1. Symmetric Spaces 15
1.1 Basic Structure Theory 15
1.2 Dual Symmetric Spaces 21
1.3 The Module Structure of To(G/H) 26
1.4 A-Subspaces 36
1.5 The Hyperboloids 38
Chapter 2. Causal Orientations 43
2.1 Convex Cones and Their Automorphisms 43
2.2 Causal Orientations 53
2.3 Semigroups 57
2.4 The Order Compactification 59
2.5 Examples 64
2.6 Symmetric Spaces Related to Tube Domains 66
Chapter 3. Irreducible Causal Symmetric Spaces 85
3.1 Existence of Causal Structures 85
3.2 The Classification of Causal Symmetric Pairs 97
Chapter 4. Classification of Invariant Cones 105
4.1 Symmetric SL(2, R) Reduction 105
4.2 The Minimal and Maximal Cones 112
4.3 The Linear Convexity Theorem 119
4.4 The Classification 124
4.5 Extension of Cones 129
Chapter 5. The Geometry 134
5.1 The Bounded Realization of H/H H K 135
5.2 The Semigroup S(C) 140
5.3 The Causal Intervals 144
5.4 Compression Semigroups 146
5.5 The Nonlinear Convexity Theorem 157
5.6 The B# - Order 166
5.7 The Affine Closure of B# 171
Chapter 6. The Order Compactification 186
6.1 Causal Galois Connections 186
6.2 An Alternative Realization of M cpt+ 192
6.3 The Stabilizers for Mcpt+ 194
6.4 The Orbit Structure of Mcpt+ 197
6.5 The Space SL(3,R)/SO(2,1) 204
Chapter 7. Holomorphic Representations 212
7.1 Holomorphic Representations of Semigroups 213
7.2 Highest-Weight Modules 217
7.3 The Holomorphic Discrete Series 223
7.4 Classical Hardy Spaces 228
7.5 Hardy Spaces 230
7.6 The Cauchy–Szegö Kernel 233
Chapter 8. Spherical Functions 236
8.1 The Classical Laplace Transform 236
8.2 Spherical Functions 238
8.3 The Asymptotics 242
8.4 Expansion Formula 244
8.5 The Spherical Laplace Transform 246
8.6 The Abel Transform 249
8.7 Relation to Representation Theory 251
Chapter 9. The Wiener-Hopf Algebra 253
Appendix A. Reductive Lie Groups 260
A.1 Notation 260
A.2 Finite-Dimensional Representations 263
A.3 Hermitian Groups 266
Appendix B. The Vietoris Topology 271
Notation 276
Bibliography 286
Index 298
Perspectives in Mathematics 301