x CONTENTS
Chapter 10. Special Case: Commutative Groups 207
10.1. The Character Group 207
10.2. The Fourier Transform and Fourier Inversion Theorems 212
10.3. Pontrjagin Duality 214
10.4. Almost Periodic Functions 216
10.5. Spectral Theorems 218
10.6. The Lie Group Case 219
Part 4.
STRUCTURE AND ANALYSIS FOR COMMUTATIVE SPACES
Chapter 11. Riemannian Symmetric Spaces 225
11.1. A Fast Tour of Symmetric Space Theory 225
11.1A. Riemannian Basics 225
11.IB. Lie Theoretic Basics 226
11.1C. Complex and Quaternionic Structures 229
11.2. Classifications of Symmetric Spaces 231
11.3. Euclidean Space 236
11.3A. Construction of Spherical Functions 236
11.3B. General Spherical Functions on Euclidean Space 238
11.3C. Positive Definite Spherical Functions on Euclidean
Space 240
11.3D. The Transitive Case 242
11.4. Symmetric Spaces of Compact Type 245
11.4A. Restricted Root Systems 245
11.4B. The Cartan-Helgason Theorem 246
11.4C. Example: Group Manifolds 249
11.4D. Examples: Spheres and Projective Spaces 250
11.5. Symmetric Spaces of Noncompact Type 252
11.5A. Restricted Root Systems 253
11.5B. Harish-Chandra's Parameterization 254
11.5C. Hyperbolic Spaces 255
11.5D. The c-Function and Plancherel Measure 257
11.5E. Example: Groups with Only One Conjugacy Class of
Cartan Subgroups 258
11.6. Appendix: Finsler Symmetric Spaces 260
Chapter 12. Weakly Symmetric and Reductive Commutative Spaces 263
12.1. Commutativity Criteria 263
12.2. Geometry of Weakly Symmetric Spaces 264
12.3. Example: Circle Bundles over Hermitian Symmetric Spaces 268
12.4. Structure of Spherical Spaces 272
12.5. Complex Weakly Symmetric Spaces 275
12.6. Spherical Spaces are Weakly Symmetric 277
12.7. Kramer Classification and the Akhiezer-Vinberg Theorem 282
12.8. Semisimple Commutative Spaces 287
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