Groups & Geometric Analysis (eBook)

Front Cover 1
Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions 4
Copyright Page 5
Contents 8
Preface 14
Suggestions to the Reader 18
Tentative Contents of the Sequel 20
Introduction: Geometric Fourier Analysis on Spaces of Constant Curvature 22
1. Harmonic Analysis on Homogeneous Spaces 22
2. The Euclidean Plane R2 25
3. The Sphere S2 37
4. The Hyperbolic Plane H2 50
Chapter I. Integral Geometry and Radon Transforms 101
1. Integration on Manifolds 102
2. The Radon Transform on Rn 117
3. A Duality in Integral Geometry. Generalized Radon Transforms and Orbital Integrals 160
4. The Radon Transform on Two-Point Homogeneous Spaces. The X-Ray Transform 171
5. Integral Formulas 201
6. Orbital Integrals 220
Exercises and Further Results 242
Notes 250
Chapter II. Invariant Differential Operators 254
1. Differentiable Functions on Rn 254
2. Differential Operators on Manifolds 260
3. Geometric Operations on Differential Operators 272
4. Invariant Differential Operators on Lie Groups and Homogeneous Spaces 295
5. Invariant Differential Operators on Symmetric Spaces 310
Exercises and Further Results 351
Notes 364
Chapter III. Invariants and Harmonic Polynomials 366
1. Decomposition of the Symmetric Algebra. Harmonic Polynomials 366
2. Decomposition of the Exterior Algebra. Primitive Forms 375
3. Invariants for the Weyl Group 377
4. The Orbit Structure of p 389
5. Harmonic Polynomials on p 401
Exercises and Further Results 403
Notes 405
Chapter IV. Spherical Functions and Spherical Transforms 406
1. Representations 406
2. Spherical Functions: Preliminaries 420
3. Elementary Properties of Spherical Functions 428
4. Integral Formulas for Spherical Functions. Connections with Representations 437
5. Harish-Chandra's Spherical Function Expansion 446
6. The c-Function 455
7. The Paley–Wiener Theorem and the Inversion Formula for the Spherical Transform 469
8. The Bounded Spherical Functions 479
9. The Spherical Transform on p, the Euclidean Type 488
10. Convexity Theorems 493
Exercises and Further Results 502
Notes 512
Chapter V. Analysis on Compact Symmetric Spaces 516
1. Representations of Compact Lie Groups 516
2. Fourier Expansions on Compact Groups 528
3. Fourier Decomposition of a Representation 550
4. The Case of a Compact Symmetric Space 555
Exercises and Further Results 564
Notes 569
Solutions to Exercises 572
Appendix 618
1. The Finite-Dimensional Representations of sl(2, C) 618
2. Representations and Reductive Lie Algebras 621
3. Some Algebraic Tools 628
Bibliography 632
Symbols Frequently Used 667
Index 671