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Tables of Contents for Groups and Geometric Analysis
Chapter/Section Title
Page #
Page Count
Preface
xiii
Preface To The 2000 Printing
xvii
Suggestions To The Reader
xix
A Sequel To The Present Volume
xxi
Introduction Geometric Fourier Analysis on Spaces of Constant Curvature
Harmonic Analysis on Homogeneous Spaces
1
3
General Problems
1
1
Notation and Preliminaries
2
2
The Euclidean Plane R2
4
12
Eigenfunctions and Eigenspace Representations
4
11
A Theorem of Paley-Wiener Type
15
1
The Sphere S2
16
13
Spherical Harmonics
16
7
Proof of Theorem 2.10
23
6
The Hyperbolic Plane H2
29
52
Non-Euclidean Fourier Analysis. Problems and Results
29
9
The Spherical Functions and spherical Transforms
38
6
The Non-Euclidean Fourier Transform. Proof of the Main Result
44
14
Eigenfunctions and Eigenspace Representations. Proofs of Theorems 4.3 and 4.4
58
11
Limit Theorems
69
3
Exercises and further Results
72
6
Notes
78
3
Integral Geometry and Radon Transforms
Integration on Manifolds
81
15
Integration of Forms. Riemannian Measure
81
4
Invoriant Measures on Coset Spaces
85
11
Haar Measure in Cononical Coordinates
96
1
The Radon Transform on Rn
96
43
Introduction
96
1
The Radon Transform of the Spaces D(Rn) and T(Rn) The Support Theorem
97
13
The Inversion Formulas
110
5
The Plancherel Formula
115
2
The Radon Transform of Distributions
117
5
Integration over d-Planes. X-Ray Transforms
122
4
Applications
126
1
Partial Differential Equations
126
4
Radiography
130
1
Appendix. distributions and Riesz Potentials
131
8
A Duality in Integral Geometry. Generalized Radon Transforms and Orbital Integrals
139
11
A Duality for Homogeneous Spaces
139
4
The Radon Transform for the Double Fibration
143
6
Orbital Integrals
149
1
The Radon Transform on Two-Point Homogeneous Spaces. The X-Ray Transform
150
30
Spaces of Constant Curvature
151
1
The Hyperbolic Space
152
9
The Spheres and the Elliptic Spaces
161
3
Compact Two-Point Homogeneous Spaces
164
13
Noncompact Two-Point Homogeneous Spaces
177
1
The X-Ray Transform on a Symmetric Space
178
2
Integral Formulas
180
19
Integral formulas Related to the Iwasawa Decomposition
181
5
Integral Formulas for the Cartan Decomposition
186
1
The Noncompact Case
186
1
The Compact Case
187
8
The Lie Algebra Case
195
1
Integral Formulas for the Bruhat Decomposition
196
3
Orbital Integrals
199
34
Pseudo-Riemannian Manifolds of Constant Curvature
199
4
Orbital Integrals for the Lorentzian Case
203
8
Generalized Riesz Potentials
211
3
Determination of a Function from Its Integrals over Lorentzian Spheres
214
4
Orbital Integrals on SL(2,R)
218
3
Exercises and Further Results
221
8
Notes
229
4
Invariant Differential Operators
Differentiable Functions on Rn
233
6
Differential Operators on Manifolds
239
12
Definition. The Spaces D(M) and &(M)
239
1
Topology of the Spaces D(M) and &(M). Distributions
239
2
Effect of Mappings. The Adjoint
241
1
The Laplace-Beltrami Operator
242
9
Geometric Operations on Differential Operators
251
23
Projections of Differential Operators
251
2
Transversal Parts and Separation of Variables for Differential Operators
253
6
Radial Parts of a Differential Operator. General Theory
259
6
Examples of Radial Parts
265
9
Invariant Differential Operators on Lie Groups and Homogeneous Spaces
274
15
Introductory Remarks. Examples. Problems
274
6
The Algebra D(G/H)
280
7
The Case of a Two-Point Homogeneous Space. The Generalized Darboux Equation
287
2
Invariant Differential Operators On Symmetric Spaces
289
56
The Action on Distributions and Commutativity
289
6
The Connection with Weyl Group Invariants
295
14
The Polar Coordinate From of the Laplacian
309
3
The Laplace-Beltrami Operator for a Symmetric Space of Rank One
312
3
The Poisson Equation Generalized
315
3
Asgeirsson's Mean-Value Theorem Generalized
318
5
Restriction of the Central Operators in D(G)
323
3
Invariant Differential Operators for Complex Semisimple Lie Algebras
326
3
Invariant Differential Operators for X=G/K, G Complex
329
1
Exercises and Further Results
330
13
Notes
343
2
Invariants and Harmonic Polynomials
Decomposition of the Symmetric Algebra. Harmonic Polynomials
345
9
Decomposition of the Exterior Algebra. Primitive Forms
354
2
Invariants for the Weyl Group
356
12
Symmetric Invariants
356
4
Harmonic Polynomials
360
3
The Exterior Invariants
363
1
Eigenfunctions of Weyl Group Invariant Operators
364
2
Restriction Properties
366
2
The Orbit Structure of p
368
12
Generalities
368
2
Nilpotent Elements
370
3
Regular Elements
373
5
Semisimple Elements
378
2
Algebro-Geometric Results on the Orbits
380
1
Harmonic Polynomials on p
380
5
Exercises and Further Results
382
2
Notes
384
1
Spherical Functions and Spherical Transforms
Representations
385
14
Generalities
385
5
Compact Groups
390
9
Spherical Functions: Preliminaries
399
8
Definition
399
3
Joint Eigenfunctions
402
1
Examples
403
4
Elementary Properties of Spherical Functions
407
9
Integral Formulas for Spherical Functions. Connections with Representations
416
9
The Compact Type
416
1
The Noncompact Type
417
7
The Euclidean Type
424
1
Harish-Chandra's Spherical Function Expansion
425
9
The General Case
425
7
The Complex Case
432
2
The c-Function
434
14
The Behavior of φλ at ∞
434
2
The Rank-One Case
436
2
Properties of H(n)
438
1
Integrals of Nilpotent Groups
439
2
The Weyl Group Acting on the Root System
441
3
The Rank-One Reduction. The Product Formula of Gindikin-Karpelevic
444
4
The Paley-Wiener Theorem and the Inversion Formula for the Spherical Transform
448
10
Normalization of Measurs
449
1
The Image of D(G) under the Spherical Transform The Paley-Wiener Theorem
450
4
The Inversion Formula
454
4
The Bounded Spherical Functions
458
9
Generalities
458
1
Convex Hulls
459
2
Boundary Components
461
6
The Spherical Transform on p the Euclidean Type
467
5
Convexity Theorems
472
23
Exercises and Further Results
481
10
Notes
491
4
Analysis on Compact Symmetric Spaces
Representations of Compact Lie Groups
495
12
The Weights
496
5
The Characters
501
6
Fourier Expansions on Compact Groups
507
22
Introduction L1(K) versus L2(K)
507
1
The Circle Group
508
2
Spectrally Continuous Operators
510
9
Absolute Convergence
519
3
Lacunary Fourier Series
522
7
Fourier Decomposition of a Representation
529
5
Generalities
529
3
Applications to Compact Homogeneous Spaces
532
2
The Case of a Compact Symmetric Space
534
17
Finite-Dimensional Spherical Representations
534
4
The Eigenfunctions and the Eigenspace Representations
538
4
The Rank-One Case
542
1
Exercises and Further Results
543
5
Notes
548
3
Solutions To Exercises
551
46
Appendix
597
14
1. The Finite-Dimensional Representations of s1(2.C)
597
3
2. Representations and Reductive Lie Algebras
600
7
1. Semisimple Representations
600
2
2. Nilpotent and Semisimple Elements
602
3
3. Reductive Lie Algebras
605
2
3. Some Algebraic Tools
607
4
Some Details
611
8
Bibliography
619
36
Symbols Frequently Used
655
4
Index
659
6
Errata
665