Groups and geometric analysis : integral geometry, invariant differential operators, and spherical functions

✍ Scribed by Sigurdur Helgason

• Publisher: Academic Press
• Year: 1984
• Tongue: English
• Leaves: 677
• Category: Library

✦ Synopsis

This volume is intended as an introduction to group-theoretic methods in analysis on spaces that possess certain amounts of mobility and symmetry. The focus is on the three topics in the subtitle. The introductory chapter deals with the three two-dimensional cases of constant curvature requiring only elementary methods and no Lie theory. Chapter I deals with modern integral geometry and Radon transforms. The second chapter deals with the interconnection between Lie groups and differential operators whereas Chapter V develops the theory of spherical functions on semisimple Lie groups with a certain degree of completeness. Each chapter concludes with "Exercises and Further Results" and solutions/references are provided at the end of the book.

✦ Table of Contents

Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions
Copyright Page
Contents
Preface
Suggestions to the Reader
Tentative Contents of the Sequel
Introduction: Geometric Fourier Analysis on Spaces of Constant Curvature
1. Harmonic Analysis on Homogeneous Spaces
2. The Euclidean Plane R2
3. The Sphere S2
4. The Hyperbolic Plane H2
Chapter I. Integral Geometry and Radon Transforms
1. Integration on Manifolds
2. The Radon Transform on Rn
3. A Duality in Integral Geometry. Generalized Radon Transforms and Orbital Integrals
4. The Radon Transform on Two-Point Homogeneous Spaces. The X-Ray Transform
5. Integral Formulas
6. Orbital Integrals
Exercises and Further Results
Notes
Chapter II. Invariant Differential Operators
1. Differentiable Functions on Rn
2. Differential Operators on Manifolds
3. Geometric Operations on Differential Operators
4. Invariant Differential Operators on Lie Groups and Homogeneous Spaces
5. Invariant Differential Operators on Symmetric Spaces
Exercises and Further Results
Notes
Chapter III. Invariants and Harmonic Polynomials
1. Decomposition of the Symmetric Algebra. Harmonic Polynomials
2. Decomposition of the Exterior Algebra. Primitive Forms
3. Invariants for the Weyl Group
4. The Orbit Structure of p
5. Harmonic Polynomials on p
Exercises and Further Results
Notes
Chapter IV. Spherical Functions and Spherical Transforms
1. Representations
2. Spherical Functions: Preliminaries
3. Elementary Properties of Spherical Functions
4. Integral Formulas for Spherical Functions. Connections with Representations
5. Harish-Chandra's Spherical Function Expansion
6. The c-Function
7. The Paley–Wiener Theorem and the Inversion Formula for the Spherical Transform
8. The Bounded Spherical Functions
9. The Spherical Transform on p, the Euclidean Type
10. Convexity Theorems
Exercises and Further Results
Notes
Chapter V. Analysis on Compact Symmetric Spaces
1. Representations of Compact Lie Groups
2. Fourier Expansions on Compact Groups
3. Fourier Decomposition of a Representation
4. The Case of a Compact Symmetric Space
Exercises and Further Results
Notes
Solutions to Exercises
Appendix
1. The Finite-Dimensional Representations of sl(2, C)
2. Representations and Reductive Lie Algebras
3. Some Algebraic Tools
Bibliography
Symbols Frequently Used
Index