**Mathematical Surveys and Monographs**

Volume: 83;
1984;
667 pp;
Hardcover

MSC: Primary 22; 43; 53; 58; 35; 44; 51;

Print ISBN: 978-0-8218-2673-7

Product Code: SURV/83

List Price: $69.00

Individual Member Price: $55.20

**Electronic ISBN: 978-1-4704-1310-1
Product Code: SURV/83.E**

List Price: $69.00

Individual Member Price: $55.20

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#### Supplemental Materials

# Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions

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*Sigurdur Helgason*

This volume, the second of Helgason's impressive three books on Lie
groups and the geometry and analysis of symmetric spaces, is an
introduction to group-theoretic methods in analysis on spaces with a
group action.

The first chapter deals with the three two-dimensional spaces of constant
curvature, requiring only elementary methods and no Lie theory. It is
remarkably accessible and would be suitable for a first-year graduate course.
The remainder of the book covers more advanced topics, including the work of
Harish-Chandra and others, but especially that of Helgason himself. Indeed,
the exposition can be seen as an account of the author's tremendous
contributions to the subject.

Chapter I deals with modern integral geometry and Radon transforms.
The second chapter examines the interconnection between Lie groups and
differential operators. Chapter IV develops the theory of spherical
functions on semisimple Lie groups with a certain degree of
completeness, including a study of Harish-Chandra's \(c\)-function. The
treatment of analysis on compact symmetric spaces (Chapter V) includes
some finite-dimensional representation theory for compact Lie groups
and Fourier analysis on compact groups. Each chapter ends with
exercises (with solutions given at the end of the book!) and
historical notes.

This book, which is new to the AMS publishing program, is an
excellent example of the author's well-known clear and careful writing
style. It has become the standard text for the study of spherical
functions and invariant differential operators on symmetric
spaces.

Sigurdur Helgason was awarded the Steele Prize for

#### Table of Contents

# Table of Contents

## Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions

Table of Contents pages: 1 2

- CONTENTS vii8 free
- PREFACE xiii14 free
- PREFACE TO THE 2000 PRINTING xvii18 free
- SUGGESTIONS TO THE READER xix20 free
- A SEQUEL TO THE PRESENT VOLUME xxi22 free
- INTRODUCTION: GEOMETRIC FOURIER ANALYSIS ON SPACES OF CONSTANT CURVATURE 124 free
- 1. Harmonic Analysis on Homogeneous Spaces 124
- 2. The Euclidean Plane R[sup(2)] 427
- 3. The Sphere S[sup(2)] 1639
- 4. The Hyperbolic Plane H[sup(2)] 2952
- 1. Non-Euclidean Fourier Analysis. Problems and Results 2952
- 2. The Spherical Functions and Spherical Transforms 3861
- 3. The Non-Euclidean Fourier Transform. Proof of the Main Result 4467
- 4. Eigenfunctions and Eigenspace Representations. Proofs of Theorems 4.3 and 4.4 5881
- 5. Limit Theorems 6992
- Exercises and Further Results 7295
- Notes 78101

- CHAPTER I: INTEGRAL GEOMETRY AND RADON TRANSFORMS 80103
- 1. Integration on Manifolds 81104
- 2. The Radon Transform on R[sup(n)] 96119
- 1. Introduction 96119
- 2. The Radon Transform of the Spaces D(R[sup(n)]) and p(R[sup(n)]). The Support Theorem 97120
- 3. The Inversion Formulas 110133
- 4. The Plancherel Formula 115138
- 5. The Radon Transform of Distributions 117140
- 6. Integration over d-Planes. X-Ray Transforms 122145
- 7. Applications 126149
- 8. Appendix. Distributions and Riesz Potentials 131154

- 3. A Duality in Integral Geometry. Generalized Radon Transforms and Orbital Integrals 139162
- 4. The Radon Transform on Two-Point Homogeneous Spaces. The X-Ray Transform 150173
- 5. Integral Formulas 180203
- 6. Orbital Integrals 199222
- 1. Pseudo-Riemannian Manifolds of Constant Curvature 199222
- 2. Orbital Integrals for the Lorentzian Case 203226
- 3. Generalized Riesz Potentials 211234
- 4. Determination of a Function from Its Integrals over Lorentzian Spheres 214237
- 5. Orbital Integrals on SL(2,R) 218241
- Exercises and Further Results 221244
- Notes 229252

- CHAPTER II: INVARIANT DIFFERENTIAL OPERATORS 233256
- 1. Differentiable Functions on R[sup(n)] 233256
- 2. Differential Operators on Manifolds 239262
- 3. Geometric Operations on Differential Operators 251274
- 4. Invariant Differential Operators on Lie Groups and Homogeneous Spaces 274297
- 5. Invariant Differential Operators on Symmetric Spaces 289312
- 1. The Action on Distributions and Commutativity 289312
- 2. The Connection with Weyl Group Invariants 295318
- 3. The Polar Coordinate Form of the Laplacian 309332
- 4. The Laplace–Beltrami Operator for a Symmetric Space of Rank One 312335
- 5. The Poisson Equation Generalized 315338
- 6. Asgeirssorfs Mean-Value Theorem Generalized 318341
- 7. Restriction of the Central Operators in D(G) 323346
- 8. Invariant Differential Operators for Complex Semisimple Lie Algebras 326349
- 9. Invariant Differential Operators for X = G/K, G Complex 329352
- Exercises and Further Results 330353
- Notes 343366

- CHAPTER III: INVARIANTS AND HARMONIC POLYNOMIALS 345368
- CHAPTER IV: SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS 385408
- 1. Representations 385408
- 2. Spherical Functions: Preliminaries 399422
- 3. Elementary Properties of Spherical Functions 407430
- 4. Integral Formulas for Spherical Functions. Connections with Representations 416439
- 5. Harish-Chandra's Spherical Function Expansion 425448
- 6. The c-Function 434457
- 7. The Paley–Wiener Theorem and the Inversion Formula for the Spherical Transform 448471
- 8. The Bounded Spherical Functions 458481
- 9. The Spherical Transform on p, the Euclidean Type 467490
- 10. Convexity Theorems 472495
- Exercises and Further Results 481504
- Notes 491514

- CHAPTER V: ANALYSIS ON COMPACT SYMMETRIC SPACES 495518
- SOLUTIONS TO EXERCISES 551574
- APPENDIX 597620
- SOME DETAILS 611634
- BIBLIOGRAPHY 619642
- SYMBOLS FREQUENTLY USED 655678
- INDEX 659682

Table of Contents pages: 1 2

#### Readership

Graduate students and research mathematicians interested in analysis on homogeneous spaces, differential geometry, and topological groups, Lie groups.

#### Reviews

The book is excellent both as a text and as a reference work; it will clearly become another instant classic.

-- American Scientist

This volume makes an excellent companion to the author's

This book, like the author's previous work on differential geometry, will no doubt inspire considerable further research and become the standard text on the subjects it covers.

-- Mathematical Reviews

Few treatises today can lay claim to being “aere
perennius”, but all of Helgason's books certainly do with a
vengeance … [He] sets a model of style and clarity that has not
been matched since Enriques's

-- The Bulletin of Mathematics Books

A most valuable contribution to Lie theory and to the interplay between geometry and analysis. It is remarkable that the beautiful theory in Chapter IV can be presented in a textbook form with complete proofs.

-- Bulletin of the London Mathematical Society

The diversity of subjects treated is great. Nevertheless the author has managed to achieve coherence of presentation by clearly putting forward a few main themes and basic problems. The first third of the book is suitable as a text for beginning graduate students; the book is also an excellent source of reference for experts. No doubt it will become a new standard in the field.

-- CWI Quarterly