**Mathematical Surveys and Monographs**

Volume: 39;
2008;
637 pp;
Hardcover

MSC: Primary 43; 53; 22; 44; 32;
Secondary 31; 35; 14; 17

Print ISBN: 978-0-8218-4530-1

Product Code: SURV/39.R

List Price: $97.00

Individual Member Price: $77.60

**Electronic ISBN: 978-1-4704-1266-1
Product Code: SURV/39.R.E**

List Price: $97.00

Individual Member Price: $77.60

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#### Supplemental Materials

# Geometric Analysis on Symmetric Spaces: Second Edition

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*Sigurdur Helgason*

This book gives the first systematic exposition of geometric analysis on Riemannian symmetric spaces and its relationship to the representation theory of Lie groups. The book starts with modern integral geometry for double fibrations and treats several examples in detail. After discussing the theory of Radon transforms and Fourier transforms on symmetric spaces, inversion formulas, and range theorems, Helgason examines applications to invariant differential equations on symmetric spaces, existence theorems, and explicit solution formulas, particularly potential theory and wave equations. The canonical multitemporal wave equation on a symmetric space is included. The book concludes with a chapter on eigenspace representations—that is, representations on solution spaces of invariant differential equations. Known for his high-quality expositions, Helgason received the 1988 Steele Prize for his earlier books Differential Geometry, Lie Groups and Symmetric Spaces and Groups and Geometric Analysis. Containing exercises (with solutions) and references to further results, this revised edition would be suitable for advanced graduate courses in modern integral geometry, analysis on Lie groups, and representation theory of Lie groups.

#### Table of Contents

# Table of Contents

## Geometric Analysis on Symmetric Spaces: Second Edition

Table of Contents pages: 1 2

- Contents vii8 free
- Preface to Second Edition xiii14 free
- Preface xv16 free
- Chapter I: A Duality in Integral Geometry 120 free
- Chapter II: A Duality for Symmetric Spaces 5978
- §1. The Space of Horocycles 6079
- §2. Invariant Differential Operators 7089
- §3. The Radon Transform and its Dual 82101
- §4. Finite-dimensional Spherical and Conical Representations 105124
- 1. Conical Distributions. Elementary Properties 105124
- 2. Conical Functions and Finite-Dimensional Representations 113132
- 3. The Finite-dimensional Spherical Representations 119138
- 4. Conical Models and Spherical Models 120139
- 5. Simultaneous Euclidean Imbeddings of X and of [omitted]. Horocycles as Plane Sections 122141
- 6. Restricted Weights 127146
- 7. The Component H(n) 131150

- §5. Conical Distributions 134153
- §6. Some Rank-One Results 157176
- Exercises and Further Results 181200
- Notes 193212

- Chapter III: The Fourier Transform on a Symmetric Space 197216
- §1. The Inversion and the Plancherel Formula 198217
- §2. Generalized Spherical Functions (Eisenstein Integrals) 227246
- §3. The Q[sup(δ)]-matrices 243262
- §4. The Simplicity Criterion 255274
- §5. The Paley-Wiener Theorem for the Fourier Transform on X = G/K 260279
- 1. Estimates of the Γ-coemcients 261280
- 2. Some Identities for C[sub(s)] 264283
- 3. The Fourier Transform and the Radon Transform. K-types 266285
- 4. Completion of the Proof of the Paley-Wiener Theorem. The Range ε'(X)~ 268287
- 5. A Topological Paley-Wiener Theorem for the K-types 273292
- 6. The Inversion Formula, the Plancherel Formula and the Range Theorem for the δ-spherical Transform 279298

- §6. Eigenfunctions and Eigenspace Representations 282301
- §7. Tangent Space Analysis 285304
- §8. Eigenfunctions and Eigenspace Representations on X[sub(o)] 300319
- §9. The Compact Case 310329
- §10. Elements of D(G/K) as Fractions 322341
- §11. The Rank-One Case 327346
- §12. The Spherical Transform Revisited 335354
- Exercises and Further Results 352371
- Notes 358377

- Chapter IV: The Radon Transform on X and on G[sub(o)]. Range Questions 363382
- §1. The Support Theorem 363382
- §2. The Ranges D(X)∧, ε'(X)∧ and ε([omitted])∨ 365384
- §3. The Range and Kernel Determined in terms of K-types 369388
- §4. The Radon Transform and its Dual for K-invariants 381400
- §5. The Radon Transform on X[sub(o)] 387406
- Exercises and Further Results 397416
- Notes 398417

- Chapter V: Differential Equations on Symmetric Spaces 401420
- §1. Solvability 401420
- §2. Mean Value Theorems 413432
- §3. Harmonic Functions on Symmetric Spaces 421440
- §4. Harmonic Functions on Bounded Symmetric Domains 442461
- 1. The Bounded Realization of a Hermitian Symmetric Space 442461
- 2. The Geodesies in a Bounded Symmetric Domain 444463
- 3. The Restricted Root Systems for Bounded Symmetric Domains 445464
- 4. The Action of G[sub(o)] on D and the Polydisk in D 451470
- 5. The Shilov Boundary of a Bounded Symmetric Domain 453472
- 6. The Dirichlet Problem for the Shilov Boundary 460479
- 7. The Hua Equations 461480
- 8. Integral Geometry Interpretation 466485

- §5. The Wave Equation on Symmetric Spaces 468487
- 1. Introduction. Huygens' Principle 468487
- 2. Huygens' Principle for Compact Groups and Symmetric Spaces X = G/K (G complex) 471490
- 3. Huygens' Principle and Cartan Subgroups 477496
- 4. Orbital Integrals and Huygens' Principle 482501
- 5. Energy Equipartition 486505
- 6. The Flat Case Revisited 490509
- 7. The Multitemporal Wave Equation on X = G/K 493512
- 8. The Multitemporal Cauchy Problem 497516
- 9. Incoming Waves and Supports 506525
- 10. Energy and Spectral Representation 511530
- 11. The Analog of the Priedlander Limit Theorem 524543

- §6. Eigenfunctions and Hyper functions 527546
- Exercises and Further Results 532551
- Notes 537556

- Chapter VI: Eigenspace Representations 539558
- §1. Generalities 539558
- §2. Irreducibility Criteria for a Symmetric Space 543562
- §3. Eigenspace Representations for the Horocycle Space G/MN 547566
- §4. Eigenspace Representations for the Complex Space G/N 562581
- §5. Two Models of the Spherical Representations 567586
- Exercises and Further Results 569588
- Notes 571590

- Solutions to Exercises 573592
- Bibliography 599618
- Symbols Frequently Used 627646
- Index 633652

Table of Contents pages: 1 2

#### Readership

Graduate students and research mathematicians interested in analysis on symmetric spaces and the representation theory of Lie groups.

#### Reviews

[This book] is a model of fine scholarship and must rank as a definitive source for the indicated material.

-- MAA Reviews

The exposition, which emphasizes the geometric and analytic side of the subject, is self-contained and very clear. It is good that this standard in the field, with its wealth of material, has become available again through a second edition.

-- EMS Newsletter

The book is written in the elegant form that was typical for the style of S. Helgason. ... The bibliography updates and completes the previous references. The book will be of interest for both students and specialists in harmonic analysis on homogeneous spaces, integral geometry and invariant differential equations on symmetric spaces.

-- Zentralblatt MATH

This monograph constitutes an important reference book for anyone working on the analysis of symmetric spaces.

-- Mathematical Reviews