**Mathematical Surveys and Monographs**

Volume: 142;
2007;
387 pp;
Hardcover

MSC: Primary 20; 22; 53;

**Print ISBN: 978-0-8218-4289-8
Product Code: SURV/142**

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

**Electronic ISBN: 978-1-4704-1369-9
Product Code: SURV/142.E**

List Price: $108.00

AMS Member Price: $86.40

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#### Supplemental Materials

# Harmonic Analysis on Commutative Spaces

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*Joseph A. Wolf*

This book starts with the basic theory of topological groups, harmonic
analysis, and unitary representations. It then concentrates on geometric
structure, harmonic analysis, and unitary representation theory in
commutative spaces. Those spaces form a simultaneous generalization of
compact groups, locally compact abelian groups, and riemannian symmetric
spaces. Their geometry and function theory is an increasingly active topic
in mathematical research, and this book brings the reader up to the
frontiers of that research area with the recent classifications of weakly
symmetric spaces and of Gelfand pairs.

Part 1, “General Theory of Topological Groups”, is an
introduction with many examples, including all of the standard
semisimple linear Lie groups and the Heisenberg groups. It presents
the construction of Haar measure, the invariant integral, the
convolution product, and the Lebesgue spaces.

Part 2, “Representation Theory and Compact Groups”,
provides background at a slightly higher level. Besides the basics, it
contains the Mackey Little-Group method and its application to
Heisenberg groups, the Peter–Weyl Theorem, Cartan's highest
weight theory, the Borel–Weil Theorem, and invariant function
algebras.

Part 3, “Introduction to Commutative Spaces”, describes
that area up to its recent resurgence. Spherical functions and
associated unitary representations are developed and applied to
harmonic analysis on \(G/K\) and to uncertainty principles.

Part 4, “Structure and Analysis for Commutative Spaces”,
summarizes riemannian symmetric space theory as a rôle model,
and with that orientation delves into recent research on commutative
spaces. The results are explicit for spaces \(G/K\) of nilpotent or
reductive type, and the recent structure and classification theory
depends on those cases.

Parts 1 and 2 are accessible to first-year graduate students. Part
3 takes a bit of analytic sophistication but generally is accessible
to graduate students. Part 4 is intended for mathematicians beginning
their research careers as well as mathematicians interested in seeing
just how far one can go with this unified view of algebra, geometry,
and analysis.

#### Readership

Graduate students and research mathematicians interested in lie groups and their representations.

#### Reviews & Endorsements

Wolf's book is an up-to-date presentation of the harmonic analysis and classification theory of commutative spaces. He needs only 360 pages and amazingly few prerequisites to give complete proofs of all the results alluded to in this review.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Harmonic Analysis on Commutative Spaces

Table of Contents pages: 1 2

- CONTENTS vii8 free
- INTRODUCTION xiii14 free
- PART 1. GENERAL THEORY OF TOPOLOGICAL GROUPS 118 free
- Chapter 1. Basic Topological Group Theory 320
- 1.1. Definition and Separation Properties 320
- 1.2. Subgroups, Quotient Groups, and Quotient Spaces 421
- 1.3. Connectedness 522
- 1.4. Covering Groups 724
- 1.5. Transformation Groups and Homogeneous Spaces 825
- 1.6. The Locally Compact Case 926
- 1.7. Product Groups 1229
- 1.8. Invariant Metrics on Topological Groups 1532

- Chapter 2. Some Examples 1936
- 2.1. General and Special Linear Groups 1936
- 2.2. Linear Lie Groups 2037
- 2.3. Groups Defined by Bilinear Forms 2138
- 2.4. Groups Defined by Hermitian Forms 2239
- 2.5. Degenerate Forms 2542
- 2.6. Automorphism Groups of Algebras 2643
- 2.7. Spheres, Projective Spaces and Grassmannians 2845
- 2.8. Complexification of Real Groups 3047
- 2.9. p–adic Groups 3249
- 2.10. Heisenberg Groups 3350

- Chapter 3. Integration and Convolution 3552

- PART 2. REPRESENTATION THEORY AND COMPACT GROUPS 5370
- Chapter 4. Basic Representation Theory 5572
- 4.1. Definitions and Examples 5673
- 4.2. Subrepresentations and Quotient Representations 5976
- 4.3. Operations on Representations 6481
- 4.4. Multiplicities and the Commuting Algebra 7087
- 4.5. Completely Continuous Representations 7289
- 4.6. Continuous Direct Sums of Representations 7592
- 4.7. Induced Representations 7794
- 4.8. Vector Bundle Interpretation 8198
- 4.9. Mackey's Little–Group Theorem 8299
- 4.10. Mackey Theory and the Heisenberg Group 87104

- Chapter 5. Representations of Compact Groups 93110
- 5.1. Finite Dimensionality 93110
- 5.2. Orthogonality Relations 96113
- 5.3. Characters and Projections 97114
- 5.4. The Peter–Weyl Theorem 99116
- 5.5. The Plancherel Formula 101118
- 5.6. Decomposition into Irreducibles 104121
- 5.7. Some Basic Examples 107124
- 5.8. Real, Complex and Quaternion Representations 113130
- 5.9. The Frobenius Reciprocity Theorem 115132

- Chapter 6. Compact Lie Groups and Homogeneous Spaces 119136
- 6.1. Some Generalities on Lie Groups 119136
- 6.2. Reductive Lie Groups and Lie Algebras 122139
- 6.3. Cartan's Highest Weight Theory 127144
- 6.4. The Peter–Weyl Theorem and the Plancherel Formula 131148
- 6.5. Complex Flag Manifolds and Holomorphic Vector Bundles 133150
- 6.6. Invariant Function Algebras 136153

- Chapter 7. Discrete Co–Compact Subgroups 141158

- PART 3. INTRODUCTION TO COMMUTATIVE SPACES 151168
- Chapter 8. Basic Theory of Commutative Spaces 153170
- 8.1. Preliminaries 153170
- 8.2. Spherical Measures and Spherical Functions 156173
- 8.3. Alternate Formulation in the Differentiable Setting 160177
- 8.4. Positive Definite Functions 165182
- 8.5. Induced Spherical Functions 168185
- 8.6. Example: Spherical Principal Series Representations 170187
- 8.7. Example: Double Transitivity and Homogeneous Trees 174191

- Chapter 9. Spherical Transforms and Plancherel Formulae 179196
- 9.1. Commutative Banach Algebras 179196
- 9.2. The Spherical Transform 184201
- 9.3. Bochner's Theorem 187204
- 9.4. The Inverse Spherical Transform 191208
- 9.5. The Plancherel Formula for K\G/K 192209
- 9.6. The Plancherel Formula for G/K 194211
- 9.7. The Multiplicity Free Criterion 197214
- 9.8. Characterizations of Commutative Spaces 198215
- 9.9. The Uncertainty Principle 199216
- 9.10. The Compact Case 204221

- Chapter 10. Special Case: Commutative Groups 207224

- PART 4. STRUCTURE AND ANALYSIS FOR COMMUTATIVE SPACES 221238
- Chapter 11. Riemannian Symmetric Spaces 225242
- Chapter 12. Weakly Symmetric and Reductive Commutative Spaces 263280
- 12.1. Commutativity Criteria 263280
- 12.2. Geometry of Weakly Symmetric Spaces 264281
- 12.3. Example: Circle Bundles over Hermit ian Symmetric Spaces 268285
- 12.4. Structure of Spherical Spaces 272289
- 12.5. Complex Weakly Symmetric Spaces 275292
- 12.6. Spherical Spaces are Weakly Symmetric 277294
- 12.7. Kramer Classification and the Akhiezer–Vinberg Theorem 282299
- 12.8. Semisimple Commutative Spaces 287304
- 12.9. Examples of Passage from the Semisimple Case 290307
- 12.10. Reductive Commutative Spaces 293310

- Chapter 13. Structure of Commutative Nilmanifolds 299316
- Chapter 14. Analysis on Commutative Nilmanifolds 329346
- Chapter 15. Classification of Commutative Spaces 345362

- BIBLIOGRAPHY 367384
- SUBJECT INDEX 373390

Table of Contents pages: 1 2