**Mathematical Surveys and Monographs**

Volume: 22;
1986;
328 pp;
Softcover

MSC: Primary 22; 35;
Secondary 43

Print ISBN: 978-0-8218-1523-6

Product Code: SURV/22

List Price: $45.00

Individual Member Price: $36.00

**Electronic ISBN: 978-1-4704-1249-4
Product Code: SURV/22.E**

List Price: $45.00

Individual Member Price: $36.00

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# Noncommutative Harmonic Analysis

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*Michael E. Taylor*

This book explores some basic roles of Lie groups in linear analysis, with particular emphasis on the generalizations of the Fourier transform and the study of partial differential equations. It began as lecture notes for a one-semester graduate course given by the author in noncommutative harmonic analysis. It is a valuable resource for both graduate students and faculty, and requires only a background with Fourier analysis and basic functional analysis, plus the first few chapters of a standard text on Lie groups.

The basic method of noncommutative harmonic analysis, a generalization of Fourier analysis, is to synthesize operators on a space on which a Lie group has a unitary representation from operators on irreducible representation spaces. Though the general study is far from complete, this book covers a great deal of the progress that has been made on important classes of Lie groups.

Unlike many other books on harmonic analysis, this book focuses on the relationship between harmonic analysis and partial differential equations. The author considers many classical PDEs, particularly boundary value problems for domains with simple shapes, that exhibit noncommutative groups of symmetries. Also, the book contains detailed work, which has not previously been published, on the harmonic analysis of the Heisenberg group and harmonic analysis on cones.

#### Table of Contents

# Table of Contents

## Noncommutative Harmonic Analysis

- Contents v6 free
- Introduction ix10 free
- 0. Some Basic Concepts of Lie Group Representation Theory 118 free
- 1. The Heisenberg Group 4259
- 1. Construction of the Heisenberg group H[sup(n)] 4259
- 2. Representations of H[sup(n)] 4663
- 3. Convolution operators on Hn and the Weyl calculus 5067
- 4. Automorphisms of H[sup(n)]; the symplectic group 5471
- 5. The Bargmann-Fok representation 5875
- 6. (Sub)Laplacians on H[sup(n)] and harmonic oscillators 6178
- 7. Functional calculus for Heisenberg Laplacians and for harmonic oscillator Hamiltonians 6784
- 8. The wave equation on the Heisenberg group 8198

- 2. The Unitary Group 87104
- 3. Compact Lie Groups 104121
- 4. Harmonic Analysis on Spheres 128145
- 5. Induced Representations, Systems of Imprimitivity, and Semidirect Products 143160
- 6. Nilpotent Lie Groups 152169
- 7. Harmonic Analysis on Cones 163180
- 8. SL(2,R) 177194
- 1. Introduction to SL(2,R) 177194
- 2. Classification of irreducible unitary representations 181198
- 3. The principal series 188205
- 4. The discrete series 193210
- 5. The complementary series 195212
- 6. The spectrum of L[sup(2)](T\PSL(2,R)), in the compact case 196213
- 7. Harmonic analysis on the Poincaré upper half plane 199216
- 8. The subelliptic operator L[sub(α)] = A[sup(2)] + B[sup(2)] + 1/2iαZ on SL(2, R) 202219

- 9. SL(2,C) and More General Lorentz Groups 204221
- 10. Groups of Conformal Transformations 226243
- 11. The Symplectic Group and the Metaplectic Group 235252
- 12. Spinors 246263
- 13. Semisimple Lie Groups 268285
- Appendixes 287304
- References 313330
- Index 327344 free

#### Reviews

Could be used as a text in a course … many people will find this book valuable as a reference and as an introduction to the literature.

-- Mathematical Reviews