This book is Number 22 in the AMS Mathematical Surveys and Monographs. It surveys a number of topics in Noncommutative Harmonic Analysis, emphasizing contacts with Partial Differential Equations.

## Contents

- 0. Some Basic Concepts of Lie Group Representation Theory
- One parameter groups of operators
- Representations of Lie groups, convolution algebras, and Lie algebras
- Representations of distributions and universal enveloping algebras
- Irreducible representations of Lie groups
- Varieties of Lie groups

- 1. The Heisenberg Group
- Construction of the Heisenberg group H
^{n} - Representations of H
^{n} - Convolution operators on H
^{n}and the Weyl calculus - Automorphisms of H
^{n}; the symplectic group - The Bargmann-Fok representation
- (Sub)Laplacians on H
^{n}and harmonic oscillators - Functional calculus for Heisenberg Laplacians and for harmonic oscillator Hamiltonians
- The wave equation on the Heisenberg group

- Construction of the Heisenberg group H
- 2. The Unitary Group
- Representation theory for SU(2), SO(3), and some variants
- Representation theory for U(n)
- The subelliptic operator X
_{2}^{2}+ X_{3}^{2}on SU(2)

- 3. Compact Lie Groups
- Weyl orthogonality relations and the Peter-Weyl theorem
- Roots, weights, and the Borel-Weil theorem
- Representations of compact groups on eigenspaces of Laplace operators

- 4. Harmonic Analysis on Spheres
- The Laplace operator in polar coordinates
- Classical PDE on spheres
- Spherical harmonics
- The subelliptic operator L
_{1}^{2}+L_{2}^{2}+iaL_{3}on S^{2}

- 5. Induced Representations, Systems of Imprimitivity, and Semidirect Products
- Induced representations and systems of imprimitivity
- The Stone-von Neumann Theorem
- Semidirect products
- The Euclidean group and the Poincare group

- 6. Nilpotent Lie Groups
- Nilpotent Lie algebras and Lie algebras with dilations
- Step 2 nilpotent Lie groups
- Representations of general nilpotent Lie groups

- 7. Harmonic Analysis on Cones
- Dilations of cones and the ax+b group
- Spectral representation and functional calculus for the Laplacian on a cone

- 8. SL(2,R)
- Introduction to SL(2,R)
- Classification of irreducible unitary representations
- The principal series
- The discrete series
- The complementary series
- The spectrum of L
^{2}(Gamma\PSL(2,R)), in the compact case - Harmonic analysis on the Poincare upper half plane
- The subelliptic operator A
^{2}+B^{2}+(1/2)iaZ on SL(2,R)

- 9. SL(2,C) and More General Lorentz Groups
- Introduction to SL(2,C)
- Representations of SL(2,C)
- The Lorentz groups SO(n,1)

- 10. Groups of Conformal Transformations
- Laplace operators and conformal changes of metric
- Conformal transformations on R
^{n}, S^{n}, and balls

- 11. The Symplectic Group and the Metaplectic Group
- Symplectic vector spaces and the symplectic group
- Symplectic inner product spaces and compact subgroups of the symplectic group
- The metaplactic representation

- 12. Spinors
- Clifford algebras and spinors
- Spinor bundles and the Dirac operator
- Spinors on four-dimensional Riemannian manifolds
- Spinors on four-dimensional Lorentz manifolds

- 13. Semisimple Lie Groups
- Introduction to semisimple Lie groups
- Some representations of semisimple Lie groups

- Appendices

A. The Fourier transform and tempered distributions

B. The spectral theorem

C. The Radon transform on Euclidean space

D. Analytic vectors, and exponentiation of Lie algebra representations