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 Hn
- Representations of Hn
- Convolution operators on Hn and the Weyl calculus
- Automorphisms of Hn; the symplectic group
- The Bargmann-Fok representation
- (Sub)Laplacians on Hn and harmonic oscillators
- Functional calculus for Heisenberg Laplacians and for harmonic oscillator Hamiltonians
- The wave equation on the Heisenberg group
- 2. The Unitary Group
- Representation theory for SU(2), SO(3), and some variants
- Representation theory for U(n)
- The subelliptic operator X22+ X32 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 L12 +L22+iaL3 on S2
- 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 L2(Gamma\PSL(2,R)), in the compact case
- Harmonic analysis on the Poincare upper half plane
- The subelliptic operator A2+B2+(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 Rn, Sn, 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