Noncommutative Harmonic Analysis

Contents

• 0. Some Basic Concepts of Lie Group Representation Theory
  1. One parameter groups of operators
  2. Representations of Lie groups, convolution algebras, and Lie algebras
  3. Representations of distributions and universal enveloping algebras
  4. Irreducible representations of Lie groups
  5. Varieties of Lie groups
• 1. The Heisenberg Group
  1. Construction of the Heisenberg group Hⁿ
  2. Representations of Hⁿ
  3. Convolution operators on Hⁿ and the Weyl calculus
  4. Automorphisms of Hⁿ; the symplectic group
  5. The Bargmann-Fok representation
  6. (Sub)Laplacians on Hⁿ and harmonic oscillators
  7. Functional calculus for Heisenberg Laplacians and for harmonic oscillator Hamiltonians
  8. The wave equation on the Heisenberg group
• 2. The Unitary Group
  1. Representation theory for SU(2), SO(3), and some variants
  2. Representation theory for U(n)
  3. The subelliptic operator X₂²+ X₃² on SU(2)
• 3. Compact Lie Groups
  1. Weyl orthogonality relations and the Peter-Weyl theorem
  2. Roots, weights, and the Borel-Weil theorem
  3. Representations of compact groups on eigenspaces of Laplace operators
• 4. Harmonic Analysis on Spheres
  1. The Laplace operator in polar coordinates
  2. Classical PDE on spheres
  3. Spherical harmonics
  4. The subelliptic operator L₁² +L₂²+iaL₃ on S²
• 5. Induced Representations, Systems of Imprimitivity, and Semidirect Products
  1. Induced representations and systems of imprimitivity
  2. The Stone-von Neumann Theorem
  3. Semidirect products
  4. The Euclidean group and the Poincare group
• 6. Nilpotent Lie Groups
  1. Nilpotent Lie algebras and Lie algebras with dilations
  2. Step 2 nilpotent Lie groups
  3. Representations of general nilpotent Lie groups
• 7. Harmonic Analysis on Cones
  1. Dilations of cones and the ax+b group
  2. Spectral representation and functional calculus for the Laplacian on a cone
• 8. SL(2,R)
  1. Introduction to SL(2,R)
  2. Classification of irreducible unitary representations
  3. The principal series
  4. The discrete series
  5. The complementary series
  6. The spectrum of L²(Gamma\PSL(2,R)), in the compact case
  7. Harmonic analysis on the Poincare upper half plane
  8. The subelliptic operator A²+B²+(1/2)iaZ on SL(2,R)
• 9. SL(2,C) and More General Lorentz Groups
  1. Introduction to SL(2,C)
  2. Representations of SL(2,C)
  3. The Lorentz groups SO(n,1)
• 10. Groups of Conformal Transformations
  1. Laplace operators and conformal changes of metric
  2. Conformal transformations on Rⁿ, Sⁿ, and balls
• 11. The Symplectic Group and the Metaplectic Group
  1. Symplectic vector spaces and the symplectic group
  2. Symplectic inner product spaces and compact subgroups of the symplectic group
  3. The metaplactic representation
• 12. Spinors
  1. Clifford algebras and spinors
  2. Spinor bundles and the Dirac operator
  3. Spinors on four-dimensional Riemannian manifolds
  4. Spinors on four-dimensional Lorentz manifolds
• 13. Semisimple Lie Groups
  1. Introduction to semisimple Lie groups
  2. 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