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Quantum-Classical Correspondences for Locally Symmetric Spaces

Joachim Hilgert

Abstract

We review the status of a program, outlined and motivated in the introduction, for the study of correspondences between spectral invariants of partially hyperbolic flows on locally symmetric spaces and their quantizations. Further we formulate a number of concrete problems which may be viewed as possible further steps to be taken in order to complete the program.

Quantum-Classical Correspondences for Locally Symmetric Spaces

Abstract

We review the status of a program, outlined and motivated in the introduction, for the study of correspondences between spectral invariants of partially hyperbolic flows on locally symmetric spaces and their quantizations. Further we formulate a number of concrete problems which may be viewed as possible further steps to be taken in order to complete the program.
Paper Structure (45 sections, 164 equations)

This paper contains 45 sections, 164 equations.

Table of Contents

  1. Introduction
  2. Some motivating examples
  3. Outline of the program and its purpose
  4. Locally symmetric spaces
  5. Symmetric spaces
  6. Equivariant differential geometry on $G/K$
  7. Invariant differential operators
  8. Representation theory and harmonic analysis
  9. Principal series representations for minimal parabolics
  10. Intertwining operators
  11. $\Gamma$-cohomology
  12. Dynamical systems
  13. Classical dynamics
  14. Anosov flows
  15. Symplectic geometry and Hamiltonian actions
  16. ...and 30 more sections