Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The theta correspondence over finite fields

Anne-Marie Aubert

Abstract

This set of lecture notes is an expanded version of a mini-course the author gave in March of 2025 for the program ``Representation Theory \& Noncommutative Geometry" at the Institut Henri Poincaré, Paris. The goal is to provide a survey of the main properties of the theta correspondence over finite fields of odd characteristic, including its compatibility with Harish-Chandra and Lusztig series, and with the Jordan decomposition of representations, as well as its full explicit description.

The theta correspondence over finite fields

Abstract

This set of lecture notes is an expanded version of a mini-course the author gave in March of 2025 for the program ``Representation Theory \& Noncommutative Geometry" at the Institut Henri Poincaré, Paris. The goal is to provide a survey of the main properties of the theta correspondence over finite fields of odd characteristic, including its compatibility with Harish-Chandra and Lusztig series, and with the Jordan decomposition of representations, as well as its full explicit description.

Paper Structure

This paper contains 37 sections, 31 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. The Weil representation
  3. The theta correspondence
  4. Reductive dual pairs
  5. Definition of the theta correspondence
  6. Witt towers of theta correspondences
  7. Theta correspondence and Harish-Chandra series
  8. Harish-Chandra series of finite groups of Lie type
  9. Application to the theta correspondence
  10. Theta correspondence and Lusztig series
  11. Deligne-Lusztig virtual characters
  12. Lusztig series
  13. Decompositions of the endoscopic groups
  14. Unitary groups
  15. Symplectic and orthogonal groups
  16. ...and 22 more sections

Key Result

Theorem 2.0.5

[Finite analogue of the Stone-von-Neumann Theorem] For every non-trivial character $\psi$ of ${\mathrm{Z}}_{H(W)}$, there exists a unique (up to equivalence) irreducible representation $(\varrho_\psi,S)$ of $H(W)$, known as the Heisenberg representation, such that

Theorems & Definitions (93)

  • Definition 2.0.2
  • Definition 2.0.3
  • Remark 2.0.4
  • Theorem 2.0.5
  • proof
  • Definition 2.0.8
  • Remark 2.0.9
  • Definition 3.1.1
  • Definition 3.1.2
  • Remark 3.1.8
  • ...and 83 more