The theta correspondence over finite fields

TL;DR

This paper provides a comprehensive survey of the theta correspondence over finite fields of odd characteristic, detailing its properties, compatibility with various series, and explicit descriptions.

Contribution

It offers an expanded, detailed overview of the theta correspondence over finite fields, including compatibility and explicit descriptions, which was not previously fully documented.

Findings

Compatibility with Harish-Chandra and Lusztig series

Full explicit description of the theta correspondence

Compatibility with Jordan decomposition of representations

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 Poincar\'e, 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.