Local Theta Correspondences between Supercuspidal Representations

TL;DR

This paper characterizes local theta correspondences between supercuspidal representations of p-adic groups using supercuspidal data, moment maps, and theta correspondences over finite fields, providing a comprehensive description of these relationships.

Contribution

It introduces a new framework linking supercuspidal data and theta correspondences via moment maps, enhancing understanding of supercuspidal representation correspondences.

Findings

Complete description of local theta correspondences between supercuspidal representations

Definition of supercuspidal data correspondence via moment maps

Short proof of Pan's depth preservation result

Abstract

By the works of Yu, Kim and Hakim-Murnaghan, we have a parameterization and construction of all supercuspidal representations of a reductive $p$-adic group in terms of supercuspidal data, when $p$ is sufficiently large. In this paper, we will define a correspondence of supercuspidal data via moment maps and theta correspondences over finite fields. Then we will show that local theta correspondences between supercuspidal representations are completely described by this notion. In Appendix B, we give a short proof of a result of Pan on "depth preservation".