Local theta correspondences between epipelagic supercuspidal representations

TL;DR

This paper investigates the local theta correspondence for epipelagic supercuspidal representations of classical dual pairs over p-adic fields, establishing a relation via moment maps with a notable exception.

Contribution

It provides a characterization of when epipelagic supercuspidal representations lift under theta correspondence, linking their data through moment maps, with a specific exceptional case.

Findings

Most epipelagic supercuspidal representations lift via theta correspondence if their data are related by moment maps.

An exceptional case where the lifting behavior differs is identified.

The results deepen understanding of the structure of supercuspidal representations in p-adic harmonic analysis.

Abstract

In this paper we study the local theta correspondences between epipelagic supercupsidal representations of a type I classical dual pair $(G,G')$ over $p$-adic fields. We show that, besides an exceptional case, an epipelagic supercupsidal representation $\pi$ of $\widetilde{G}$ lifts to an epipelagic supercupsidal representation $\pi'$ of $\widetilde{G}'$ if and only if the epipelagic data of $\pi$ and $\pi'$ are related by the moment maps.