Shallow Characters and Supercuspidal Representations

TL;DR

This paper extends the construction of supercuspidal representations for split reductive p-adic groups by classifying characters on deeper Moy-Prasad subgroups and analyzing stability conditions.

Contribution

It generalizes Reeder-Yu's methods to deeper Moy-Prasad subgroups and clarifies the role of stability conditions in constructing supercuspidal representations.

Findings

Classified all characters vanishing on certain Moy-Prasad subgroups.

Naive stability conditions are sufficient but not necessary for supercuspidal construction.

Extended methods apply to split reductive p-adic groups.

Abstract

In 2014, Reeder and Yu constructed epipelagic representations of a reductive $p$-adic group $G$ from stable functions on shallowest Moy-Prasad quotients. In this paper, we extend these methods when $G$ is split. In particular, we classify all complex-valued characters vanshing on a slightly deeper Moy-Prasad subgroup and show that, while sufficient, a naive extension of Reeder-Yu's stability condition is not necessary for constructing supercuspidal representations.