Stable vectors in Moy-Prasad filtrations

TL;DR

This paper characterizes when the dual of the first Moy-Prasad filtration quotient contains stable vectors, extending previous classifications and enabling new supercuspidal representations of p-adic groups.

Contribution

It provides necessary and sufficient conditions for stability in Moy-Prasad filtrations, generalizing earlier results to all primes and constructing new supercuspidal representations.

Findings

Necessary and sufficient conditions for stable vectors in the dual of the filtration quotient

Extension of Reeder and Yu's classification to all primes

Construction of new supercuspidal representations

Abstract

Let k be a finite extension of Q_p, let G be an absolutely simple split reductive group over k, and let K be a maximal unramified extension of k. To each point in the Bruhat-Tits building of G_K, Moy and Prasad have attached a filtration of G(K) by bounded subgroups. In this paper we give necessary and sufficient conditions for the dual of the first Moy-Prasad filtration quotient to contain stable vectors for the action of the reductive quotient. Our work extends earlier results by Reeder and Yu, who gave a classification in the case when p is sufficiently large. By passing to a finite unramified extension of k if necessary, we obtain new supercuspidal representations of G(k).