On the Moy-Prasad filtration

TL;DR

This paper investigates the Moy-Prasad filtration representations for reductive groups over local fields, demonstrating their independence from the residual characteristic p and providing explicit descriptions and conditions for stable vectors, leading to new supercuspidal representations.

Contribution

It introduces p-independent descriptions of Moy-Prasad filtrations and extends conditions for stable vectors, enabling the construction of new supercuspidal representations.

Findings

Filtration representations are p-independent in a certain sense.

Explicit descriptions of these representations as Weyl modules.

Necessary and sufficient conditions for stable vectors in the filtrations.

Abstract

Let K be a maximal unramified extension of a nonarchimedean local field with arbitrary residual characteristic p. Let G be a reductive group over K which splits over a tamely ramified extension of K. We show that the associated Moy-Prasad filtration representations are in a certain sense independent of p. We also establish descriptions of these representations in terms of explicit Weyl modules and as representations occurring in a generalized Vinberg-Levy theory. As an application, we use these results to provide necessary and sufficient conditions for the existence of stable vectors in Moy-Prasad filtration representations, which extend earlier results by Reeder and Yu (which required p to be large) and by Romano and the author (which required G to be absolutely simple and split). This yields new supercuspidal representations. We also treat reductive groups G that are not necessarily split over a tamely ramified field extension.