On the Moy–Prasad filtration

Abstract

Let $K$ be a maximal unramified extension of a non-archimedean 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 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 present 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.