Stable vectors in dual Vinberg representations of $F_4$

TL;DR

This paper classifies stable vectors in dual Vinberg representations of type F4, linking them to Moy-Prasad filtrations, and constructs new supercuspidal representations for small primes.

Contribution

It provides a field-independent classification of stable vectors in F4 Vinberg representations and applies this to construct new supercuspidal representations.

Findings

Classification of stable vectors in F4 Vinberg representations

Connection to Moy-Prasad filtrations and epipelagic supercuspidal representations

New supercuspidal representations for small primes

Abstract

This paper gives a classification of stable vectors in dual Vinberg representations coming from a graded Lie algebra of type $F_4$ in a way that is independent of the field of definition. Relating these gradings to Moy-Prasad filtrations, we obtain the input for Reeder-Yu's construction of epipelagic supercuspidal representations. As a corollary, this construction gives new supercuspidal representations of $F_4(\mathbb{Q}_p)$ when $p$ is small.