Stable Vectors in Dual Vinberg Representations of F4

Abstract

AbstractThis paper gives a classification of stable vectors in dual Vinberg representations coming from a graded Lie algebra of type F4 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 $F_{4}(\mathbb {Q}_{p})$ F 4 ( ℚ p ) when p is small.