Stable Vectors in Dual Vinberg Representations of F4

Abstract This paper gives a classification of stable vectors in dual Vinberg representations coming from graded Lie algebra type F 4 way that is independent the field definition. Relating these gradings to Moy–Prasad filtrations, we obtain input for Reeder–Yu’s construction epipelagic supercuspidal representations. As corollary, this new $F_{4}(\mathbb {Q}_{p})$ F 4 ( ℚ p ) when p small.