The wavefront sets of unipotent supercuspidal representations

Abstract

We prove that the double (or canonical unramified) wavefront set of an irreducible depth-0 supercuspidal representation of a reductive $p$-adic group is a singleton provided $p>3(h-1)$, where $h$ is the Coxeter number. We deduce that the geometric wavefront set is also a singleton in this case, proving a conjecture of Mœglin and Waldspurger. When the group is inner to split and the representation belongs to Lusztig’s category of unipotent representations, we give an explicit formula for the double and geometric wavefront sets. As a consequence, we show that the nilpotent part of the Deligne–Langlands–Lusztig parameter of a unipotent supercuspidal representation is precisely the image of its geometric wavefront set under Spaltenstein’s duality map.