Wavefront sets and descent method for finite classical groups

Abstract

In this paper, we complete the explicit description of rational nilpotent orbits in wavefront sets for irreducible representations of classical groups over a finite field $\mathbbm{k}$ of characteristic $p>3(h-1)$, where $h$ is the Coxeter number of the corresponding root system. In particular, for unipotent representations, we describe the rational nilpotent orbits in their wavefront sets in terms of Lusztig canonical quotients and establish the Alvis-Curtis duality for these orbits. Our approach is based on the descent method. We develop a new composition law for nilpotent orbits, which reduces the computation of rational wavefront sets to the finite Gan-Gross-Prasad problem established in our previous work.