Rational wavefront sets and the descent method for finite classical groups

Abstract

Let \(\mathbbm{k}\) be a finite field of sufficiently large odd characteristic, let \(F\) be the Frobenius map, and let \(G\) be a symplectic or orthogonal group defined over \(\mathbbm{k}\). For an irreducible representation \(\pi\) of \(G^F\), its rational wavefront set \(\ScO(\pi)\) is the set of \(F\)-rational nilpotent orbits \(\mathcal O\) such that \(\pi\) occurs in the generalized Gelfand--Graev representation \(\Gamma_{\mathcal O}\). We prove that all maximal rational orbits in \(\ScO(\pi)\) lie in a single \(F\)-stable nilpotent orbit, and that the whole rational wavefront set is obtained from its maximal rational orbits by rational closure. We also give an explicit description of these maximal rational orbits in terms of Lusztig parameters and compute the dimensions of the corresponding generalized Whittaker models. For unipotent representations, we prove that the maximal rational wavefront set is a single fiber of Lusztig's canonical quotient. For special unipotent representations, this fiber identifies the kernel of the canonical quotient map, giving a finite-field representation-theoretic realization of the canonical quotient. We also describe the effect of Alvis--Curtis duality on maximal rational wavefront sets.