Generalised Gelfand-Graev Representations in Small Characteristics

TL;DR

This paper extends Lusztig's formula for Generalised Gelfand-Graev Representations to smaller primes, establishing new conditions under which the formula holds and analyzing implications for character theory in algebraic groups.

Contribution

It proves Lusztig's formula for GGGRs under milder prime conditions and demonstrates the uniqueness of wave front sets and unipotent supports for characters and character sheaves.

Findings

Lusztig's formula valid for acceptable primes, including very good primes

Uniqueness of wave front sets for irreducible characters when p is good

Uniqueness of unipotent support for character sheaves when p is good

Abstract

Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraic closure $\overline{\mathbb{F}_p}$ of the finite field of prime order $p$ and let $F : \mathbf{G} \to \mathbf{G}$ be a Frobenius endomorphism with $G = \mathbf{G}^F$ the corresponding $\mathbb{F}_q$-rational structure. One of the strongest links we have between the representation theory of $G$ and the geometry of the unipotent conjugacy classes of $\mathbf{G}$ is a formula, due to Lusztig, which decomposes Kawanaka's Generalised Gelfand-Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, Lusztig's results are only valid under the assumption that $p$ is large enough. In this article we show that Lusztig's formula for GGGRs holds under the much milder assumption that $p$ is an acceptable prime for $\mathbf{G}$ ($p$ very good is sufficient but not necessary). As an application we show that every irreducible character of $G$, resp., character sheaf of $\mathbf{G}$, has a unique wave front set, resp., unipotent support, whenever $p$ is good for $\mathbf{G}$.