Title: Generalised Gelfand-Graev Representations in Small Characteristics

Author: Jay Taylor

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 both
$p$ and $q$ are 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 $\mathbf{G}$.