GENERALIZED GELFAND–GRAEV REPRESENTATIONS IN SMALL CHARACTERISTICS

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$F:\mathbf{G}\rightarrow \mathbf{G}$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 (Adv. Math.94(2) (1992), 139–179), which decomposes Kawanaka’s Generalized Gelfand–Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, the formula given in Lusztig (Adv. Math.94(2) (1992), 139–179) is 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$G$, respectively, character sheaf of$\mathbf{G}$, has a unique wave front set, respectively, unipotent support, whenever$p$is good for $\mathbf{G}$.