Generalized Gelfand–Graev representations in small characteristics

Let G be a connected reductive algebraic group over an algebraic closure
¯
Fp
of the finite field of prime order p and let F:G→G be a Frobenius endomorphism with G=GF the corresponding Fq -rational structure. One of the strongest links we have between the representation theory of G and the geometry of the unipotent conjugacy classes of 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 G ( p very good is sufficient but not necessary). As an application we show that every irreducible character of G , respectively, character sheaf of G , has a unique wave front set, respectively, unipotent support, whenever p is good for G .