Geck's conjecture and the generalized Gelfand-Graev representations in bad characteristic

Abstract

For a connected reductive algebraic group G defined over a finite field Fq, Kawanaka introduced the generalized Gelfand-Graev representations (GGGRs for short) of the finite group G(Fq) in the case where q is a power of a good prime for G. This representation has been widely studied and used in various contexts. Recently, Geck proposed a conjecture, characterizing Lusztig's special unipotent classes in terms of weighted Dynkin diagrams. Based on this conjecture, he gave a guideline for extending the definition of GGGRs to the case where q is a power of a bad prime for G. Here, we will give a proof of Geck's conjecture. Combined with Geck's pioneer work, our proof verifies Geck's conjectural characterization of special unipotent classes, and completes his definition of GGGRs in bad characteristics.