Geck's Conjecture and the Generalized Gelfand-Graev Representations in Bad Characteristic

TL;DR

This paper proves Geck's conjecture, confirming the characterization of special unipotent classes and extending the definition of generalized Gelfand-Graev representations to bad characteristic cases in algebraic groups.

Contribution

It provides a proof of Geck's conjecture, completing the definition of GGGRs in bad characteristic and confirming the classification of special unipotent classes.

Findings

Proof of Geck's conjecture established

Verification of Lusztig's classification in bad characteristic

Extension of GGGRs to bad primes completed

Abstract

For a connected reductive algebraic group $G$ defined over a finite field $\mathbb F_q$, Kawanaka introduced the generalized Gelfand-Graev representations (GGGRs for short) of the finite group $G(\mathbb F_q)$ 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.