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

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.