  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.
