  Let $G$ be a connected reductive algebraic group defined over a finite field
with $q$ elements. In the 1980's, Kawanaka introduced generalised Gelfand-Graev
representations of the finite group $G(F_q)$, assuming that $q$ is a power of a
good prime for $G$. These representations have turned out to be extremely
useful in various contexts. Here we investigate to what extent Kawanaka's
construction can be carried out when we drop the assumptions on~$q$. As a
curious by-product, we obtain a new, conjectural characterisation of Lusztig's
concept of special unipotent classes of $G$ in terms of weighted Dynkin
diagrams.
