  Let $G$ be a connected reductive group defined over a finite field
$\mathbb{F}_q$ of characteristic $p$, with Deligne--Lusztig dual $G^\ast$. We
show that, over $\overline{\mathbb{Z}}[1/pM]$ where $M$ is the product of all
bad primes for $G$, the endomorphism ring of a Gelfand--Graev representation of
$G(\mathbb{F}_q)$ is isomorphic to the Grothendieck ring of the category of
finite-dimensional $\overline{\mathbb{F}}_q$-representations of
$G^\ast(\mathbb{F}_q)$.
