  For a connected reductive group $G$ defined over $\mathbb{F}_q$ and equipped
with the induced Frobenius endomorphism $F$, we study the relation among the
following three $\mathbb{Z}$-algebras: (i) the $\mathbb{Z}$-model
$\mathsf{E}_G$ of endomorphism algebras of Gelfand-Graev representations of
$G^F$; (ii) the Grothendieck group $\mathsf{K}_{G^\ast}$ of the category of
representations of $G^{\ast F^\ast}$ over $\overline{\mathbb{F}_q}$
(Deligne-Lusztig dual side); (iii) the ring $\mathsf{B}_{G^\vee}$ of the scheme
$(T^\vee/\!\!/ W)^{F^\vee}$ over $\mathbb{Z}$ (Langlands dual side). The
comparison between (i) and (iii) is motivated by recent advances in the local
Langlands program.
