On endomorphism algebras of Gelfand-Graev representations

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