On endomorphism algebras of Gelfand-Graev representations

Abstract

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.

43 pages