On endomorphism algebras of Gelfand-Graev representations

TL;DR

This paper explores the relationships among endomorphism algebras of Gelfand-Graev representations, the Grothendieck group of dual group representations, and the Langlands dual ring, linking these concepts within the context of the local Langlands program.

Contribution

It establishes connections between endomorphism algebras, dual group representations, and Langlands dual structures, advancing understanding in the local Langlands correspondence.

Findings

Identifies the algebraic relations among the three studied structures.

Provides a framework linking Gelfand-Graev endomorphisms with Langlands dual data.

Motivates future research in the local Langlands program through these connections.

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.