On endomorphism algebras of Gelfand-Graev representations II

TL;DR

This paper establishes an isomorphism between the endomorphism ring of Gelfand-Graev representations of a finite reductive group and the Grothendieck ring of its dual group's representations over an algebraic closure, revealing deep structural connections.

Contribution

It proves a new isomorphism linking endomorphism rings of Gelfand-Graev representations to the Grothendieck ring of dual group representations over specific rings.

Findings

Endomorphism ring is isomorphic to the Grothendieck ring over certain rings.

The result holds over $ar{Z}[1/pM]$, excluding bad primes.

Provides a structural link between representations of dual groups.

Abstract

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 $G^\ast(\mathbb{F}_q)$.