G-companions on algebraic stacks and applications to canonical $\ell$-adic local systems on Shimura stacks

TL;DR

This paper extends Deligne's companion conjecture results to smooth Artin stacks, proving compatibility of canonical $ ext{ell}$-adic local systems on Shimura stacks, thus broadening the scope of previous work on algebraic stacks and local systems.

Contribution

It generalizes Drinfeld's theorem for reductive groups to smooth Artin stacks and applies this to study $ ext{ell}$-adic local systems on Shimura stacks.

Findings

Extended Drinfeld's theorem to smooth Artin stacks.

Proved compatibility of $ ext{ell}$-adic local systems on Shimura stacks.

Broadened the applicability of Deligne's companion conjecture.

Abstract

Cases of Deligne's companion conjecture for normal schemes over finite fields have been proven by L. Lafforgue, Drinfeld, and Zheng in recent years: L. Lafforgue proved the conjecture for curves, Drinfeld proved the conjecture for all smooth schemes and later also for representations valued in a reductive group, and Zheng proved Deligne's conjecture for smooth Artin stacks. In this paper, we extend Drinfeld's theorem for general reductive groups to smooth Artin stacks of finite presentation and apply the result to the study of compatibility of the canonical $\ell$-adic local systems on Shimura stacks.