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

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.