G-rigid local systems are integral

TL;DR

This paper proves that certain rigid local systems with specific properties are integral, generalizing previous results for general linear groups and confirming a conjecture of Simpson.

Contribution

It extends integrality results to G-rigid local systems with finite abelianization and quasi-unipotent monodromies, and shows the semisimplicity of their monodromy Zariski-closure.

Findings

G-rigid local systems are integral under specified conditions

Connected component of monodromy Zariski-closure is semisimple

Generalizes previous results for GL_n and confirms Simpson's conjecture

Abstract

Let $G$ be a reductive group, and let $X$ be a smooth quasi-projective complex variety. We prove that any $G$-irreducible, $G$-cohomologically rigid local system on $X$ with finite order abelianization and quasi-unipotent local monodromies is integral. This generalizes work of Esnault and Groechenig when $G= \mathrm{GL}_n$, and it answers positively a conjecture of Simpson for $G$-cohomologically rigid local systems. Along the way we show that the connected component of the Zariski-closure of the monodromy group of any such local system is semisimple.