A remark on a conjecture of Buzzard-Gee and the cohomology of Shimura varieties

TL;DR

This paper explores the connections between Buzzard-Gee's conjecture on Galois representations, Kottwitz's cohomology description of Shimura varieties, and Arthur's reciprocity law, by extending Langlands's L-group representation to the C-group.

Contribution

It extends Langlands's L-group representation to the C-group of Buzzard-Gee, providing insights into the Tate twist in Kottwitz's cohomology description.

Findings

Extended Langlands's L-group to the C-group of Buzzard-Gee.

Provided an explanation for the Tate twist in Kottwitz's description.

Linked conjectures of Buzzard-Gee, Kottwitz, and Arthur through this extension.

Abstract

We compare the conjecture of Buzzard-Gee on the association of Galois representations to C-algebraic automorphic representations with the conjectural description of the cohomology of Shimura varieties due to Kottwitz, and the reciprocity law at infinity due to Arthur. This is done by extending Langlands's representation of the L-group associated with a Shimura datum to a representation of the C-group of Buzzard-Gee. The approach offers an explanation of the explicit Tate twist appearing in Kottwitz's description.