Global B(G) with adelic coefficients and transfer factors at non-regular elements

TL;DR

This paper extends Kottwitz's theory of B(G) with adelic coefficients from tori to all connected reductive groups and constructs transfer factors for non-regular elements, aiding in the stabilization of Shimura and Igusa varieties.

Contribution

It generalizes the definition of B(G) with adelic coefficients to all connected reductive groups and provides explicit transfer factors for non-regular semisimple elements.

Findings

Extended B(G) theory to all connected reductive groups.

Constructed transfer factors for non-regular semisimple elements.

Applied formulas to stabilize cohomology of Shimura and Igusa varieties.

Abstract

The goal of this paper is extend Kottwitz's theory of $B(G)$ for global fields. In particular, we show how to extend the definition of "$B(G)$ with adelic coefficients" from tori to all connected reductive groups. As an application, we give an explicit construction of certain transfer factors for non-regular semisimple elements of non-quasisplit groups. This generalizes some results of Kaletha and Taibi. These formulas are used in the stabilization of the cohomology of Shimura and Igusa varieties.