Rigid inner forms of real and p-adic groups

TL;DR

This paper introduces a new cohomology set for algebraic groups over local fields, aiding the normalization of transfer factors and extending the understanding of L-packets and endoscopy beyond quasi-split groups, with validation in the real case.

Contribution

It defines a novel cohomology set for algebraic groups over local fields and applies it to normalize transfer factors and describe L-packets for all connected reductive groups.

Findings

New cohomology set extends Galois cohomology.

Normalization of endoscopic transfer factors achieved.

Conjectural description validated in the real case.

Abstract

We define a new cohomology set for an affine algebraic group G and a multiplicative finite central subgroup Z, both defined over a local field of characteristic zero, which is an enlargement of the usual first Galois cohomology set of G. We show how this set can be used to normalize the Langlands-Shelstad endoscopic transfer factors and to give a conjectural description of the internal structure and endoscopic transfer of L-packets for arbitrary connected reductive groups that extends the well-known conjectural description for quasi-split groups. In the real case, we show that this description is correct using Shelstad's work.