On the local Langlands conjectures for disconnected groups

TL;DR

This paper extends the local Langlands conjectures to certain disconnected groups with non-abelian components, introducing normalized transfer factors and establishing a new correspondence, including a proof for the case where the identity component is a torus.

Contribution

It generalizes the local Langlands conjectures to disconnected groups with non-abelian components and introduces normalized transfer factors and correspondences.

Findings

Proved the conjecture when the identity component is a torus.

Recast aspects of twisted endoscopy in this new framework.

Introduced normalized twisted transfer factors.

Abstract

We extend the local Langlands conjectures to a certain class of disconnected groups, allowing non-abelian component groups, and recast in this language some aspects of twisted endoscopy. We further introduce normalized twisted transfer factors and a normalized correspondence between an $L$-packet for a disconnected group and the set of representations of the centralizer groups of its Langlands parameter. We prove the first instance of this conjecture, in which the identity component of the (possibly non-abelian) disconnected group is a torus.