On the refined local Langlands conjecture for discrete $L$-parameters of inner forms of quasi-split disconnected real reductive groups

TL;DR

This paper proves the refined local Langlands conjecture for inner forms of quasi-split disconnected real reductive groups by constructing L-packets for discrete L-parameters and verifying endoscopic character identities.

Contribution

It constructs L-packets for discrete L-parameters of inner forms of disconnected groups and confirms the refined local Langlands correspondence for these cases.

Findings

Constructed L-packets for each discrete L-parameter.

Proved these L-packets satisfy endoscopic character identities.

Confirmed the conjectural refined local Langlands correspondence.

Abstract

Given a quasi-split connected reductive $\mathbb{R}$-group $G$ and a finite group $A$ acting on $G$ by $\mathbb{R}$-automorphisms that preserve an $\mathbb{R}$-pinning, we construct for each discrete $L$-parameter for $G$ a corresponding $L$-packet of irreducible discrete series representations on each inner forms $\tilde G_z(\mathbb{R})$ of the disconnected group $\tilde G = G \rtimes A$. We prove that these $L$-packets satisfy the endoscopic character identities with respect to normalized transfer factors. This proves the conjectural refined local Langlands correspondence for inner forms of quasi-split disconnected real reductive groups, as recently formulated by the first author.