Twisted endoscopic character relation for toral supercuspidal L-packets of classical groups

TL;DR

This paper proves that Kaletha's toral supercuspidal L-packets satisfy the twisted endoscopic character relation for certain classical groups, confirming their compatibility with Arthur's local Langlands correspondence.

Contribution

It extends Kaletha's results to the twisted setting and verifies the compatibility of his construction with Arthur's framework for classical groups.

Findings

Kaletha's L-packets satisfy the twisted endoscopic character relation in specific cases.

Verification of the compatibility between Kaletha's and Arthur's local Langlands correspondences.

Application of Waldspurger's framework to emulate the proof in the twisted setting.

Abstract

We prove that Kaletha's toral supercuspidal L-packets satisfy the twisted endoscopic character relation in some cases, including the case of general linear groups equipped with an involution. Consequently, we verify that Kaletha's construction of the local Langlands correspondence for toral supercuspidal representations of quasi-split symplectic or special orthogonal groups coincides with Arthur's. The strategy is to emulate Kaletha's proof of the standard endoscopic character relation in the twisted setting by appealing to Waldspurger's framework ``l'endoscopie tordue n'est pas si tordue''.