The B(G)-parametrization of the local Langlands correspondence

TL;DR

This paper advances the understanding of the local Langlands correspondence for non-quasi-split groups by establishing a parametrization framework linked to the full Kottwitz set, extending previous models and discussing endoscopic identities.

Contribution

It introduces a new parametrization indexed by the full Kottwitz set for non-quasi-split groups, generalizing existing models and connecting Galois representations with algebraic representations of centralizer groups.

Findings

Parametrization indexed by the basic part implies one indexed by the full Kottwitz set.

Discussion of a generalization of the endoscopic character identity for p-adic fields.

Extension of the local Langlands correspondence to non-quasi-split groups.

Abstract

This article is on the parametrization of the local Langlands correspondence over local fields for non-quasi-split groups according to the philosophy of Vogan. We show that a parametrization indexed by the basic part of the Kottwitz set (which is an extension of the set of pure inner twists) implies a parametrization indexed by the full Kottwitz set. On the Galois side, we consider irreducible algebraic representations of the full centralizer group of the $L$-parameter (i.e not a component group). When $F$ is a $p$-adic field, we discuss a generalization of the endoscopic character identity.