Centre de Bernstein dual pour les groupes classiques

TL;DR

This paper explores the relationship between parabolic induction and the local Langlands correspondence for classical groups, proposing a conjecture and verifying it in known cases, leading to a Bernstein-style decomposition of parameters.

Contribution

It introduces a conjecture linking supercuspidal representations and Langlands parameters, constructs a Bernstein-like decomposition for classical groups, and verifies compatibility with parabolic induction.

Findings

Verification of the conjecture for linear and classical groups

Construction of a Bernstein-style decomposition of Langlands parameters

Demonstration of compatibility between Langlands correspondence and parabolic induction

Abstract

In this article, we consider the links between parabolic induction and the local Langlands correspondence. We enunciate a conjecture about the (enhanced) Langlands parameters of supercuspidal representation of split reductives $p$-adics groups. We are able to verify this in those known cases of the local Langlands correspondence for linear groups and classical groups. Furthermore, in the case of classical groups, we can construct the "cuspidal support" of an enhanced Langlands parameter and get a decomposition of the set of enhanced Langlands parameters a la Bernstein. We check that these constructions match under the Langlands correspondence and as consequence, we obtain the compatibility of the Langlands correspondence with parabolic induction.