Sur les paquets d'Arthur des groupes classiques et unitaires non quasi-d\'eploy\'es

TL;DR

This paper extends Arthur's results to non quasi-split orthogonal and unitary groups, providing a complete Langlands classification for tempered representations and establishing multiplicity one for unipotent packets in both p-adic and archimedean cases.

Contribution

It generalizes known results from quasi-split to non quasi-split groups, offering a full classification and multiplicity one results for unipotent packets.

Findings

Complete Langlands classification for tempered representations in the p-adic case.

Multiplicity one for unipotent packets in non quasi-split groups.

Extension of results to archimedean cases using global methods.

Abstract

Nous \'etendons aux groupes orthogonaux et unitaires non quasi-d\'eploy\'es sur un corps local des r\'esultats de J. Arthur et de la premi\`ere auteure \'etablis dans le cas quasi-d\'eploy\'e. En particulier, nous obtenons une classification de Langlands compl\`ete pour les repr\'esentations temp\'er\'ees dans le cas $p$-adique. Nous en d\'eduisons en utilisant l'involution d'Aubert-Schneider-Stuhler un r\'esultat de multiplicit\'e un dans les paquets unipotents, et par des m\'ethodes globales, le m\^eme r\'esultat pour les paquets unipotents dans le cas archim\'edien. We extend to non quasi-split orthogonal and unitary groups over a local field some results of J. Arthur and the first author established in the quasi-split case. In particular, we obtain a full Langlands classification for tempered representations in the $p$-adic case. Using Aubert-Schneider-Stuhler involution, we deduce from this a multiplicity one result for unipotent packets, and by global methods, the same result for unipotent packets in the archimedean case.