Paquets stables des s\'eries discr\`etes accessibles par endoscopie tordue; leur param\`etre de Langlands

TL;DR

This paper explicitly determines the Langlands parameters for packets of discrete series representations across various classical groups using twisted endoscopy, simplifying the proof and illustrating the effectiveness of the doubling method.

Contribution

Provides a straightforward proof of Langlands parameters for discrete series packets via twisted endoscopy, connecting representation theory with L-group parameters for classical groups.

Findings

Explicit Langlands parameters for discrete series packets

Simplified proof using twisted endoscopy and doubling method

Enhanced understanding of reducibility points in induced representations

Abstract

In this paper we gives the Langlands parameters of Langlands' packets of discrete series using the twisted endoscopy as explained by Arthur; this holds for orthogonal, symplectic, unitary and G-Spin groups and gives the most simple proof available. We have assume that the groups are quasi-split but this is just for simplicity. The proof explaines first what is the classification from the representation's theory point of view; this gives the Langlands' packets purely in terms of representation theory. And then using the theory of L-function of Shahidi and the doubling method of Rallis and Piatetskii-Shapiro, we translate this result in term of the $L$-group. Only the first part differs at some places of Arthur's point of view and gives more results about reducibility points of induced representations. We hope that this paper will make very clear how fruitful is the doubling method.