Paquets d'Arthur des groupes classiques et unitaires

TL;DR

This paper proves the equivalence of two different constructions of packets of irreducible unitary cohomological representations for classical groups, linking Adams-Johnson and Arthur's frameworks through endoscopic transfer computations.

Contribution

It establishes the coincidence of Adams-Johnson and Arthur packets for classical groups by explicit endoscopic transfer calculations.

Findings

Confirmed the equivalence of Adams-Johnson and Arthur packets for classical groups.

Computed endoscopic transfer of stable distributions to twisted GL_N.

Showed the transfer matches Arthur's prescribed twisted trace.

Abstract

Let $G=\mathbf{G}(\mathbb{R})$ be the group of real points of a quasi-split connected reductive algebraic group defined over $\mathbb{R}$. Assume furthermore that $G$ is a classical group (symplectic, special orthogonal or unitary). We show that the packets of irreducible unitary cohomological representations defined by Adams and Johnson in 1987 coincide with the ones defined recently by J. Arthur in his work on the classification of the discrete automorphic spectrum of classical groups (C.-P. Mok for unitary groups). For this, we compute the endoscopic transfer of the stable distributions on $G$ supported by these packets to twisted $\mathbf{GL}_N$ in terms of standard modules and show that it coincides with the twisted trace prescribed by Arthur.