Sur la contribution unipotente dans la formule des traces d'Arthur pour les groupes g\'en\'eraux lin\'eaires

TL;DR

This paper investigates the unipotent contribution in Arthur's trace formula for general linear groups, focusing on Richardson orbits induced by Levi subgroups with distinct blocks, and introduces new integral formulas and constructions.

Contribution

It provides a detailed analysis of Richardson orbits' contributions and introduces a new construction of Arthur's local unipotent weighted orbital integral.

Findings

Contribution of Richardson orbits expressed via global unipotent weighted orbital integrals

Derived integral formulas for certain Arthur's global coefficients

New construction method for Arthur's local unipotent weighted orbital integrals

Abstract

The theme of the article is the study of the unipotent part of Arthur's trace formula for general linear groups. The case of regular (or "regular by blocks") unipotent orbits has been essentially done in a previous paper. Here we are interested by the contribution of Richardson orbits that are induced by Levi subgroups with two-by-two distinct blocks. In this case, the contribution is remarkably given by a global unipotent weighted orbital integral. As a corollary, we get integral formulas for some of Arthur's global coefficients. We also present a new construction of Arthur's local unipotent weighted orbital integral.