G\'eom\'etrisation du lemme fondamental pour l'alg\`ebre de Hecke

TL;DR

This paper proves the geometric stabilization for the fundamental lemma in the group case, establishing an identity between orbital integrals and deriving transfer factors, advancing the understanding of the Langlands program.

Contribution

It extends Ngo's geometric stabilization from Lie algebras to groups, providing new formulas for transfer factors and confirming conjectures by Frenkel and Ngo.

Findings

Proved geometric stabilization in the group case.

Established an identity between orbital integrals.

Derived formulas for Langlands-Shelstad transfer factors.

Abstract

This article is the third one of the series \cite{Bt1}-\cite{Bt2} on Hitchin-Frenkel-Ngo fibration and Vinberg semigroup. Ngo \cite{N} proved the fundamental lemma for Lie algebras in equal characteristics as a consequence of geometric stabilization. This article show the geometric stabilization in the group case which was conjectured by Frenkel and Ngo \cite{FN}. Along the proof, we establish an identity between orbital integrals, which is analog to Langlands-Shelstad fundamental lemma. From this equality, we deduce a formula for Langlands-Shelstad transfer factors which was previously only known for Lie algebras.