Le lemme fondamental pour l'endoscopie tordue: le cas o{\`u} le groupe endoscopique non ramifi{\'e} est un tore

TL;DR

This paper proves the fundamental lemma for twisted endoscopy in the specific case where the underlying group is a torus, confirming the lemma for all elements in spherical Hecke algebras in characteristic zero.

Contribution

It establishes the fundamental lemma for twisted endoscopy for non ramified elliptic tori, completing the proof for all spherical Hecke algebra elements in characteristic zero.

Findings

Fundamental lemma proven for non ramified elliptic tori

Validates the lemma for all spherical Hecke algebra elements

Applicable in characteristic zero and any residue characteristic

Abstract

We prove the fundamental lemma for twisted endoscopy, for the unit elements of the spherical Hecke algebras, in the case of a non ramified elliptic endo- scopic datum whose underlying group is a torus. This implies that the fundamental lemma for twisted endoscopy is now proved, for all elements in the spherical Hecke algebras, in characteristic zero and any residue characteristic.