Le lemme fondamental pour l'endoscopie tordue: r\'eduction aux \'el\'ements unit\'es

TL;DR

This paper demonstrates that the fundamental lemma for twisted endoscopy, proven for unit elements, extends to all elements of the spherical Hecke algebras, simplifying the proof process.

Contribution

It shows that the fundamental lemma for all elements follows from the case of unit elements, leveraging the transfer method and ideas from Arthur.

Findings

Fundamental lemma for units implies the lemma for all elements.

Transfer method is key to extending the proof.

Simplifies the proof of the fundamental lemma for twisted endoscopy.

Abstract

We show here that the fundamental lemma for twisted endoscopy, now proved for the unit elements in the spherical Hecke algebras, implies the fundamental lemma for all elements of these Hecke algebras. The proof, whose idea is due to Arthur, uses the transfer, which is known as a consequence of the fundamental lemma for the units.