Le lemme fondamental pour les algebres de Lie

TL;DR

This paper presents a proof of the fundamental lemma for Lie algebras and the non-standard fundamental lemma, using the decomposition of l-adic cohomology of the Hitchin fibration into simple perverse sheaves.

Contribution

It provides a new proof of key conjectures in the Langlands program by analyzing the Hitchin fibration's cohomology decomposition.

Findings

Proof of the fundamental lemma for Lie algebras

Verification of the non-standard fundamental lemma

Advancement in understanding Hitchin fibration cohomology

Abstract

We propose a proof for conjectures of Langlands, Shelstad and Waldspurger known as the fundamental lemma for Lie algebras and the non-standard fundamental lemma. The proof is based on a study of the decomposition of the l-adic cohomology of the Hitchin fibration into direct sum of simple perverse sheaves.