Le lemme fondamental pour les groupes unitaires

TL;DR

This paper proves the Langlands Fundamental Lemma for unramified unitary groups over certain local fields using Hitchin fibration techniques, advancing the understanding of orbital integrals and endoscopic transfer.

Contribution

It establishes the lemma for unitary groups over local fields of characteristic p, employing Hitchin fibration and deformation of orbital integrals, extending previous results.

Findings

Proof of the fundamental lemma for specific unitary groups

Introduction of a deformation of orbital integrals via Hitchin fibration

Implications for Langlands program and endoscopic transfer

Abstract

Let G be an unramified reductive group over a non archimedian local field F. The so-called "Langlands Fundamental Lemma" is a family of conjectural identities between orbital integrals for G(F) and orbital integrals for endoscopic groups of G. In this paper we prove the Langlands fundamental lemma in the particular case where F is a finite extension of F_p((t)), G is a unitary group and p>rank(G). Waldspurger has shown that this particular case implies the Langlands fundamental lemma for unitary groups of rank <p when F is any finite extension of Q_p. We follow in part a strategy initiated by Goresky, Kottwitz and MacPherson. Our main new tool is a deformation of orbital integrals which is constructed with the help of the Hitchin fibration for unitary groups over projective curves.