Le lemma fondamental pour les groupes unitaires

Abstract

Let $\scriptstyle G$ be an unramified reductive group over a nonarchimedian 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 ${\Bbb F}_{p}((t))$, $G$ is a unitary group and $ p>\,\hbox{rank}(G)$. Waldspurger has shown that this particular case implies the Langlands fundamental lemma for unitary groups of rank $\lt p$ when $\scriptstyle F$ is any finite extension of ${\Bbb Q}_{p}$.

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.