Fibration de Hitchin et endoscopie

We propose a geometric interpretation of the theory of elliptic endoscopy, due to Langlands and Kottwitz, in terms of the Hitchin fibration. As applications, we prove a global analog of a purity conjecture, due to Goresky, Kottwitz and MacPherson. For unitary groups, this global purity statement has been used, in a joint work with G. Laumon, to prove the fundamental lemma over local fields of equal characteristics.