2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/100240We 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.published version, Latex fileAlgebraic GeometryRepresentation Theory14H60, 22E35 (Primary) 14F20 (Secondary)text