# On the Truncated Hitchin Fibration and the Weighted Fundamental Lemma (joint work with G\'erard Laumon)

 SHIMURA VARIETIES AND TRACE FORMULA SEMINAR Topic: On the Truncated Hitchin Fibration and the Weighted Fundamental Lemma (joint work with G\'erard Laumon) Speaker: P.-H. Chaudouard Affiliation: Universit'e Paris-Sud 11 and Member, School of Mathematics Date: Monday, October 13 Time/Room: 10:30am - 12:00pm/West Bldg. Lecture Hall

The fundamental lemma of Langlands and Shelstad is a family of combinatorial identities between local orbital integrals. It is necessary to the stabilization of the elliptic part of the Arthur-Selberg trace formula. In order to stabilize the whole trace formula, one also needs the weighted fundamental lemma conjectured by Arthur. It is a combinatorial statement involving the local weighted orbital integrals which are the local components of the hyperbolic terms in the geometric part of the trace formula. It is a generalization of the fundamental lemma. Ng\^o Bao Ch\^au recently proved the fundamental lemma of Langlands and Shelstad by a cohomological study of the elliptic part of the Hitchin fibration. In this talk, based on a joint work with G\'erard Laumon, I shall introduce a truncated version of the Hitchin fibration. The number of rational points in a truncated fiber is a global weighted orbital which appears in the trace formula. I shall explain how to extend the results of Ng\^o to our situation and how to deduce the weighted fundamental lemma of Arthur.