Endoscopy and harmonic analysis on reductive groups

Abstract: Let G be a connected reductive group over a number field and let H be an endoscopic group of G. A conjecture of Langlands predicts that there exists a correspondence between automorphic representations of H(A) and automorphic representations of G(A), where A is the ring of adeles of the ground field. Langlands’ idea of proof is to compare the Arthur-Selberg’s trace for- mulas of H and G. It’s necessary to solve many problems, in particular two problems of harmonic analysis over a local field: the transfer conjecture and the fundamental lemma. These two questions have remained open until the decisive result of Ngô Bao Châu, two years ago. In my talk, I’ll try to explain what is the endoscopic transfer and what is the fundamental lemma. I will give several statements of that lemma, more or less sophisticated. I’ll try to explain the situation at the present time. In fact, all the useful problems are resolved even if certain related questions of harmonic analysis remain open.