Report on the Fundamental Lemma

摘要

The statement known as the fundamental lemma, a collection of identities in harmonic analysis on reductive groups over local fields, evolved in the course of Langlands' attempt to use the Arthur-Selberg trace formula to establish important special cases of his functoriality conjecture, the heart of his conjectural reorganization of number theory, representation theory and automorphic forms under a single heading. Thanks to the work of many mathematicians (see Section 8), and especially Bao Chau Ngo, the fundamental lemma is now a theorem. In this article, we explain Langlands' approach to the trace formula and the central role of the fundamental lemma in completing what might be considered the first stage of the Langlands program.