Mathematical Sciences: Topological Methods in Representation Theory

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9626616 Goresky During the last decade, many apparently insurmountable difficulties in the Langlands program have been overcome. Recently, attention has turned to the so-called "Fundamental lemma," which appears to be the single most difficult unproven conjecture in the field, and which threatens to delay indefinitely the completion of the program. Nevertheless, the lemma is so widely believed to be true that papers are regularly published that depend on its validity. In the simplest cases, the fundamental lemma is a conjectural equality between an orbital integral on a p-adic algebraic group, and another orbital integral on an endoscopic group. It is an amazing formula, which has been verified (with great difficulty) in a number of special cases. Professors Goresky, Kottwitz, and MacPherson have found a new approach to this problem that adds a number of geometric ideas and methods; by a sequence of reductions, applications of the Weil conjectures and the Lefschetz fixed point formula, each case of the fundamental lemma may be restated as an equality between the equivariant cohomology groups of two naturally occurring complex projective algebraic varieties. These two varieties apparently have little or nothing to do with each other except that they both admit an action by the same torus. Using their newly developed formula for equivariant cohomology, Professors Goresky, Kottwitz, and MacPherson have found it possible to compare both sides of the fundamental lemma, which in many cases suffices for a proof. They are refining this technique with the hopes that it may eventually lead to a complete proof. To put the foregoing in perspective, during the 1970's, R. Langlands (of the Institute for Advanced Study in Princeton) outlined a series of conjectures and ideas of enormous scope and depth that, when fully explored and verified, will result in a "grand unification" of several branches of mathematics, including number theory, representation theory of Lie groups, and harmonic analysis. During the last twenty years enormous progress has been made on this program, and very difficult obstacles have been overcome. Nevertheless, it is commonly believed that it may take another twenty years (or more) before Langlands' ideas are fully explored. The "fundamental lemma" remains the single most outstanding difficulty in the program. It has been proven in many special cases, and, as mentioned above, it is so widely believed to be true that papers which depend on it are regularly published in refereed journals. Professors Goresky, Kottwitz and MacPherson have discovered new geometric techniques in the study of the fundamental lemma that will lead to its proof in many new cases and may eventually lead to a complete proof. Although these geometric techniques have been developed primarily to carry out this step in Langlands' program, they have already been applied to problems in other areas of mathematics. ***