Geometric Methods in Automorphic Forms

Goresky, R. Mark, Princeton NJ

Investigators

Abstract

The Langlands-Shelstad conjecture (and its special case, the so-called fundamental lemma) has emerged as one of the most pressing and stubborn problems in the modern approach to automorphic forms and representation theory. The principal investigator, together with his colleagues Robert MacPherson and Robert Kottwitz, have discovered that the kappa-orbital integrals which occur in the fundamental lemma may be expressed as the trace of Frobenius acting on the cohomology of an "affine Springer fiber". So the (conjectured) fundamental lemma is equivalent to a (fairly complicated) statement concerning the structure of the cohomology groups of affine Springer fibers. (An affine Springer fiber is the fixed point set, on the flag manifold of a loop group, or of a Kac-Moody Lie group, of the vectorfield which is determined by a semisimple element in the Lie algebra of the group. These researchers have been able to prove the required cohomological statement for affine Springer fibers which are associated to elements in unramified tori in the loop group. They are addressing the many technical problems associated with understanding the homology of affine Springer fibers associated to elements of ramified tori. In the 1970's, R. Langlands (of the Institute for Advanced Study in Princeton N.J.) developed an elaborate theory, indicating that there should be deep and hidden connections between several widely separated areas in mathematics: number theory, representation theory, algebraic geometry, and automorphic forms. He showed, for example, how results from representation theory could be used to deduce results in number theory. This vision was so far-reaching and broad in scope that it became known as "Langlands' program", and it is perhaps the mathematician's version of "grand unification". However, most of this program was conjectural and to some degree, even speculative. Progress on these conjectures was slow at first, as research in this subject demands an understanding of several different, highly technical branches of mathematics. Nevertheless, after decades of research by scores of dedicated and talented mathematicians worldwide, enormous progress has been made on Langlands' conjectures. For example, Andrew Wiles' celebrated proof of "Fermat's Last Theorem" depends in an essential way on some of these results. However, one step in this program, which was originally felt to be a relatively minor one, has turned out to be one of the most difficult questions in the area: the so-called "fundamental lemma" (and its generalization, the Langlands-Shelstad conjecture). While the supporting evidence for this conjecture is overwhelming, the conjecture has only been proven, after Herculean efforts, in a handful of special cases. It is a stubborn obstacle which threatens to indefinitely delay further progress in the area. The principal investigator and his colleagues Robert MacPherson (Institute for Advanced Study) and Robert Kottwitz (University of Chicago) have discovered that the Langlands-Shelstad conjecture may be restated in terms of the geometrical properties of certain objects ("affine Springer fibers") which have recently attracted the attention of mathematicians for completely different reasons. Using these geometric techniques, the investigator and his colleagues expect to outline a proof for the Langlands-Shelstad conjecture in a broad class of cases, the so-called "unramified" cases. They are also addressing the many difficulties involved with the remaining "ramified" cases. It is expected that this exciting connection between Langlands' program and "Springer theory" will lead to new developments in both subjects.