Mathematical Sciences: Topological Methods in Representation Theory

小行星9626616 在过去十年中，许多显然无法克服的困难 朗兰兹纲领中的问题已经被克服了 最近，人们的注意力转向了所谓的“基本引理”，这似乎是该领域中最困难的未经证明的猜想，并且有可能无限期地推迟该计划的完成。 尽管如此，引理被广泛认为是正确的，以至于依赖于它的有效性的论文定期发表。 在最简单的情况下，基本引理是一个p-adic代数群上的轨道积分和一个endoscopic群上的另一个轨道积分之间的代数等式。 这是一个惊人的公式，它已经在许多特殊情况下得到了验证（非常困难）。 Goresky、Kottwitz和MacPherson教授发现了一种新的方法来解决这个问题，增加了一些几何思想和方法;通过一系列的简化，Weil定理和Lefschetz不动点公式的应用，基本引理的每一种情况都可以被重述为两个自然出现的复射影代数簇的等变上同调群之间的等式。 这两个变种显然没有什么或没有任何关系，除了他们都承认一个行动，由同一环面。 利用他们新开发的等变上同调公式，Goresky、Kottwitz和MacPherson教授发现可以比较基本引理的两边，这在许多情况下足以证明。 他们正在改进这项技术，希望最终能得出一个完整的证明。 为了正确地看待上述情况，在20世纪70年代，R。朗兰兹（普林斯顿高等研究院）概述了一系列具有巨大广度和深度的理论和思想，当充分探索和验证时，将导致数学的几个分支的“大统一”，包括数论，李群表示论和调和分析。 在过去的二十年里，这一计划取得了巨大的进展，克服了非常困难的障碍。 尽管如此，人们普遍认为，可能还需要20年（或更长时间）才能充分探索朗兰兹的思想。 “基本引理”仍然是程序中最突出的困难。 它已在许多特殊情况下得到证明，而且，如上所述，它是如此广泛地被认为是真实的，依赖于它的论文定期发表在参考期刊上。 教授Goresky，Kottwitz和MacPherson已经发现了新的几何技术在研究的基本引理，将导致其证明在许多新的情况下，并可能最终导致一个完整的证明。 虽然这些几何技巧主要是为了实现朗兰兹程序中的这一步而开发的，但它们已经被应用于其他数学领域的问题。 ***

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. ***