Mathematical Sciences: Topological Methods in Representation Theory

在过去的十年中，朗兰兹纲领中许多看来不可克服的困难已经被克服了。最近，人们的注意力转向了所谓的“基本引理”，这似乎是该领域最难被证明的猜想，而且有可能无限期地推迟该计划的完成。然而，这个引理被广泛认为是正确的，以至于定期发表的论文都依赖于它的有效性。在最简单的情况下，基本引理是p进代数群上的一个轨道积分与内窥镜群上的另一个轨道积分之间的推测等式。这是一个惊人的公式，在许多特殊情况下（很困难地）得到了验证。Goresky、Kottwitz和MacPherson教授发现了一种解决这个问题的新方法，它增加了许多几何思想和方法；通过一系列的约化，Weil猜想和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存在的复射影代数变体的等变上同调群之间的等式。这两种变体显然彼此之间几乎没有关系，除了它们都承认同一个环面的作用。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，希望最终能得到完整的证明。从长远来看，在20世纪70年代，r·朗兰兹（普林斯顿高等研究院的朗兰兹）概述了一系列范围和深度巨大的猜想和想法，这些猜想和想法一旦得到充分的探索和验证，将导致数学几个分支的“大统一”，包括数论、李群的表示理论和谐波分析。在过去的二十年里，这个计划取得了巨大的进展，克服了非常困难的障碍。然而，人们普遍认为，要充分探索朗兰兹的思想可能还需要20年（或更长时间）。“基本引理”仍然是程序中唯一最突出的困难。它已经在许多特殊情况下得到了证明，而且，正如上面提到的，它被广泛认为是正确的，以至于依赖于它的论文经常发表在有评审的期刊上。Goresky、Kottwitz和MacPherson教授在研究基本引理时发现了新的几何技术，这些技术将导致在许多新情况下对引理的证明，并可能最终导致一个完整的证明。虽然这些几何技术主要是为了实现朗兰兹计划中的这一步而开发的，但它们已经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. ***