Mathematical Sciences: Topological Methods in Representation Theory

9626616戈雷斯基在过去的十年里，朗兰兹计划中的许多明显无法克服的困难已经被克服。最近，人们的注意力转向了所谓的“基本引理”，这似乎是该领域最困难的未经证实的猜想，它可能会无限期地推迟该计划的完成。然而，这一引理被广泛认为是正确的，因此定期发表的论文取决于它的有效性。在最简单的情况下，基本引理是一个猜想的等式，它是p-进代数群上的轨道积分和内窥镜群上的另一个轨道积分之间的相等。这是一个令人惊叹的公式，它已经在许多特殊情况下得到了(非常困难的)验证。Goresky，Kottwitz和MacPherson教授发现了一种新的方法来解决这个问题，它增加了一些几何思想和方法；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产生的复射影代数簇的等变上同调群之间的相等。这两个变种显然彼此几乎没有关系，除了它们都承认由同一环所做的动作。Goresky、Kottwitz和MacPherson教授使用他们新开发的等变上同调公式，发现比较基本引理的两边是可能的，这在许多情况下足以作为证明。他们正在改进这项技术，希望最终能得出一个完整的证据。为了正确认识上述问题，在1970年的《S》中，(普林斯顿高等研究院的朗兰兹)概述了一系列猜想和想法，这些猜想和想法具有巨大的广度和深度，当充分探索和验证时，将导致包括数论、李群表示理论和调和分析在内的几个数学分支的“大一统”。在过去的二十年里，这一计划取得了巨大的进步，克服了非常困难的障碍。然而，人们普遍认为，朗兰兹的思想可能还需要20年(或更长时间)才能得到充分的探索。“基本引理”仍然是该计划中最突出的一个难点。它已经在许多特殊情况下得到了证明，如上所述，人们普遍认为它是正确的，因此依赖于它的论文经常发表在被引用的期刊上。Goresky，Kottwitz和MacPherson教授在基本引理的研究中发现了新的几何技术，这些技术将在许多新的情况下得到它的证明，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在朗兰兹的程序中执行这一步而开发的，但它们已经被应用于数学的其他领域的问题。***

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. ***