Geometric Methods in Automorphic Forms

朗兰兹-谢尔斯塔德猜想(及其特例，即所谓的基本引理)已成为现代自同构形式和表示理论中最紧迫和最顽固的问题之一。这位首席研究员与他的同事罗伯特·麦克弗森和罗伯特·科特维茨发现，在基本引理中出现的卡帕轨道积分可以表示为Frobenius作用于“仿射Springer纤维”的上同调的迹。因此，(猜想的)基本引理等价于(相当复杂的)关于仿射Springer纤维上同调群结构的陈述。(仿射Springer纤维是环群或Kac-Moody李群的标志流形上的不动点集，它是由群的李代数中的半单元确定的向量场。这些研究人员已经能够证明仿射Springer纤维所需的上同调声明，该声明与环群中未分叉的环面中的元素相关。他们正在解决许多与理解仿射Springer纤维与分叉的元素相关联的同源性相关的技术问题。在1970年代，R·朗兰兹(新泽西州普林斯顿高等研究所的S)发展了一套详尽的理论，指出在数论、表示论、代数几何和自同构形式这几个广泛分离的数学领域之间应该有深刻而隐蔽的联系。例如，他展示了如何使用表象理论的结果来推导数论中的结果。这一愿景是如此深远和广泛，以至于它成为众所周知的“朗兰兹计划”，它可能是数学家版本的“大统一”。然而，这个项目的大部分内容都是猜测，在某种程度上，甚至是投机性的。起初，对这些猜想的研究进展缓慢，因为这门学科的研究需要理解几个不同的、高度技术性的数学分支。尽管如此，经过世界各地数十位敬业而有才华的数学家几十年的研究，朗兰兹猜想已经取得了巨大的进展。例如，安德鲁·怀尔斯著名的“费马大定理”证明在很大程度上依赖于其中的一些结果。然而，这个最初被认为是相对次要的程序中的一个步骤，后来被证明是该领域最困难的问题之一：所谓的“基本引理”(及其推广，朗兰兹-谢尔斯塔德猜想)。虽然支持这一猜想的证据是压倒性的，但这一猜想只是在经过艰苦的努力后，才在少数几个特殊情况下得到证实。这是一个顽固的障碍，有可能无限期拖延该领域的进一步进展。首席研究人员和他的同事罗伯特·麦克弗森(高级研究所)和罗伯特·科特维茨(芝加哥大学)发现，朗兰兹-谢尔斯塔德猜想可以用某些物体(仿射施普林格纤维)的几何性质来重述，这些物体最近因为完全不同的原因引起了数学家的注意。利用这些几何技术，这位研究人员和他的同事们希望在一大类案件中勾勒出朗兰兹-谢尔斯塔德猜想的证明，即所谓的“未分枝”案件。他们还在解决剩余的“分支”案件所涉及的许多困难。预计朗兰兹的计划和“斯普林格理论”之间的这种激动人心的联系将导致这两个学科的新发展。

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 inthe 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.