Geometric Methods in Automorphic Forms

Langlands-Shelstad猜想（及其特殊情况，即所谓的基本引理）已经成为现代自同构形式和表征理论研究中最紧迫和最顽固的问题之一。首席研究员和他的同事Robert MacPherson和Robert Kottwitz发现，在基本引理中出现的kappa-轨道积分可以表示为Frobenius作用于“仿射施普林格纤维”上同调的痕迹。因此，（推测的）基本引理等价于一个（相当复杂的）关于仿射施普林格纤维上同调群结构的陈述。(1)仿射光纤是由群的李代数中的半单元决定的向量场在环群或Kac-Moody李群的标志流形上的不动点集。这些研究人员已经能够证明与环群中未分枝环面元素相关的仿射施普林格纤维所需的上同源性陈述。他们正在解决与理解与分支环面元素相关的仿射施普林格纤维同源性相关的许多技术问题。在20世纪70年代，r·朗兰兹（新泽西州普林斯顿高等研究所的）发展了一个详尽的理论，指出数学中几个广泛分离的领域之间应该有深刻而隐藏的联系：数论、表示论、代数几何和自同态形式。例如，他展示了如何用表征理论的结果来推导数论的结果。这一设想是如此深远和广泛，以至于它被称为“朗兰兹纲领”，它可能是数学家版本的“大统一”。然而，这个计划的大部分都是推测性的，在某种程度上，甚至是推测性的。这些猜想的进展一开始是缓慢的，因为研究这一课题需要了解数学中几个不同的、高度技术性的分支。尽管如此，经过几十年的研究，在世界范围内许多有献身精神和才华的数学家，朗兰兹的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”的证明在本质上就依赖于其中的一些结果。然而，这个计划的一个步骤，最初被认为是一个相对较小的步骤，已经成为该领域最困难的问题之一：所谓的“基本引理”（以及它的推广，Langlands-Shelstad猜想）。虽然支持这一猜想的证据是压倒性的，但经过巨大的努力，这一猜想只在少数特殊情况下得到了证明。这是一个顽固的障碍，有可能无限期地拖延该地区的进一步进展。首席研究员和他的同事Robert MacPherson（高级研究所）和Robert Kottwitz（芝加哥大学）发现，Langlands-Shelstad猜想可以在某些物体（“仿射施普林格纤维”）的几何特性方面得到重申，这些物体最近因完全不同的原因引起了数学家的注意。利用这些几何技术，研究者和他的同事们希望在一种广泛的情况下，即所谓的“非分支”情况下，勾勒出朗兰兹-谢尔斯塔德猜想的证明。他们还在处理其余“分支”案件所涉及的许多困难。人们期望朗兰兹的纲领和“施普林格理论”之间的这种令人兴奋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.