Geometric Methods in Automorphic Forms

自守形式的几何方法

• 批准号: 0139986
• 负责人: R. Mark Goresky
• 金额: $ 8.43万
• 依托单位: Goresky, R. Mark
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0139986&HistoricalAwards=false
• 关键词: Geometric; Methods; Automorphic; Forms

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.

Langlands-Shelstad猜想（及其特例，所谓的基本引理）已经成为自守形式和表示论的现代方法中最紧迫和最顽固的问题之一。 主要研究者，连同他的同事罗伯特·麦克弗森和罗伯特·科特维茨，发现了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”的上同调的迹。 因此，（固定的）基本引理等价于关于仿射施普林格纤维的上同调群的结构的（相当复杂的）陈述。 (An仿射Springer纤维是指在一个loop群或Kac-Moody李群的旗流形上，由该群的李代数中的半单元素所确定的向量场的不动点集。 这些研究人员已经能够证明所需的上同调声明仿射施普林格纤维是相关的元素在unramified环面的循环组。 他们致力于解决与理解与分支环面元素相关的仿射Springer纤维的同源性相关的许多技术问题。朗兰兹（新泽西州普林斯顿高等研究院）他提出了一个详尽的理论，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，代数几何和自守形式。 例如，他展示了如何用表象论的结果来推导数论的结果。 这一设想是如此深远和广泛的范围，它被称为“朗兰兹纲领”，它可能是数学家版本的“大统一”。 然而，这一计划的大部分内容是理论性的，在某种程度上甚至是投机性的。 起初，这些数学的研究进展缓慢，因为这一主题的研究需要了解几个不同的、高度技术性的数学分支。 尽管如此，经过几十年的研究，由几十个专门和有才华的数学家在世界各地，巨大的进展已经取得了朗兰兹的地图。 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，这个程序中的一个步骤，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”（及其推广，Langlands-Shelstad猜想）。 虽然支持这个猜想的证据是压倒性的，但经过巨大的努力，这个猜想只在少数特殊情况下被证明。 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，这些物体最近因完全不同的原因引起了数学家的注意。 利用这些几何技巧，研究者和他的同事们希望在广泛的情况下，即所谓的“非分歧”情况下，概述朗兰兹-谢尔斯塔德猜想的证明。 他们还在处理与其余“分歧”案件有关的许多困难。 预期朗兰兹纲领和“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。