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的猜想（及其特殊情况，即所谓的基本引理）已成为现代自动形式和代表理论的现代方法中最紧迫，最顽固的问题之一。 首席研究员以及他的同事罗伯特·麦克弗森（Robert Macpherson）和罗伯特·科特维茨（Robert Kottwitz）发现，基本引理中发生的Kappa-轨道积分可能被表示为frobenius的痕迹，这些痕迹是frobenius的痕迹，这些痕迹是对“仿生弹性纤维”的共同体作用。 因此，（猜想的）基本引理相当于（相当复杂的）陈述，陈述了仿生弹性纤维的同种学组的结构。 （仿生的弹簧纤维是循环组的旗不同或vectorfield的kac-moody lie群体的固定点集，由该组的谎言代数中的半密布元素确定。这些研究人员能够证明与较大的spripner fibers相关的同类式词组，这些元素与群落相关联的元素是与他们相关联的元素，这些元素与群落相关联的元素是与他们相关联的，这些元素与他们相关联的群体中的群体中，他们在这些元素中与他们有关。植物弹簧纤维与毛刺的元素相关。推断出数字理论的结果。 但是，该计划的大部分是猜想，在某种程度上，甚至是投机性的。 这些猜想的进展起初是缓慢的，因为该主题的研究需要了解几个不同技术的数学分支。 然而，经过数十年的数十年研究，全球敬业和才华横溢的数学家进行了研究，兰兰兹的猜想已经取得了巨大进步。 例如，安德鲁·威尔斯（Andrew Wiles）著名的“费马特（Fermat）最后一个定理”的证明取决于其中的一些结果。 但是，该计划最初被认为是相对较小的一步，事实证明是该地区最困难的问题之一：所谓的“基本引理”（及其概括，兰兰兹·塞尔斯塔德的猜想）。 尽管这一猜想的支持证据是压倒性的，但在少数特殊情况下，猜想仅在艰苦的努力之后才得到证明。 这是一个顽固的障碍，威胁要无限期地延迟该地区的进一步进展。 首席调查员及其同事罗伯特·麦克弗森（Robert Macpherson）（高级研究所）和罗伯特·科特威茨（Robert Kottwitz）（芝加哥大学）发现，兰兰兹·塞尔斯塔德（Langlands-Shelstad）的猜想可能会根据某些物体的几何特性来重申，这些物体的几何特性（“仿生的弹簧纤维纤维”最近吸引了完全不同的理由吸引了数学原因的注意力。 使用这些几何技术，研究人员及其同事们希望在广泛的案例中概述兰格兰斯 - 塞尔斯塔德猜想的证明，即所谓的“未遭受的”案件。 他们还解决了其余的“受损”案件涉及的许多困难。 可以预期，兰兰兹计划与“施普林格理论”之间的这种令人兴奋的联系将导致这两个主题的新发展。