P-adic Representation Theory and Geometry of the Lubin-Tate Tower

鲁宾-泰特塔的P进表示理论和几何

• 批准号: 1748706
• 负责人: Sarah Kitchen
• 金额: $ 2.4万
• 依托单位: Michigan Technological University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2018-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1748706&HistoricalAwards=false
• 关键词: adic; Representation; Theory; Geometry; Lubin

The principal investigator studies two outstanding problems in p-adic representation theory. The first problem is analyzing a conjectural geometric construction of supercuspidal representations of p-adic groups proposed by Lusztig in 1979. Lusztig's construction can be viewed as a special case of automorphic induction. While automorphic induction has been constructed in many special cases, the existing approaches are quite complicated and often rely on global methods. Lusztig's construction is much more elegant, but the questions of formalizing it and comparing it to the more classical constructions of p-adic representation theory remain completely open. It is expected that the results obtained by the principal investigator will shed light on the geometry that underlies the known cases of automorphic induction. The second problem was formulated by M. Harris in 2002. It asks for a construction of Bushnell-Kutzko types for the general linear group of a local field K in the cohomology of suitable analytic subspaces of the Lubin-Tate tower of K. The principal investigator studies a family of open affinoids in the Lubin-Tate tower whose cohomology is expected to realize various special cases of the local Langlands and Jacquet-Langlands correspondences (which will partially answer Harris's question). The relevant cohomology computations have much in common with the examples of p-adic Lusztig induction that have so far been understood.The Langlands Program has dominated much of research in algebra during the last 40 years. It has connections to some of the most prominent results in number theory and other areas of mathematics, such as Fermat's Last Theorem. The principal investigator works in a branch of this field known as the local Langlands program. It is concerned with the representation theory of the so-called p-adic groups, and the main driving force is the search for a general proof of the local Langlands correspondence. Various special cases of this correspondence have been obtained by Henniart, Harris, Taylor and many other mathematicians. However, most of the existing proofs are not explicit and do not provide sufficient information for the desirable applications of the local Langlands correspondence. The principal investigator uses methods of geometric representation theory for unipotent groups, developed in his previous works, to give new explicit constructions of the local Langlands correspondence, and to simplify and clarify the existing ones. One of the main tools is a conjectural geometric construction of representations of p-adic groups formulated by Lusztig in 1979. Until recently this construction has remained relatively unknown because it was not clear how to compare it to the more classical constructions. The principal investigator developed general techniques for analyzing this construction, and is currently using it to shed light on the geometry behind the local Langlands correspondence.

主要研究了p-进表示理论中的两个突出问题。第一个问题是分析由Lusztig在1979年提出的p-Add群的超尖球面表示的一个猜想几何结构。Lusztig的构造可以看作是自同构归纳法的特例。虽然在许多特殊情况下已经构造了自同构归纳法，但现有的方法相当复杂，而且往往依赖于全局方法。Lusztig的构造要优雅得多，但将其形式化并将其与p-进表示理论的更经典的构造进行比较的问题仍然完全悬而未决。预计首席研究者所获得的结果将有助于阐明构成自同构归纳法的已知案例的几何原理。第二个问题是M.Harris在2002年提出的。它要求在K的Lubin-Tate塔的适当解析子空间的上同调中构造局部域K的一般线性群的Bushnell-Kutzko类型。主要研究者研究了Lubin-Tate塔中的一族开仿射，它的上同调有望实现局部Langland和Jacquet-Langland对应的各种特殊情况(这将部分回答Harris的问题)。相关的上同调计算与迄今已被理解的p-进Lusztig归纳的例子有许多共同之处。在过去的40年里，朗兰兹计划主导了代数的大部分研究。它与数论和其他数学领域的一些最突出的结果有关，例如费马大定理。首席研究员在这一领域的一个分支--当地朗兰兹计划--工作。它涉及到所谓的p-add群的表示理论，其主要驱动力是寻找局部朗兰兹对应的一般证明。亨尼亚特、哈里斯、泰勒和许多其他数学家已经得到了这种通信的各种特殊情况。然而，现有的证明大多是不明确的，并且不能为局部朗兰兹通信的理想应用提供足够的信息。主要研究人员利用他以前的工作中发展的关于酉群的几何表示理论的方法，给出了局部朗兰兹对应的新的显式构造，并简化和澄清了已有的构造。其中一个主要的工具是由Lusztig在1979年提出的p-进群表示的猜想几何构造。直到最近，这种结构仍然相对鲜为人知，因为人们不清楚如何将它与更经典的结构进行比较。首席研究员开发了分析这种结构的一般技术，目前正在用它来阐明当地朗兰兹通信背后的几何学。