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

主要研究者研究了p-adic表示理论中的两个突出问题。 第一个问题是分析Lusztig在1979年提出的p-adic群的超尖点表示的几何构造。 Lusztig的构造可以看作是自守归纳法的一个特例。 虽然在许多特殊情况下已经构造了自守归纳法，但现有的方法相当复杂，并且通常依赖于全局方法。 Lusztig的构造要优雅得多，但将其形式化并将其与p-adic表示理论的更经典的构造进行比较的问题仍然是完全开放的。 预计主要研究者获得的结果将揭示已知自守归纳法的几何基础。 第二个问题是由M. 2002年，哈里斯。 它要求在局部域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型。 首席研究员研究了一个家庭的开放仿射在鲁宾-泰特塔的上同调预计将实现各种特殊情况下的地方朗兰兹和雅克-朗兰兹对应（这将部分回答哈里斯的问题）。 相关的上同调计算与迄今为止已经理解的p进Lusztig归纳法的例子有很多共同之处。在过去的40年里，朗兰兹纲领主导了代数学的许多研究。它与数论和其他数学领域的一些最突出的结果有联系，例如费马大定理。首席研究员在该领域的一个分支工作，该分支被称为当地的朗兰兹项目。 它关注的是所谓的p-adic群的表示理论，主要的驱动力是寻找局部朗兰兹对应的一般证明。 各种特殊情况下，这种对应关系已获得亨尼亚特，哈里斯，泰勒和许多其他数学家。 然而，大多数现有的证明是不明确的，并没有提供足够的信息，本地朗兰兹对应的理想应用。 主要研究人员使用的方法的几何表示理论的幂幺群，发展在他以前的作品，给新的明确建设的地方朗兰兹对应，并简化和澄清现有的。 主要工具之一是Lusztig在1979年提出的p-adic群表示的几何构造。 直到最近，这种结构仍然相对未知，因为不清楚如何将其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，目前正在使用它来阐明当地朗兰兹对应背后的几何结构。

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.