Parahoric Character Sheaves and Representations of p-Adic Groups

在过去的半个世纪里，数学的前沿发现出现在三个主要学科的交界处：数论(研究素数)、表示论(使用线性代数研究对称性)和几何(研究多项式方程的解集)。这些学科之间的相互作用在朗兰兹计划的背景下特别有影响力，该计划可以说是现代数学研究中规模最大的单一项目。这项拟议的研究旨在通过探索表征理论中的几何技术来进一步推动这些进展，特别是在朗兰兹猜想的背景下受到问题的激励。这个项目还为本科生和研究生提供了研究培训的机会。更详细地说，局部域上的约化代数群(局部群)及其表示控制着朗兰兹计划中对称的行为。这个项目的目的是建立局部群的表示与两个基本几何结构之间的联系：Deligne-Lusztig簇和特征标组。在过去的十年里，人们构造和研究了这些几何对象的类比，导致了局部朗兰兹的(猜想)代数结构与几何现象之间的联系，从而将开放的代数问题转化为代数几何中的容易处理的问题。在这个项目中，PI将使用这些新颖的Deligne-Lusztig变体和字符集合体的正深度复述类似物来攻击当地朗兰兹计划中的杰出猜想。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

In the past half century, cutting-edge discoveries in mathematics have occurred at the interface of three major disciplines: number theory (the study of prime numbers), representation theory (the study of symmetries using linear algebra), and geometry (the study of solution sets of polynomial equations). The interactions between these subjects has been particularly influential in the context of the Langlands program, arguably the most expansive single project in modern mathematical research. The proposed research aims to further these advances by exploring geometric techniques in representation theory, especially motivated by questions within the context of the Langlands conjectures. This project also provides research training opportunities for undergraduate and graduate students.In more detail, reductive algebraic groups over local fields (local groups) and their representations control the behavior of symmetries in the Langlands program. This project aims to develop connections between representations of local groups and two fundamental geometric constructions: Deligne-Lusztig varieties and character sheaves. Over the past decade, parahoric analogues of these geometric objects have been constructed and studied, leading to connections between (conjectural) algebraic constructions of the local Langlands correspondence to geometric phenomena, and thereby translating open algebraic questions to tractable problems in algebraic geometry. In this project, the PI will wield these novel positive-depth parahoric analogues of Deligne-Lusztig varieties and character sheaves to attack outstanding conjectures in the local Langlands program.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.