Representations of Reductive Groups and Étale Hessenberg Varieties

还原群和 ätale Hessenberg 簇的表示

• 批准号: 1601282
• 负责人: Cheng-Chiang Tsai
• 金额: $ 13.2万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2017-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601282&HistoricalAwards=false
• 关键词: Representations; Reductive; Groups; tale; Hessenberg

The Langlands program, a central theme in contemporary number theory, comprises several predictions that provide an approach to the analytic study of solution sets of Diophantine equations using techniques from linear algebra to exploit hidden symmetries in these solution sets. Geometric representation theory brings geometric tools to bear in studying problems in linear algebra. This project will apply methods from geometric representation theory to first reduce the computation of certain integrals that arise in the Langlands program to a discrete problem in combinatorial geometry, and then to study the solution of this latter problem. The results are expected to deepen and extend understanding in number theory and geometric representation theory, areas of mathematics with connections to many other subjects, including cyber security and physics.In work already completed, the investigator has succeeded in reducing the computation of certain orbital integrals to the problem of counting rational points of étale Hessenberg varieties over finite fields. This project will extend and refine the earlier computation, as well as study étale Hessenberg varieties in their own right, specifically as strata in affine Springer fibers, and place them in a broader context of relative Springer theory.

朗兰兹纲领是当代数论的一个中心主题，它包括几个预言，这些预言提供了一种方法来分析研究丢番图方程的解集，使用线性代数的技术来利用这些解集中隐藏的对称性。几何表示理论带来了几何工具，以承担在线性代数研究问题。这个项目将应用几何表示理论的方法，首先将朗兰兹程序中出现的某些积分的计算减少到组合几何中的离散问题，然后研究后者问题的解决方案。 这些结果有望加深和扩展对数论和几何表示论的理解，这些数学领域与许多其他学科有联系，包括网络安全和物理。在已经完成的工作中，研究人员成功地将某些轨道积分的计算减少到有限域上的étale Hessenberg簇的有理点的计数问题。该项目将扩展和完善早期的计算，以及研究étale Hessenberg品种本身，特别是仿射Springer纤维中的层，并将它们置于相对Springer理论的更广泛背景下。