Metaplectic automorphic forms and matrix coefficients

Metaplectic 自守形式和矩阵系数

• 批准号: 1406238
• 负责人: Benjamin Brubaker
• 金额: $ 18万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406238&HistoricalAwards=false
• 关键词: Metaplectic; automorphic; forms; matrix; coefficients

This research project will explore topics in number theory and representation theory with connections to geometry, algebraic combinatorics, and statistical mechanics. Over the past several decades it has become clear that some of the deepest questions and conjectures in number theory, most notably those connected with the Langlands program, have powerful analogs in geometry and physics. However, the mechanism behind this relationship and associated conjectures remains largely mysterious. This project aims to explore possible sources of the connections by broadening the class of objects under consideration and using the winnowed set of techniques that apply to this larger class.In particular, many of the projects proposed center around the investigation of matrix coefficients for p-adic algebraic groups and their arithmetic covers. These matrix coefficients play a key role in the construction of automorphic L-functions. Their explicit computation in the context of metaplectic covers leads to surprising connections with geometry of Schubert varieties, to various specializations of Macdonald polynomials, and to quantum groups via both canonical bases and lattice models. These will be further developed in the proposed work and a framework for classifying matrix coefficients on algebraic groups as intertwining operators for Hecke algebra modules will be pursued. New distribution results for arithmetic functions will be another byproduct of these investigations.

该研究项目将探讨数字理论和表示理论的主题，并与几何形状，代数组合和统计力学联系。在过去的几十年中，很明显，数字理论中最深层的问题和猜想，最值得注意的是与兰兰兹计划相关的人，在几何和物理学方面具有强大的类似物。但是，这种关系和相关猜想背后的机制在很大程度上仍然是神秘的。 该项目的目的是通过扩大所考虑的对象，并使用适用于该较大类别的机器的技术集来探索连接的可能来源。特别是，许多项目围绕调查P-Adic代数组的基质系数的许多项目中心。这些矩阵系数在构建自动形态L功能中起着关键作用。它们在元容器覆盖范围的上下文中的明确计算导致与舒伯特品种的几何形状，与麦当劳多项式的各种专业相关的连接，并通过规范底座和晶格模型到量子组。这些将在拟议的工作中进一步开发，并将追求代数群体的矩阵系数分类为Hecke代数模块的交互操作员的框架。算术功能的新分布结果将是这些研究的另一个副产品。