Local Geometric Langlands Correspondence and Representation Theory

局部几何朗兰兹对应与表示理论

• 批准号: 2416129
• 负责人: Sam Raskin
• 金额: $ 18万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-01-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2416129&HistoricalAwards=false
• 关键词: Local; Geometric; Langlands; Correspondence; Representation

Representation theory studies the realization of groups as linear symmetries. There are two typical stages: 1) finding the general structure of representations of a given group (e.g., classifying irreducible representations), and 2) applying this to representations of particular interest (e.g., functions on a homogeneous space). This project aims to study higher representation theory, which studies the realization of groups as categorical symmetries. The emphasis of the proposal focuses on loop groups, where the theory remarkably mirrors classical harmonic analysis for p-adic groups. In particular, one finds Langlands-style decompositions here. This project focuses on understanding some key categories of interest in this framework. The investigator will study 3d mirror symmetry conjectures, representations of affine Lie algebras, and moduli spaces of bundles arising in the global geometric Langlands program. This project provides training opportunities for graduate students.In more detail, 3d mirror symmetry, representations of (reductive) affine Lie algebras, and the geometric Langlands program are the three primary ways actions of loop groups of reductive groups on categories arise. A large class of 3d mirror symmetry conjectures concerns the categorical Plancherel formula for loop group actions on categories of sheaves on loop spaces of particular varieties with group actions. The PI will establish first cases of 3d mirror symmetry and apply the results to give coherent descriptions of some categories of primary interest in geometric representation theory. Representations of Lie algebras concern the action of a group on its category of Lie algebra representations. The PI will extend previous work on critical level localization theory and develop a substitute for Soergel modules that will apply to poorly understood categories in the local geometric Langlands program. The applications to global geometric Langlands concern actions of loop groups of reductive groups on moduli spaces of a global nature, namely bundles with a level structure. The PI will extend the Satake theorem and apply the result to study Eisenstein series in the global geometric 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.

表示论研究群作为线性对称性的实现。有两个典型的阶段：1）找到给定组的表示的一般结构（例如，对不可约表示进行分类），2）将其应用于特别感兴趣的表示（例如，同质空间上的函数）。该项目旨在研究更高表示理论，该理论研究群作为范畴对称性的实现。该提案的重点集中在循环群上，该理论显着地反映了 p 进群的经典调和分析。特别是，人们在这里发现了朗兰兹式的分解。该项目的重点是了解该框架中一些关键的兴趣类别。研究者将研究 3d 镜像对称猜想、仿射李代数的表示以及全局几何朗兰兹纲领中出现的丛的模空间。该项目为研究生提供培训机会。更详细地说，3d 镜像对称、（还原）仿射李代数的表示以及几何朗兰兹纲领是还原群的循环群在类别上产生作用的三种主要方式。一大类 3d 镜像对称猜想涉及特定类型环空间上具有群作用的滑轮类别上的环群作用的分类 Plancherel 公式。 PI 将建立 3D 镜像对称的第一个案例，并应用结果对几何表示理论中主要感兴趣的一些类别进行连贯的描述。李代数的表示涉及群对其李代数表示范畴的作用。 PI 将扩展之前关于临界水平定位理论的工作，并开发 Soergel 模块的替代品，该模块将应用于局部几何 Langlands 程序中不太了解的类别。全局几何朗兰兹的应用涉及约简群的环群在全局性质的模空间（即具有水平结构的丛）上的作用。 PI 将扩展 Satake 定理，并将结果应用于全球几何朗兰兹计划中的爱森斯坦级数研究。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。