Local Geometric Langlands Correspondence and Representation Theory

表示论研究群作为线性对称的实现。有两个典型的阶段：1)找到给定群的表示的一般结构(例如，对不可约表示进行分类)，以及2)将其应用于特别感兴趣的表示(例如，齐次空间上的函数)。这个项目的目的是学习更高表示理论，它研究群作为范畴对称的实现。该建议的重点集中在环群上，其中该理论显著地反映了对p-Add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。这个项目的重点是了解这个框架中感兴趣的一些关键类别。研究人员将研究3D镜像对称猜想，仿射李代数的表示，以及在全球几何朗兰兹计划中产生的丛的模空间。这个项目为研究生提供了培训机会。更详细地说，三维镜像对称性、(约化)仿射李代数的表示和几何朗兰兹程序是约化群的环群对范畴产生作用的三种主要方式。一大类3D镜像对称猜想涉及具有群作用的特殊变元的环空间上的叶范畴上的环群作用的范畴Plancherel公式。PI将建立3D镜像对称的第一个案例，并应用其结果给出几何表示理论中主要感兴趣的一些范畴的连贯描述。李代数的表示涉及群在其李代数表示范畴上的作用。PI将扩展之前在临界水平本地化理论方面的工作，并开发一种替代索尔盖尔模块，该模块将应用于当地几何朗兰兹计划中理解较少的类别。在整体几何朗兰兹中的应用涉及还原群的循环群在具有整体性质的模空间上的作用，即具有水平结构的丛。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，被认为是值得支持的。

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.