Local Geometric Langlands Correspondence and Representation Theory

表示理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，分类不可约表示），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）。本项目旨在研究高级表征理论，该理论研究群作为绝对对称的实现。该建议的重点集中在环路群，其中的理论显著反映了p进群的经典谐波分析。特别地，我们在这里发现朗兰兹式分解。这个项目的重点是理解这个框架中感兴趣的一些关键类别。研究者将研究三维镜像对称猜想，仿射李代数的表示，以及在全局几何朗兰兹规划中产生的束的模空间。这个项目为研究生提供了训练的机会。更详细地说，三维镜像对称、（约化）仿射李代数的表示和几何朗兰兹规划是约化群的环群作用于范畴的三种主要方式。一大类三维镜像对称猜想涉及到具有群作用的特定变异的环空间上的束的范畴上的环群作用的范畴Plancherel公式。PI将建立三维镜像对称的第一个案例，并将结果应用于对几何表示理论中主要感兴趣的一些类别的连贯描述。李代数的表示涉及群对其李代数表示范畴的作用。PI将扩展先前在临界水平定位理论方面的工作，并开发一个Soergel模块的替代品，该模块将适用于局部几何朗兰兹程序中理解不透彻的类别。在全局几何朗兰的应用涉及到约化群的环群在全局性质的模空间上的作用，即具有水平结构的束。PI将推广Satake定理，并将结果应用于全局几何Langlands规划中的Eisenstein级数的研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.