Local Geometric Langlands Correspondence and Representation Theory

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.