D-modules on the affine flag variety and representations of affine Kac-Moody algebras

TL;DR

This paper establishes deep connections between modules over affine Kac-Moody algebras at the critical level, D-modules on affine flag varieties, and sheaves on Miura opers, advancing the geometric Langlands program.

Contribution

It proves a localization equivalence for Kac-Moody modules and D-modules, and confirms a conjecture relating modules to sheaves on Miura opers.

Findings

Established an equivalence between certain Kac-Moody modules and D-modules on affine flag varieties.

Proved a correspondence between Kac-Moody modules and quasi-coherent sheaves on Miura opers.

Confirmed a conjecture linking representation theory and geometric Langlands duality.

Abstract

We study the connection between the category of modules over the affine Kac-Moody Lie algebra at the critical level, and the category of D-modules on the affine flag scheme G((t))/I, where I is the Iwahori subgroup. We prove a localization-type result, which establishes an equivalence between certain subcategories on both sides. We also establish an equivalence between a certain subcategory of Kac-Moody modules, and the category of quasi-coherent sheaves on the scheme of Miura opers for the Langlands dual group, thereby proving a conjecture of [FG2].