An Extension of the Kazhdan-Lusztig Equivalence

Abstract

We establish an equivalence between the DG category of Iwahori–integrable affine Lie algebra representations and the DG category of representations of the “mixed” quantum group, thus confirming a conjecture made by D. Gaitsgory. This result is a tamely ramified version of an equivalence established by D. Kazhdan and G. Lusztig in the spherically integrable setting. Our proof, which utilizes the notion of factorization algebras introduced by A. Beilinson and V. Drinfeld, is independent from the original Kazhdan–Lusztig equivalence. Along the way we establish several results in the quantum local geometric Langlands program.

The work reported here was done jointly with L. Chen.