An Extension of the Kazhdan-Lusztig Equivalence

TL;DR

This paper proves a tamely ramified version of the Kazhdan-Lusztig equivalence, establishing a new connection between affine Lie algebra representations and quantum group representations using factorization algebras.

Contribution

It extends the Kazhdan-Lusztig equivalence to the tamely ramified case, confirming a conjecture by Gaitsgory and employing factorization algebra techniques.

Findings

Established an equivalence between Iwahori-integrable affine Lie algebra and mixed quantum group representations.

Confirmed Gaitsgory's conjecture on tamely ramified Kazhdan-Lusztig equivalence.

Utilized factorization algebras to prove the equivalence.

Abstract

We prove a tamely ramified version of the Kazhdan-Lusztig equivalence using factorization algebras. More precisely, 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. This confirms a conjecture by D. Gaitsgory in arXiv:1810.09054 [math.RT].