Kazhdan-Lusztig Equivalence at the Iwahori Level

We construct an equivalence between Iwahori-integrable representations of affine Lie algebras and representations of the "mixed" quantum group, thus confirming a conjecture by Gaitsgory. Our proof utilizes factorization methods: we show that both sides are equivalent to algebraic/topological factorization modules over a certain factorization algebra, which can then be compared via Riemann-Hilbert. On the quantum group side this is achieved via general machinery of homotopical algebra, whereas the affine side requires inputs from the theory of (renormalized) ind-coherent sheaves as well as compatibility with global geometric Langlands over P1. This is joint work with Lin Chen.