Semi-infinite Cohomology, Quantum Group Cohomology, and the Kazhdan-Lusztig Equivalence

The Kazhdan-Lusztig tensor equivalence is a monoidal functor which sends modules over an affine Lie algebra at negative level to modules over the associated quantum group. A positive level Kazhdan-Lusztig functor is defined using Arkhipov-Gaitsgory's duality between positive and negative level affine Lie algebras. Our main result proves that under the positive level Kazhdan-Lusztig functor, the semi-infinite cohomology functor corresponds to the quantum group cohomology functor with respect to the positive part of Lusztig's quantum group. Monoidal structures of a category can be interpreted as factorization data on the associated global category. We describe a conjectural reformulation of the Kazhdan-Lusztig tensor equivalence in factorization terms. In this reformulation, the semi-infinite cohomology functor at positive level is naturally factorizable, and it is conjectured that the factorizable semi-infinite cohomology functor is essentially the positive level Kazhdan-Lusztig tensor functor modulo the Riemann-Hilbert correspondence. Our main result provides an important technical tool in a proposed approach to a proof of this conjecture.Ph.D