  A positive level Kazhdan-Lusztig functor is defined using Arkhipov-Gaitsgory
duality for affine Lie algebras. The functor sends objects in the DG category
of G(O)-equivariant positive level affine Lie algebra modules to objects in the
DG category of modules over Lusztig's quantum group at a root of unity. We
prove that the semi-infinite cohomology functor for positive level modules
factors through the Kazhdan-Lusztig functor at positive level and the quantum
group cohomology functor with respect to the positive part of Lusztig's quantum
group.
