Kazhdan-Lusztig conjecture for affine Lie algebras via Uhlenbeck spaces

We present a new geometric approach to Kazhdan-Lusztig type conjectures for finite and affine Lie algebras (the actual results are new only in the case of affine Lie algebras and critical level). One nice property of this approach is that allows to treat the cases of representation of affine Lie algebras with positive, negative and critical level in a uniform way.
This is a joint work in progress with M.Finkelberg and H.Nakajima.