This is an old joint work with Finkelberg and Nakajima, which we finally decided to write down. We realize the universal enveloping algebra of a simple Lie algebra g inside the algebra of Ext's between IC sheaves on various strata on the so called Zastava space associated with the Langlands dual Lie algebra. This implies the Kazhdan-Lusztig conjecture for g by a very simple geometric argument. In the talk we shall also discuss the affine case: the above geometric realization of the universal enveloping algebra is still conjectural at the moment, but the argument which implies the Kazhdan-Lusztig conjecture goes through; in particular, it provides a uniform approach to the proof of the Kazhdan-Lusztig conjecture for representations of affine Lie algebras of all levels (while in the traditional approach via (an affine generalization of) Beilinson-Bernstein localization one has to treat the cases of positive, negative and critical level separately).

This is a part of the "GRT at home" seminar series, see grt-home.org