Perverse sheaves, Koszul IC-modules, and the quiver for the category O

For a stratified topological space we introduce the category of IC-modules, which are linear algebra devices with the relations described by the equation d^2=0. We prove that the category of (mixed) IC-modules is equivalent to the category of (mixed) perverse sheaves for flag varieties. As an application, we describe an algorithm calculating the quiver underlying the BGG category O for arbitrary simple Lie algebra, thus answering a question which goes back to I. M. Gelfand.Comment: NEW: final version to appear in Inventiones Mathematicae. Some exposition revisions. Quiver computed explicitly for A_2 in the appendix. 29 page

Quiver varieties, affine Lie algebras, algebras of BPS states, and semicanonical basis

We suggest a (conjectural) construction of a basis in the plus part of the affine Lie algebra of type ADE indexed by irreducible components of certain quiver varieties. This construction is closely related to a string-theoretic construction of a Lie algebra of BPS states. We then study the new combinatorial questions about the (classical) root systems naturally arising from our constructions and Lusztig's semicanonical basis.Comment: 16 page