Perverse sheaves, Koszul IC-modules, and the quiver for the category \mathcal{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 \(\mathcal{O}\) for arbitrary simple Lie algebra, thus answering a question which goes back to I. M. Gelfand.