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 for arbitrary simple Lie algebra, thus answering a question which goes back to I. M. Gelfand. Dedicated to George Lusztig on the occasion of his 60-th birthdayMathematics Subject Classification (1991)  14F43, 17B10, 32S60