Koszul duality patterns in representation theory

The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to representation theory. The paper consists of three parts relatively independent of each other. The first part gives a reasonably selfcontained introduction to Koszul rings and Koszul duality. Koszul rings are certain Z-graded rings with particularly nice homological properties which involve a kind of duality. Thus, to a Koszul ring one associates naturally the dual Koszul ring. The second part is devoted to an application to representation theory of semisimple Lie algebras. We show that the block of the Bernstein-Gelfand-Gelfand category O that corresponds to any fixed central character is governed by the Koszul ring. Moreover, the dual of that ring governs a certain subcategory of the category O again. This generalizes the selfduality theorem conjectured by Beilinson and Ginsburg in 1986 and proved by Soergel in 1990. In the third part we study certain categories of mixed perverse sheaves on a variety stratified by affine linear spaces. We provide a general criterion for such a category to be governed by a Koszul ring. In the flag variety case this reduces to the setup of part two. In the more general case of affine flag manifolds and affine Grassmannians the criterion should yield interesting results about representations of quantum groups and affine Lie algebras.