Koszul Duality Patterns in Representation Theory

The aim of this paper is to work out a concrete example as wellas to provide the general pattern of applications of Koszul duality to repre-sentation theory. The paper consists of three parts relatively independent ofeach other.The first part gives a reasonably selfcontained introduction to Koszul ringsand Koszul duality. Koszul rings are certain Z-graded rings with particularlynice homological properties which involve a kind of duality. Thus, to a Koszulring one associates naturally the dual Koszul ring. The second part is devotedto an application to representation theory of semisimple Lie algebras. We show License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use KOSZUL DUALITY PATTERNS527 that the block of the Bernstein-Gelfand-Gelfand category O that correspondsto any fixed central character is governed by the Koszul ring. Moreover, thedual of that ring governs a certain subcategory of the category O again. Thisgeneralizes the selfduality theorem conjectured by Beilinson and Ginsburg in1986 and proved by Soergel in 1990. In the third part we study certain cate-gories 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 Koszulring. In the flag variety case this reduces to the setup of part two. In the moregeneral case of affine flag manifolds and affine Grassmannians the criterionshould yield interesting results about representations of quantum groups andaffine Lie algebras. Department of Mathematics, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139E-mail address: [email protected] Department of Mathematics, The University of Chicago, Chicago, Illinois 60637E-mail address: [email protected] Max-Planck-Institut für Mathematik, Gottfried-Claren-Straße 26, D-53 Bonn 3,GermanyCurrent address: Mathematisches Institut, Universität Freiburg, Albertstraße 23b, D-79104Freiburg, GermanyE-mail address: [email protected] License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use