Koszul duality for parabolic and singular category $\mathcal O$

Abstract:

This paper deals with a generalization of the “Koszul duality theorem” for the Bernstein-Gelfand-Gelfand category $\mathcal O$ over a complex semi-simple Lie-algebra, established by Beilinson, Ginzburg and Soergel in Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473–527. In that paper it was proved that any “block” in $\mathcal O$, determined by an integral, but possibly singular weight, is Koszul (i.e. equivalent to the category of finitely generated modules over some Koszul ring) and, moreover, that the “Koszul dual” of such a block is isomorphic to a “parabolic subcategory” of the trivial block in $\mathcal O$. We extend these results to prove that a parabolic subcategory of an integral and (possibly) singular block in $\mathcal O$ is Koszul and we also calculate the Koszul dual of such a category.