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

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.