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

Author: Erik Backelin
Journal: Represent. Theory 3 (1999), 139-152
MSC (1991): Primary 17B10, 18G15, 17B20
Published electronically: July 19, 1999
MathSciNet review: 1703324
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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.

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Erik Backelin
Affiliation: Department of Mathematics, Albert-Ludwigs-Universitat, Eckerstr. 1, D-79104 Freiburg im Briesgau, Germany

Received by editor(s): August 24, 1998
Received by editor(s) in revised form: January 31, 1999
Published electronically: July 19, 1999
Article copyright: © Copyright 1999 American Mathematical Society