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Israel Journal of Mathematics

, Volume 219, Issue 2, pp 661–706 | Cite as

Dualities and derived equivalences for category \(\mathcal{O}\)

  • Kevin Coulembier
  • Volodymyr Mazorchuk
Article

Abstract

We determine the Ringel duals for all blocks in the parabolic versions of the BGG category \(\mathcal{O}\) associated to a reductive finite-dimensional Lie algebra. In particular, we find that, contrary to the original category \(\mathcal{O}\) and the specific previously known cases in the parabolic setting, the blocks are not necessarily Ringel self-dual. However, the parabolic category \(\mathcal{O}\) as a whole is still Ringel self-dual. Furthermore, we use generalisations of the Ringel duality functor to obtain large classes of derived equivalences between blocks in parabolic and original category \(\mathcal{O}\). We subsequently classify all derived equivalence classes of blocks of category \(\mathcal{O}\) in type A which preserve the Koszul grading.

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Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Department of Mathematical AnalysisGhent UniversityGentBelgium
  2. 2.Department of MathematicsUniversity of UppsalaUppsalaSweden

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