Dualities and derived equivalences for category $$\mathcal{O}$$ ℒ

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.