Dualities and derived equivalences for category O

TL;DR

This paper explores the structure of parabolic category O for reductive Lie algebras, revealing new dualities and classifying derived equivalences, especially in type A, enhancing understanding of its algebraic and geometric properties.

Contribution

It determines the Ringel duals for all blocks in parabolic category O and classifies derived equivalence classes in type A that preserve Koszul grading, extending prior geometric approaches.

Findings

Parabolic blocks are not necessarily Ringel self-dual.

The entire parabolic category O remains Ringel self-dual.

Classified all derived equivalence classes of blocks in type A that preserve Koszul grading.

Abstract

We determine the Ringel duals for all blocks in the parabolic versions of the BGG category O associated to a reductive finite dimensional Lie algebra. In particular we find that, contrary to the original category O and the specific previously known cases in the parabolic setting, the blocks are not necessarily Ringel self-dual. However, the parabolic category 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 O. It seems that only some special cases were known thus far, where the proof relied purely on a geometric approach to category O. We subsequently classify all derived equivalence classes of blocks of category O in type A which preserve the Koszul grading.