The classification of blocks in BGG category $${\mathcal {O}}$$

Abstract

We classify all equivalences between the indecomposable abelian categories which appear as blocks in BGG category $${\mathcal {O}}$$ for reductive Lie algebras. Our classification implies that a block in category $${\mathcal {O}}$$ only depends on the Bruhat order of the relevant parabolic quotient of the Weyl group. As part of the proof, we observe that any finite dimensional algebra with simple preserving duality admits at most one quasi-hereditary structure.