BGG complexes in singular blocks of category O

Authors: Volodymyr Mazorchuk, Rafael Mrđen

Journal of Pure and Applied Algebra 224 (2020), Issue 12, 106449

Abstract: Using translation from the regular block, we construct and analyze properties of BGG complexes in singular blocks of BGG category \mathcal{O}. We provide criteria, in terms of the Kazhdan-Lusztig-Vogan polynomials, for such complexes to be exact. In the Koszul dual picture, exactness of BGG complexes is expressed as a certain condition on a generalized Verma flag of an indecomposable projective object in the corresponding block of parabolic category \mathcal{O}.

In the second part of the paper, we construct BGG complexes in a more general setting of balanced quasi-hereditary algebras and show how our results for singular blocks can be used to construct BGG resolutions of simple modules in \mathcal{S}-subcategories in \mathcal{O}.