BGG complexes in singular blocks of category O

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 ${mathcal{O}}$.