# BGG complexes in singular blocks of category O

## TL;DR

This paper extends the theory of BGG complexes to singular blocks of category O, providing criteria for their exactness and constructing resolutions using Kazhdan-Lusztig-Vogan polynomials and generalized Verma modules.

## Contribution

It introduces new criteria for the exactness of BGG complexes in singular blocks and generalizes the construction to balanced quasi-hereditary algebras.

## Key findings

- Criteria for BGG complex exactness using Kazhdan-Lusztig-Vogan polynomials
- Construction of BGG complexes in balanced quasi-hereditary algebras
- Resolutions of simple modules in subcategories of category O

## 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 ${\mathcal{O}}$.