BGG complexes in singular blocks 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 $\mathcal{O}$.

References in corpus (2)

• Singular BGG complexes over isotropic 2-Grassmannian
• Singular BGG complexes for the symplectic case

Cited by in corpus (3)

• A computer algorithm for the BGG resolution
• Some homological properties of category $\mathcal{O}$, VII
• Non symmetric Cauchy kernel, crystals and last passage percolation