BGG complexes in singular blocks of category O

Authors: Volodymyr Mazorchuk, Rafael Mrđen

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

https://doi.org/10.1016/j.jpaa.2020.106449

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 $\mathcal{O}$ .

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