Singular blocks of the category \({\mathcal O}\) (Q752154)

Let \({\mathcal O}\) denote the Bernstein-Gelfand-Gelfand category of modules for a complex semisimple Lie algebra. Then \({\mathcal O}\) decomposes into blocks corresponding to central characters. In this paper it is proved that several results known to hold for regular blocks also hold for singular blocks. For instance, it is proved that the Verma modules in a singular block have unique Loewy series and that their composition factor multiplicities are given by certain Kazhdan-Lusztig polynomials. The key to the proofs is \textit{W. Soergel}'s recent purity result [Invent. Math. 98, 565-580 (1989)].