In 1976 I. N. Bernstein, I. M. Gel’fand and S. I. Gel’fand introduced the category \(\mathcal{O}\) for a complex semisimple Lie algebra \(\mathfrak{g}\) [Sel. Math. Sov. 1, 121-142 (1981); translation from Tr. Semin. Im. I. G. Petrovskogo 2, 3-21 (1976; Zbl 0499.22004)]. Two basic properties were established there, namely the block decomposition of \(\mathcal{O}\) (these blocks being equivalent to a module category over a quasi-hereditary algebra) and the BGG reciprocity principle between Verma, projective and simple modules.
In this paper the authors analyze subcategories of \(\mathcal{O}\) consisting of complete modules admitting quasi-Verma flags. It is proven that the subcategories are admissible in the sense used in another paper by the the authors [Algebr. Represent. Theory 5, 259-276 (2002; Zbl 1031.17008)]. It is proven that these categories are Abelian (this property is not inherited by the category \(\mathcal{O}\)). A combinatorial description for these subcategories analogous to that existing for \(\mathcal{O}\) is obtained. Using the double centralizer property of Soergel, \(\mathcal{S}\)-subcategories are introduced, and new proofs of the character formula for the tilting module of \(\mathcal{O}\) [W. Soergel, Represent. Theory 2, 432-448 (1998; Zbl 0964.17018)] and the self-duality of Ringel for the principal block of \(\mathcal{O}\) are obtained. The proof use the Enright completion functor [T. J. Enright, Ann. Math. (2) 110, 1-82 (1979; Zbl 0417.17005)].