A generalization of the category O of Bernstein–Gelfand–Gelfand

In the study of simple modules over a simple complex Lie algebra, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting for highest weight modules. In this note, we define a family of categories which generalizes the BGG category. We classify the simple modules for some of these categories. As a consequence we show that these categories are semisimple. To cite this article: 1. Weight modules and Generalized Verma Modules Let g denote a simple Lie algebra over C and U(g) denote its universal enveloping algebra. Let h be a fixed Cartan subalgebra and denote by R the corresponding set of roots. For α ∈ R we denote by Hα ∈ h the corresponding co–root and by g the rootspace for the root α. We will denote by ∆ a set of simple roots. We write R for the corresponding set of positive roots. For a subset θ ⊂ ∆, we denote by 〈θ〉 the set of all roots which are linear combination of elements of θ and set 〈θ〉 = 〈θ〉 ∩ R. We consider the following subalgebras of g : lθ = h ⊕ ∑