On the structure and characters of weight modules

Abstract

Let 𝔤 be a classical Lie superalgebra of type I or a Cartan-type Lie superalgebra W(n). We study weight 𝔤-modules using a method inspired by Mathieu's classification of the simple weight modules with finite weight multiplicities over reductive Lie algebras, [Mathieu O.: Classification of irreducible weight modules. Ann. Inst. Fourier 50 (2000), 537–592]. Our approach is based on the fact that every simple weight 𝔤-module with finite weight multiplicities is obtained via a composition of a twist and localization from a highest weight module. This allows us to transfer many results for category 𝒪 modules to the category of weight modules with finite weight multiplicities. As a main application of the method we reduce the problems of finding a 𝔤0-composition series and a character formula for all simple weight modules to the same problems for simple highest weight modules. In this way, using results of Serganova we obtain a character formula for all simple weight W(n)-modules and all simple atypical nonsingular -modules. Some of our results are new already in the case of a classical reductive Lie algebra 𝔤.