On the structure and characters of weight modules

Abstract

Let $\mathfrak{g}$ be a classical Lie superalgebra of type I or a Cartan-type Lie superalgebra {\bf W}$(n)$. We study weight $\mathfrak{g}$-modules using a method inspired by Mathieu's classification of the simple weight modules with finite weight multiplicities over reductive Lie algebras, \cite{M}. Our approach is based on the fact that every simple weight $\mathfrak{g}$-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 $\mathcal{O}$ 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 ${\mathfrak{g}}_0$-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 {\bf W}$(n)$-modules and all simple atypical nonsingular $\mathfrak{s}\mathfrak{l}(m|1)$-modules. Some of our results are new already in the case of a classical reductive Lie algebra $\mathfrak{g}$.

18 pages