On the Structure of Weight

Given any simple Lie superalgebra g, we investigate the structure of an arbitrary simple weight g-module. We introduce two invariants of simple weight modules: the shadow and the small Weyl group. Generalizing results of Fernando and Futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a Levi subsuper-algebra. Then we classify the cuspidal Levi subsuperalgebras of all simple classical Lie superalgebras and of the Lie superalgebra W(n). Most of them are simply Levi subalgebras of g 0 , in which case the classiication of all nite cuspidal representations has recently been carried out by one of us in M]. Our results reduce the classiication of the nite simple weight modules over all classical simple Lie superalgebras to classifying the nite cuspidal modules over certain Lie superalgebras which we list explicitly. Table of Contents 1. Notations and conventions. 2. Generalities about generalized weight modules. 3. The shadow and a generalization of Fernando-Futorny's theorem. 4. A cohomological characterization of nite cuspidal modules. 5. The small Weyl group. 6. Reduction of the classiication problem. 7. A classiication of cuspidal Levi subsuperalgebras. 8. Conclusion. Introduction In order to be able to explain the topic of this paper and to state the results, we need to start with a few deenitions. The remaining deenitions needed are given in section 1. Some of the results apply to arbitrary nite dimensional Lie superalgebras and will be stated in this way.