On the structure of weight modules

Given any simple Lie superalgebra ${\mathfrak{g}}$, we investigate the structure of an arbitrary simple weight ${\mathfrak{g}}$-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 subsuperalgebra. 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 ${\mathfrak{g}}_{0}$, in which case the classification of all finite cuspidal representations has recently been carried out by one of us (Mathieu). Our results reduce the classification of the finite simple weight modules over all classical simple Lie superalgebras to classifying the finite cuspidal modules over certain Lie superalgebras which we list explicitly.