On finite-dimensional representations of finite W-superalgebras

We first formulate and prove a version of Premet's conjecture for finite W-superalgebras associated with basic Lie superalgebras. As in the case of W-algebras, Premet's conjecture is very close to giving a classification of finite-dimensional simple modules of finite W-superalgebras. In the case of basic type I Lie superalgebras, we classify the finite-dimensional simple supermodules with the integral central character and give an algorithm to compute their characters based on the g((0) over bar)-rough structure of g-modules.