Coherent families of weight modules of Lie superalgebras and an explicit description of the simple admissible -modules

Let be a classical Lie superalgebra of type I. We introduce coherent families of weight -modules with bounded weight multiplicities, and establish a correspondence between cuspidal and highest weight submodules of these families by extending Mathieuʹs work [Ann. Inst. Fourier 50 (2000) 537]. This enables us to reduce the description of the -module structure of arbitrary simple weight -modules with bounded weight multiplicities to the -module structure of highest weight modules with bounded weight multiplicities. We then construct tensor coherent families for which yield an explicit description of the -structure of an arbitrary simple weight module with bounded weight multiplicities. In particular, we show that for , the maximal length of an indecomposable -submodule of a simple weight module with bounded multiplicities equals 5.