Let g be the Lie superalgebra of polynomial vector fields on Formula: see text and Formula: see text. In this paper, we first study the Kac module over a finite-dimensional Lie superalgebra Formula: see text, which is an induced module from a simple module over a certain finite-dimensional Lie algebra. We determine the simplicity of such module, and its submodules and quotient modules when it is not simple. Then based on these results, we classify all simple smooth Formula: see text-modules. We also give a characterization of simple smooth Formula: see text-modules. In addition, we describe all Whittaker vectors and submodules of the universal Whittaker Formula: see text-module.
Lü et al. (Fri,) studied this question.