Simple smooth modules over the Lie superalgebra of polynomial vector fields ℂ1|1

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.