On the variety of 1-dimensional representations of finite W-algebras in low rank

Let g be a simple Lie algebra over C and let e∈g be nilpotent. We consider the finite W-algebra U(g,e) associated to e and the problem of determining the variety E(g,e) of 1-dimensional representations of U(g,e). For g of low rank, we report on computer calculations that have been used to determine the structure of E(g,e), and the action of the component group Γe of the centralizer of e on E(g,e). As a consequence, we provide two examples where the nilpotent orbit of e is induced, but there is a 1-dimensional Γe-stable U(g,e)-module which is not induced via Losev's parabolic induction functor. In turn this gives examples where there is a “non-induced” multiplicity free primitive ideal of U(g).