On the Premet conjecture for finite W-superalgebras

Abstract

Let $\bullet^†$ be the map in sense of the Losev, which sends the set of two sided ideals of a finite W-algebras to that of the universal enveloping algebra of corresponding Lie algebras. The Premet conjecture which was proved in \cite{Lo11}, says that, restricted to the set of primitive ideals with finite codimension, any fiber of the map $\bullet^†$ is a single orbit under an action of a finite group. In this article we formulate and prove a similar fact in the super case. This will give a classification to the set of finite dimensional irreducible representations of W-superalgebras provided $C_{e}$ is a trivial group and the set of primitive ideals of the corresponding universal enveloping algebra of Lie superalgebra is known.