Representations of shifted super Yangians and finite $W$-superalgebras of type A

Abstract

In this article, we study the representation theory of shifted super Yangians and finite $W$-superalgebras of type A. A criterion for the finite dimensionality of irreducible modules is obtained in the standard parity case. Furthermore, we provide an explicit Gelfand-Tsetlin character formula for Verma modules of finite $W$-superalgebras. As an application, we show that the centers of the finite $W$-superalgebras associated to any even nilpotent elements belonging to the same general linear Lie superalgebra are all isomorphic to the center of the universal enveloping superalgebra.