Translation for finite 𝑊-algebras

Abstract

A finite W W -algebra U ( g , e ) U(\mathfrak {g},e) is a certain finitely generated algebra that can be viewed as the enveloping algebra of the Slodowy slice to the adjoint orbit of a nilpotent element e e of a complex reductive Lie algebra g \mathfrak {g} . It is possible to give the tensor product of a U ( g , e ) U(\mathfrak {g},e) -module with a finite dimensional U ( g ) U(\mathfrak {g}) -module the structure of a U ( g , e ) U(\mathfrak {g},e) -module; we refer to such tensor products as translations. In this paper, we present a number of fundamental properties of these translations, which are expected to be of importance in understanding the representation theory of U ( g , e ) U(\mathfrak {g},e) .