A monoidal category viewpoint for translation functors for finite W-algebras

Abstract

We re-interpret Goodwin's translation functors for a finite W-algebra $H_{\ell }$ as an action of a monoidal subcategory of $U\left(g\right)$-mod on the category of finitely generated $H_{\ell }$-modules. This action is obtained by transporting the tensor product of $U\left(g\right)$-modules through Skryabin's equivalence. We apply this interpretation to show that the Skryabin equivalence by stages introduced by Genra and Juillard is an equivalence of $U\left(g\right)$-module categories.