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

We re-interpret Goodwin’s translation functors for a finite $W$-algebra $H_{\mathrm{\ell }}$ as an action of a monoidal subcategory of $U(\mathfrak{g})$-mod on the category of finitely generated $H_{\mathrm{\ell }}$-modules. This action is obtained by transporting the tensor product of $U(\mathfrak{g})$-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 $U(\mathfrak{g})$-module categories.