Minimal nilpotent finite $W$-algebra and cuspidal module category of ${\mathfrak{s}\mathfrak{p}}_{2n}$

Let $U_S$ be the localization of $U({\mathfrak{s}\mathfrak{p}}_{2n})$ with respect to the Ore subset $S$ generated by the root vectors $X_{{\epsilon }_1-{\epsilon }_2},\dots ,X_{{\epsilon }_1-{\epsilon }_n},X_{2{\epsilon }_1}$. We show that the minimal nilpotent finite $W$-algebra $W({\mathfrak{s}\mathfrak{p}}_{2n},e)$ is isomorphic to the centralizer $C_{U_S}(B)$ of some subalgebra $B$ in $U_S$, and it can be identified with a tensor product factor of $U_S$. As an application, we show that the category of weight ${\mathfrak{s}\mathfrak{p}}_{2n}$-modules with injective actions of all root vectors and finite-dimensional weight spaces is equivalent to the category of finite-dimensional modules over $W({\mathfrak{s}\mathfrak{p}}_{2n},e)$, explaining the coincidence that both of them are semi-simple.