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

TL;DR

This paper establishes an isomorphism between a minimal nilpotent finite W-algebra associated with sp_{2n} and a centralizer in a localized universal enveloping algebra, linking it to a specific module category.

Contribution

It provides a new description of the minimal nilpotent finite W-algebra for sp_{2n} as a tensor factor in a localized algebra and relates module categories to this algebra.

Findings

W(rak{sp}_{2n}, e) is isomorphic to a centralizer in U_S.

The category of certain weight modules is equivalent to modules over W(rak{sp}_{2n}, e).

Both categories are shown to be semi-simple.

Abstract

Let $U_S$ be the localization of $U(\mathfrak{sp}_{2n})$ 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 $W(\mathfrak{sp}_{2n}, e)$ 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 $\mathfrak{sp}_{2n}$-modules with injective actions of all root vectors and finite-dimensional weight spaces is equivalent to the category of finite-dimensional modules over $W(\mathfrak{sp}_{2n}, e)$, explaining the coincidence that both of them are semi-simple.