An algebra isomorphism on $U(\mathfrak{gl}_n)$

Abstract: For each positive integer $n$, let $\mathfrak{s}n=\mathfrak{gl}_n\ltimes \mathbb{C}^n$. We show that $U(\mathfrak{s}{n}){X{n}}\cong \mathcal{D}{n}\otimes U(\mathfrak{s}{n-1})$ for any $n\in\mathbb{Z}{\geq 2}$, where $U(\mathfrak{s}{n}){X{n}}$ is the localization of $U(\mathfrak{s}{n})$ with respect to the subset $X_n:={e_1^{i_1}\cdots e{n}^{i_{n}}\mid i_1,\dots,i_{n}\in \mathbb{Z}+}$, and $\mathcal{D}{n}$ is the Weyl algebra $\mathbb{C}[x_1^{\pm 1}, \cdots, x_{n}^{\pm 1}, \frac{\partial}{\partial x_1},\cdots, \frac{\partial}{\partial x_{n}}]$. As an application, we give a new proof of the Gelfand-Kirillov conjecture for $\mathfrak{s}n$ and $\mathfrak{gl}_n$. Moreover we show that the category of Harish-Chandra $U(\mathfrak{s}{n}){X_n}$-modules with a fixed weight support is equivalent to the category of finite dimensional $\mathfrak{s}{n-1}$-modules whose representation type is wild, for any $n\in \mathbb{Z}_{\geq 2}$.