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

Abstract

For each positive integer $n$, let ${\mathfrak{s}}_n={\mathfrak{gl}}_n⋉{\mathbb{C}}^n$. We show that $U({\mathfrak{s}}_n{)}_{X_n}\cong {\mathcal{D}}_n\otimes U({\mathfrak{s}}_{n-1})$ for any $n\in {\mathbb{Z}}_{\ge 2}$, 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 $n\in {\mathbb{Z}}_{\ge 2}$.