Quantized Weyl algebras, the double centralizer property, and a new First Fundamental Theorem for $U_q({\mathfrak{g}\mathfrak{l}}_n)$

Let $\mathcal{P}:={\mathcal{P}}_{m\times n}$ denote the quantized coordinate ring of the space of $m\times n$ matrices, equipped with natural actions of the quantized enveloping algebras $U_q({\mathfrak{g}\mathfrak{l}}_m)$ and $U_q({\mathfrak{g}\mathfrak{l}}_n)$. Let $\mathcal{L}$ and $\mathcal{R}$ denote the images of $U_q({\mathfrak{g}\mathfrak{l}}_m)$ and $U_q({\mathfrak{g}\mathfrak{l}}_n)$ in $\mathrm{E}\mathrm{n}\mathrm{d}(\mathcal{P})$, respectively. We define a $q$-analogue of the algebra of polynomial-coefficient differential operators inside $\mathrm{E}\mathrm{n}\mathrm{d}(\mathcal{P})$, henceforth denoted by $\mathcal{P}\mathcal{D}$, and we prove that $\mathcal{L}\cap \mathcal{P}\mathcal{D}$ and $\mathcal{R}\cap \mathcal{P}\mathcal{D}$ are mutual centralizers inside $\mathcal{P}\mathcal{D}$. Using this, we establish a new First Fundamental Theorem of invariant theory for $U_q({\mathfrak{g}\mathfrak{l}}_n)$. We also compute explicit formulas in terms of $q$-determinants for generators of the intersections with $\mathcal{P}\mathcal{D}$ of the images of the Cartan subalgebras of $U_q({\mathfrak{g}\mathfrak{l}}_m)$ and $U_q({\mathfrak{g}\mathfrak{l}}_n)$.