Mixed tensor products and Capelli-type determinants

In this paper we study properties of a homomorphism $rho$ from the universal enveloping algebra $U=U(mathfrak{gl}(n+1))$ to a tensor product of an algebra $mathcal D(n)$ of differential operators and $U(mathfrak{gl}(n))$. We find a formula for the image of the Capelli determinant of $mathfrak{gl}(n+1)$ under $rho$, and, in particular, of the images under $rho$ of the Gelfand generators of the center $Z(mathfrak{gl}(n+1))$ of $U$. This formula is proven by relating $rho$ to the corresponding Harish-Chandra isomorphisms, and, alternatively, by using a purely computational approach. Furthermore, we define a homomorphism from $mathcal D(n) otimes U(mathfrak{gl}(n))$ to an algebra containing $U$ as a subalgebra, so that $sigma (rho (u)) - u in G_1 U$, for all $u in U$, where $G_1 = sum_{i=0}^{n} E_{ii}$.