Mixed tensor products and Capelli-type determinants

In this paper we study properties of a homomorphism $ρ$ 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 $ρ$, and, in particular, of the images under $ρ$ of the Gelfand generators of the center $Z(\mathfrak{gl}(n+1))$ of $U$. This formula is proven by relating $ρ$ 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 $σ(ρ(u)) - u \in G_1 U$, for all $u \in U$, where $G_1 = \sum_{i=0}^{n} E_{ii}$.

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