Mixed tensor products and Capelli-type determinants

Abstract

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 $G_1 = \sum_{i=0}^{n} E_{ii}$.