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 ${\mathcal{D}}^{\prime }(n)$ 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 ${\mathcal{D}}^{\prime }(n)\otimes U(\mathfrak{gl}(n))$ to an algebra containing $U$ as a subalgebra, so that $\sigma (\rho (u))-u\in G_{1}U$, for all $u\in U$, where $G_1={\sum }_{i=0}^n{E}_{ii}$.

21 pages