Mixed Tensor Products, Capelli Berezinians, and Newton's Formula for $\mathfrak{gl}(m|n)$

Abstract

In this paper, we extend the results of Grantcharov and Robitaille in 2021 on mixed tensor products and Capelli determinants to the superalgebra setting. Specifically, we construct a family of superalgebra homomorphisms ${\phi }_R:U(\mathfrak{gl}(m+1|n))\to {\mathcal{D}}^{\prime }(m|n)\otimes U(\mathfrak{gl}(m|n))$ for a certain space of differential operators ${\mathcal{D}}^{\prime }(m|n)$ indexed by a central element $R$ of ${\mathcal{D}}^{\prime }(m|n)\otimes U(\mathfrak{gl}(m|n))$. We then use this homomorphism to determine the image of Gelfand generators of the center of $U(\mathfrak{gl}(m+1|n))$. We achieve this by first relating ${\phi }_R$ to the corresponding Harish-Chandra homomorphisms and then proving a super-analog of Newton's formula for $\mathfrak{gl}(m)$ relating Capelli generators and Gelfand generators. We also use the homomorphism ${\phi }_R$ to obtain representations of $U(\mathfrak{gl}(m+1|n))$ from those of $U(\mathfrak{gl}(m|n))$, and find conditions under which these inflations are simple. Finally, we show that for a distinguished central element $R_1$ in ${\mathcal{D}}^{\prime }(m|n)\otimes U(\mathfrak{gl}(m|n))$, the kernel of ${\phi }_{R_1}$ is the ideal of $U(\mathfrak{gl}(m+1|n))$ generated by the first Gelfand invariant $G_1$.

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