The refined solution to the Capelli eigenvalue problem for $\mathfrak{gl}(m|n)\oplus \mathfrak{gl}(m|n)$ and $\mathfrak{gl}(m|2n)$

Abstract

Let $\mathfrak{g}$ be either the Lie superalgebra $\mathfrak{gl}(V)\oplus \mathfrak{gl}(V)$ where $V:={\mathbb{C}}^{m|n}$ or the Lie superalgebra $\mathfrak{gl}(V)$ where $V:={\mathbb{C}}^{m|2n}$. Furthermore, let $W$ be the $\mathfrak{g}$-module defined by $W:=V\otimes V^{\ast }$ in the former case and $W:={\mathcal{S}}^2(V)$ in the latter case. Associated to $(\mathfrak{g},W)$ there exists a distinguished basis of Capelli operators ${\left\{D^{\hat{I}}»\right\}}_{\hat{I}»\in \hat{I}\text{c◯}}$, naturally indexed by a set of hook partitions $\hat{I}\text{c◯}$, for the subalgebra of $\mathfrak{g}$-invariants in the superalgebra $\mathcal{PD}(W)$ of superdifferential operators on $W$. Let $\mathfrak{b}$ be a Borel subalgebra of $\mathfrak{g}$. We compute eigenvalues of the $D^{\hat{I}}»$ on the irreducible $\mathfrak{g}$-submodules of $\mathcal{P}(W)$ and obtain them explicitly as the evaluation of the interpolation super Jack polynomials of Sergeev--Veselov at suitable affine functions of the $\mathfrak{b}$-highest weight. While the former case is straightforward, the latter is significantly more complex. This generalizes a result by Sahi, Salmasian and Serganova for these cases, where such formulas were given for a fixed choice of Borel subalgebra.