The refined solution to the Capelli eigenvalue problem for gl(m|n)⊕gl(m|n) and gl(m|2n)

Abstract

Let $g$ be either the Lie superalgebra $\mathrm{gl}\left(V\right)\oplus \mathrm{gl}\left(V\right)$ where $V≔{ℂ}^{m|n}$ or the Lie superalgebra $\mathrm{gl}\left(V\right)$ where $V≔{ℂ}^{m|2n}$. Furthermore, let $W$ be the $g$-module defined by $W≔V\otimes V^{\ast }$ in the former case and $W≔S^2\left(V\right)$ in the latter case. Associated to $(g,W)$ there exists a distinguished basis of Capelli operators ${\left\{D^{\lambda }\right\}}_{\lambda \in \Omega }$, naturally indexed by a set of hook partitions $\Omega$, for the subalgebra of $g$-invariants in the superalgebra $\mathrm{PD}\left(W\right)$ of superdifferential operators on $W$.

Let $b$ be a Borel subalgebra of $g$. We compute eigenvalues of the $D^{\lambda }$ on the irreducible $g$-submodules of $P\left(W\right)$ and obtain them explicitly as the evaluation of the interpolation super Jack polynomials of Sergeev–Veselov at suitable affine functions of the $b$-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.