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

TL;DR

This paper computes explicit eigenvalues of Capelli operators for certain Lie superalgebras, generalizing previous results by expressing them via interpolation super Jack polynomials evaluated at specific weights.

Contribution

It provides a refined and explicit formula for Capelli eigenvalues in the context of $rak{gl}(m|n)$ superalgebras, extending prior work to more general Borel subalgebras.

Findings

Eigenvalues expressed via interpolation super Jack polynomials

Explicit formulas for $rak{gl}(m|n)$ and $rak{gl}(m|2n)$ cases

Generalization of previous fixed Borel subalgebra results

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 $W:=V\otimes V^*$ 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 $\left\{D^\lambda\right\}_{\lambda\in\Omega}$, naturally indexed by a set of hook partitions $\Omega$, 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 $D^\lambda$ 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.