The Capelli problem for $\mathfrak{gl}(m|n)$ and the spectrum of invariant differential operators

TL;DR

This paper extends the Capelli problem to Lie superalgebras, connecting invariant differential operators with Jack polynomials in the supersymmetric setting, and provides solutions for new algebraic pairs.

Contribution

It generalizes the Capelli problem to supersymmetric pairs involving $rak{gl}(m|n)$ and $rak{osp}(m|2n)$, establishing new links with Jack polynomials.

Findings

Extended the Capelli problem to supersymmetric pairs.

Connected invariant differential operators with Jack polynomials.

Provided affirmative solutions for the abstract Capelli problem in new cases.

Abstract

The "Capelli problem" for the symmetric pairs $(\mathfrak{gl}\times \mathfrak{gl},\mathfrak{gl})$ $(\mathfrak{gl},\mathfrak{o})$, and $(\mathfrak{gl},\mathfrak{sp})$ is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter. In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs $(\mathfrak{g},\mathfrak{g}):=(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$ and $(\mathfrak{gl}(m|n)\times\mathfrak{gl}(m|n),\mathfrak{gl}(m|n))$, acting on $W:=S^2(\mathbb C^{m|2n})$ and $\mathbb C^{m|n}\otimes(\mathbb C^{m|n})^*$. We also give an affirmative answer to the abstract Capelli problem for these cases.