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

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 ${\mathbb{C}}^{m|n}\otimes ({\mathbb{C}}^{m|n}{)}^{\ast }$. We also give an affirmative answer to the abstract Capelli problem for these cases.

References in corpus (1)

• Spherical representations of Lie supergroups

Cited by in corpus (1)

• Quadratic Capelli operators and Okounkov polynomials