Explicit formulas for Eigenvalues of Capelli operators for the Lie superalgebra $\frak{osp}(1|2n)$

TL;DR

This paper derives explicit formulas for the eigenvalues of Capelli operators associated with the Lie superalgebra rak{osp}(1|2n), solving a previously unaddressed problem in the representation theory of orthosymplectic superalgebras.

Contribution

It provides explicit eigenvalue formulas for Capelli operators on rak{osp}(1|2n), completing the solution to the eigenvalue problem for these superalgebras.

Findings

Explicit eigenvalue formulas for Capelli operators derived.

The eigenvalue problem for rak{osp}(1|2n) addressed and solved.

Main technique involves calculating a determinant with polynomial entries.

Abstract

We define a natural basis for the algebra of $\frak{gosp}(1|2n)$-invariant differential operators on the affine superspace $\mathbb{C}^{1|2n}$. We prove that these operators lie in the image of the centre of the enveloping algebra of $\frak{gosp}(1|2n)$. Using this result, we compute explicit formulas for the eigenvalues of these operators on irreducible summands of $\mathcal{P}(\mathbb{C}^{1|2n})$. This settles the Capelli eigenvalue problem for orthosymplectic Lie superalgebras in the cases that were not addressed in Sahi-Salmasian-Serganova. Our main technique relies on an explicit calculation of a certain determinant with polynomial entries.