Capelli operators for spherical superharmonics and the Dougall-Ramanujan identity

Abstract

Let $(V,\omega )$ be an orthosympectic ${\mathbb{Z}}_2$-graded vector space and let $\mathfrak{g}:=\mathfrak{gosp}(V,\omega )$ denote the Lie superalgebra of similitudes of $(V,\omega )$. When the space $\mathcal{P}(V)$ of superpolynomials on $V$ is \emph{not} a completely reducible $\mathfrak{g}$-module, we construct a natural basis $D_{\lambda }$ of Capelli operators for the algebra of $\mathfrak{g}$-invariant superpolynomial superdifferential operators on $V$, where the index set $\mathcal{P}$ is the set of integer partitions of length at most two. We compute the action of the operators $D_{\lambda }$ on maximal indecomposable components of $\mathcal{P}(V)$ explicitly, in terms of Knop-Sahi interpolation polynomials. Our results show that, unlike the cases where $\mathcal{P}(V)$ is completely reducible, the eigenvalues of a subfamily of the $D_{\lambda }$ are \emph{not} given by specializing the Knop-Sahi polynomials. Rather, the formulas for these eigenvalues involve suitably regularized forms of these polynomials. In addition, we demonstrate a close relationship between our eigenvalue formulas for this subfamily of Capelli operators and the Dougall-Ramanujan hypergeometric identity. We also transcend our results on the eigenvalues of Capelli operators to the Deligne category $\mathsf{Rep}(O_t)$. More precisely, we define categorical Capelli operators $\{{\mathbf{D}}_{t,\lambda }{\}}_{\lambda \in \mathcal{P}}^{}$ that induce morphisms of indecomposable components of symmetric powers of ${\mathsf{V}}_t$, where ${\mathsf{V}}_t$ is the generating object of $\mathsf{Rep}(O_t)$. We obtain formulas for the eigenvalue polynomials associated to the ${\left\{{\mathbf{D}}_{t,\lambda }\right\}}_{\lambda \in \mathcal{P}}$ that are analogous to our results for the operators $\{D_{\lambda }{\}}_{\lambda \in \mathcal{P}}^{}$.

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