CAPELLI OPERATORS FOR SPHERICAL SUPERHARMONICS AND THE DOUGALL–RAMANUJAN IDENTITY

Abstract

Let (V, ω) be an orthosymplectic ℤ₂-graded vector space and let 𝔤:= 𝔤𝔬𝔰𝔭 (V, ω) denote the Lie superalgebra of similitudes of (V, ω). It is known that as a 𝔤-module, the space (V ) of superpolynomials on V is completely reducible, unless dim ${V}_{\overline{\mathrm{o}}}$ and dim ${V}_{\overline{1}}$ are positive even integers and dim ${V}_{\overline{\mathrm{O}}}\le \dim\ {V}_{\overline{1}}$. When (V ) is not a completely reducible 𝔤-module, we construct a natural basis ${\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}}$ of “Capelli operators” for the algebra (V ) ^𝔤 of 𝔤 -invariant superpolynomial superdifferential operators on V , where the index set 𝒯 is the set of integer partitions of length at most two. We compute the action of the operators ${\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}}$ on maximal indecomposable components of (V ) explicitly, in terms of Knop–Sahi interpolation polynomials. Our results show that, unlike the cases where (V ) is completely reducible, the eigenvalues of a subfamily of the {D_⋋} are not given by specializing the Knop–Sahi polynomials. Rather, the formulas for these eigenvalues involve suitably regularized forms of these polynomials. This is in contrast with what occurs for previously studied Capelli operators. 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 Rep (O_t). More precisely, we define categorical Capelli operators ${\left\{{D}_{t,\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}}$ that induce morphisms of indecomposable components of symmetric powers of V_t, where V_t is the generating object of Rep (O_t). We obtain formulas for the eigenvalue polynomials associated to the ${\left\{{D}_{t,\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}}$ that are analogous to our results for the operators ${\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}}$.