CAPELLI OPERATORS FOR SPHERICAL SUPERHARMONICS AND THE DOUGALL–RAMANUJAN IDENTITY

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 [InlineMediaObject not available: see fulltext.] (V) of superpolynomials on V is completely reducible, unless dim V_{o ¯} and dim V_{1 ¯} are positive even integers and dim (Formula presented.). When [InlineMediaObject not available: see fulltext.] (V) is not a completely reducible g-module, we construct a natural basis (Formula presented.) of “Capelli operators” for the algebra [InlineMediaObject not available: see fulltext.] (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 (Formula presented.) on maximal indecomposable components of [InlineMediaObject not available: see fulltext.] (V) explicitly, in terms of Knop–Sahi interpolation polynomials. Our results show that, unlike the cases where [InlineMediaObject not available: see fulltext.] (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 (Formula presented.) 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 (Formula presented.) that are analogous to our results for the operators (Formula presented.).