The refined solution to the Capelli eigenvalue problem for [formula omitted] and [formula omitted]

Abstract

Let g be either the Lie superalgebra gl(V)⊕gl(V) where V≔ℂm|n or the Lie superalgebra gl(V) where V≔ℂm|2n. Furthermore, let W be the g-module defined by W≔V⊗V∗ in the former case and W≔S2(V) in the latter case. Associated to (g,W) there exists a distinguished basis of Capelli operators{Dλ}λ∈Ω, naturally indexed by a set of hook partitions Ω, for the subalgebra of g-invariants in the superalgebra PD(W) of superdifferential operators on W.Let b be a Borel subalgebra of g. We compute eigenvalues of the Dλ on the irreducible g-submodules of P(W) and obtain them explicitly as the evaluation of the interpolation super Jack polynomials of Sergeev–Veselov at suitable affine functions of the b-highest weight. While the former case is straightforward, the latter is significantly more complex. This generalizes a result by Sahi, Salmasian and Serganova for these cases, where such formulas were given for a fixed choice of Borel subalgebra.