The refined solution to the Capelli eigenvalue problem for gl(m|n) ⊕ gl(m|n) and gl(m|2n )

Let g be either the Lie superalgebra gl(V) circle plus gl(V) where V := C m | n or the Lie superalgebra gl(V) where V := C m| 2 n . Furthermore, let W be the g-module defined by W := V circle times 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 { D lambda } lambda is an element of ohm , naturally indexed by a set of hook partitions ohm , 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 D lambda 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. (c) 2024 The Authors. Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).