the image of the shifted Jack polynomials under a certain *shifted Kerov map*, and the fact that the image of the shifted Kerov map is a polynomial is indeed a nontrivial statement which is proved in [@SerVes] indirectly. However, our definition of $c_\lambda$ is more conceptual, and the proofs are more straightforward, as they are based on the Harish–Chandra homomorphism and the solution of the abstract Capelli problem. Furthermore, the work of Sergeev and Veselov does not address the relation with spherical representations of Lie superalgebras. Our paper establishes this connection.

We now outline the structure of this article. Section \[section1\] defines the basic notation that is used throughout the rest of the paper. The solution to the abstract Capelli problem for $\gl(m|n)$ acting on $\sS^2(\C^{m|n})$ is given in Section [prfof571]{}. In Section \[Secsuperpr\] we study spherical highest weight modules of $\gl(m|2n)$. We prove in Proposition \[le-dlam\] and Remark \[rmk-v\*\] that every irreducible $\gl(m|2n)$-submodule of $\sS(\sS^2(\C^{m|2n}))$ or $\sP(\sS^2(\C^{m|2n}))$ has a unique (up to scalar) nonzero $\g{osp}(m|2n)$-fixed vector. It is worth mentioning that Proposition \[le-dlam\] does not follow from the work of Alldridge and Schmittner [@AllSch], since they need to assume that the highest weight is “high enough” in some sense. 