---
abstract: |
    The “Capelli problem” for the symmetric pairs $(\gl\times \gl,\gl)$ $(\gl,\g{o})$, and $(\gl,\g{sp})$ is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter (see [@KostantSahi1], [@KostantSahi2], [@KnopSahi], [@Okounkov]). In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs $(\g g,\g k):=(\gl(m|2n),\g{osp}(m|2n))$ and $(\gl(m|n)\times\gl(m|n),\gl(m|n))$, acting on $W:=\sS^2(\C^{m|2n})$ and $\C^{m|n}\otimes(\C^{m|n})^*$.

    To achieve this goal, we first prove that the center of the universal enveloping algebra of the Lie superalgebra $\g g$ maps surjectively onto the algebra $\sPD(W)^{\g g}_{}$ of $\g g$-invariant differential operators on the superspace $W$, thereby providing an affirmative answer to the “abstract” Capelli problem for $W$. Our proof works more generally for $\gl(m|n)$ acting on $\sS^2(\C^{m|n})$ and is new even for the “ordinary” cases ($m=0$ or $n=0$) considered by Howe and Umeda in [@HoweUmeda].

    We next describe a natural basis $\left\{ D_{\lambda }\right\} $ of $\sPD(W)^{\g g}$, that we call the Capelli basis. Using the above result on the abstract Capelli problem, we generalize the work of Kostant and Sahi [@KostantSahi1], [@KostantSahi2], [@Sahi] by showing that the spectrum of $D_{\lambda }$ is given by a polynomial $c_{\lambda }$, which is characterized 