symmetric pairs $$(\gl(n)\times \gl(n),\gl(n)),\
(\gl(n),\g{o}(n)),\
(\gl(2n),\g{sp}(2n)).$$ In the Lie superalgebra setting the last two come together and there are only two pairs to consider, namely $$(\gl(m|n)\times \gl(m|n),\gl(m|n))\text{ and }(\gl(m|2n),\g{osp}(m|2n)).$$ The extension of the theory to the first case is easier and, as we show in Appendix \[appxB\], can be achieved by combining the work of Molev [@Molev] with that of Sergeev–Veselov [@SerVes]. Therefore in the body of the paper we concentrate on the second case, which corresponds to the action of $\gl(m|2n)$ on $\sS^2(\C^{m|2n})$.

In the extension of the aforementioned theory to Lie superalgebras, at least two serious difficulties arise. The first issue is that complete reducibility of finite dimensional representations fails in the case of Lie superalgebras. Complete reducibility is crucial in the approach to the abstract Capelli problem in [@HoweUmeda] and [@GoodmanWallach], where it is needed to split off the $\g g$-invariants in $\bfU(\g g)$ and $\sPD(W)^{\g g}$.

The second issue that arises is that the Cartan–Helgason Theorem is not known in full generality for Lie superalgebras. In particular, it is not immediately obvious that every $\gl(m|2n)$-module that appears in $\sP(\sS^2(\C^{m|2n}))$ has an $\g{osp}(m|2n)$-invariant vector. Furthermore, in the purely even case the fact that a spherical vector is 