after a *Frobenius transform* (see Definition \[FrobT\]), the polynomial $c_\lambda^*$ satisfies the supersymmetry and vanishing properties of the shifted super Jack polynomials of [@SerVes]. It then follows that the two polynomials coincide up to a scalar multiple.

We now elaborate on some of the new techniques and ideas introduced in our paper. Our method of proof of Theorem \[prpgw\] yields a recursive procedure for expressing a given invariant differential operator explicitly as the image of an element of $\bfZ(\gl(m|n))$ using Gelfand elements. In the setting of ordinary Lie algebras, Howe and Umeda [@HoweUmeda] obtain such a formula for the *generators* of the algebra of invariant differential operators. By contrast, even in this setting, our construction is more general and gives explicit pre-images for a *basis* of the algebra.

Another new idea is the construction of spherical vectors in tensor representations of $\gl(m|2n)$ using symbols of the Capelli operators $D_\lambda$ (see Lemma \[lemZnZk=0\]).\
**Acknowledgement.** We thank the referee for useful comments and pointing to references for Lemma \[gelfzhel\]. Throughout this project, we benefited from discussions with Alexander Alldridge, Ivan Dimitrov, Roe Goodman, Roger Howe, Friedrich Knop, James Lepowsky, Nolan Wallach, and Tilmann Wurzbacher. We thank them for fruitful and encouraging discussions. 