In Section \[Sec-Sec5\], we prove Theorem \[MAINTHM\] and Theorem \[thm-unqclam\] (see the second goal above). Section \[SecRelSer\] is devoted to connecting the eigenvalue polynomials $c_\lambda$ to the shifted super Jack polynomials of Sergeev and Veselov [@SerVes]. Appendix \[sec-pflem\] contains the proof of Proposition \[DGBRVVV\]. Finally, in Appendix B we outline the proofs of our main theorems for the case of $\gl(m|n)\times\gl(m|n)$ acting on $\C^{m|n}\otimes (\C^{m|n})^*$.

We now briefly describe the structure of our proofs. There is no mystery in the formulation of the statement of Theorem \[prpgw\], but its proof is *not* a simple generalization of any of the existing proofs in the purely even case (i.e., when $n=0$) that we are aware of. Our approach is inspired by the proof given by Goodman and Wallach in [@GoodmanWallach Sec. 5.7.1], but it diverges quickly because their argument relies heavily on the complete reducibility of rational representations of a reductive algebraic group. Our proof proceeds by induction, after we show that the symbol of an invariant differential operator is in the image of $\bfZ(\gl(m|n))$. This symbol is in the span of invariant tensors $\bft_\sigma$ defined in where $\sigma$ is a permutation. The next step is to reduce the latter problem to 