g$ is $%
\Phi:=\Phi^+\cup \Phi^-$ where $$\Phi^+:=\Big\{
\eps_k^{}-\eps_l^{} \Big\}_{1\leq k<l\leq m} \cup \left\{ \eps_k^{}-\eps_{%
\oline l}^{} \right\} _{ 1\leq k\leq m,\,1\leq l\leq n} \cup \left\{ \eps_{%
\oline k}^{}-\eps_{\oline l}^{} \right\}_{1\leq k<l\leq n}$$ and $\Phi^-:=-\Phi^+$. For $\alpha\in\Phi$, we set $\g g_\alpha:=\{x\in\g g\,:\, [h,x]=\alpha(h)x\text{ for every }h\in\g h\}$. Then $\g g=\g n^-\oplus\g h\oplus\g n^+$, where $\g n^\pm:=\bigoplus_{\alpha\in\Phi^\pm}\g g_\alpha$. Finally, the supersymmetric bilinear form $$\label{kappa}
\kappa:\g g\times\g g\to\C\ ,\ \kappa(X,Y):=\mathrm{str}(XY)\text{ for }
X,Y\in\g g,$$ is nondegenerate and $\g g$-invariant.

Set $W:=\sS^2(V)$. The standard representation of $\g g$ on $V$ gives rise to canonically defined representations on $V^{\otimes d}$, $W$, $W^*$, $\sS(W)$, and $\sP(W)$. From now on, we denote the latter representations of $\g g$ on $\sS(W)$ and $\sP(W)$ by $\rho$ and $\check\rho$ respectively.

The goal of the rest of this section is to prove Theorem \[prpgw\], which gives an affirmative answer to the abstract Capelli problem for the $\g g$-module $W$. Set $$\bfZ(\g g):=\{x\in\bfU(\g g)\,:\,[x,y]=0\text{ for every }y\in\bfU(\g g)\},$$ where $[\,\cdot,\cdot\,]$ denotes the standard superbracket of $\bfU(\g g)$.

\[gelfzhel\] Let $\mathbf E\in\mathrm{Mat}_{m,n}(\bfU(\g g))_\eev$ be the matrix with entries $$\mathbf E_{i,j}
:=
(-1)^{|i|\cdot|j|}E_{j,i}
\text{ for }i,j\in\mathcal I_{m,n}.$$ Then $\mathrm{str}(\mathbf E^d)\in\bfZ(\g g)$ for every $d\geq 1$.

This lemma can be found for example in [@sergeev82] 