.$$ Furthermore, $\left(\sP(W)\otimes \sS(W)\right)^\g g
=\bigoplus_{d\geq 0}\left(
\sP^d(W)\otimes \sS^d(W)\right)^\g g$.

For every $k,l\geq 0$, the action of $\g g$ on $\sP(W)\otimes\sS(W)$ leaves $\sP^k(W)\otimes \sS^l(W)$ invariant. It follows that $\left(\sP(W)\otimes \sS(W)\right)^\g g=\bigoplus_{k,l\geq 0}\left(\sP^k(W)\otimes \sS^l(W)\right)^\g g$. By considering the action of the centre of $\g g$ we obtain that $$\label{sSksSlk=l}
\left(\sP^k(W)\otimes \sS^l(W)\right)^\g g=\{0\} \text{ unless }k=l.$$ Next we prove . Recall the definition of $\psf_d^{}$ and $\psf_d^*$ from . It suffices to prove that the canonical projection $$\label{eq-split}
V^{*\otimes 2d}\otimes V^{\otimes 2d}
\xrightarrow{\psf^*_d\otimes\psf_d^{}}
\sS^d(W^*)\otimes \sS^d(W)$$ is surjective on $\g g$-invariants. This follows from $
\sS^d(W^*)\otimes \sS^d(W)=
\left(
V^{*\otimes 2d}\otimes V^{\otimes 2d}
\right)^{H_{2d}\times H_{2d}}
$ and the fact that $\psf_d^*\otimes \psf_d^{}$ restricts to the identity map on $\sS^d(W^*)\otimes \sS^d(W)$.

For every $i,j\in\mathcal{I}_{m,n}$, let $\varphi_{i,j} \in \sP(W)\otimes\sS
(W)$ be defined by $$\varphi_{i,j}:=(-1)^{|i|\cdot|j|} \sum_{r\in\mathcal{I}%
_{m,n}}(-1)^{|r|}y_{r,j}\otimes x_{r,i}.$$ Note that $$\label{sfmPHIEIJ}
\sfm(\varphi_{i,j})=-\check\rho(E_{i,j}) .$$

\[lem2..8\] Fix $d\geq 1$ and let $\sigma:=(2,4,\ldots,2d)\in S_{2d}$, that is, $\sigma(2r)=2r+2$ for $1\leq r\leq d-1$, $\sigma(2d)=2$, and $\sigma(2r-1)=2r-1$ for $1\leq r\leq d$. Then $$\bft_{\sigma}
=(-2)^d\,\widehat\sfs_d\big(
\check\rho\big(\mathrm{str}\big(\mathbf E^d\big)\big)
\big),$$ where $\mathbf E$ is the matrix defined in Lemma \[gelfzhel\].

For $i_1,\ldots,i_{2d}\in\mathcal I_{m,n}$, we set $$\seps'(i_1,\ldots,i_{2d}):={\sum_{s=1}^{2d}|i_s|+\sum_{s=3}^{2d}
|i_2|\cdot|i_s|+\sum_{s=1}^{d-1}|i_{2s+1}|\cdot|i_{2s+2}|}
.$$ By a straightforward but tedious sign calculation, we obtain from that $$\begin{aligned}
\bft_\sigma&=
\frac{1}{2^d}
\sum_{i_1,\ldots,i_{2d}\in\mathcal I_{m,n}}
(-1)^{\seps'(i_1,\ldots,i_{2d})}_{}y_{i_{2d-1},i_{2d}}\cdots y_{i_{1},i_{2}}
