we index the rows (from left to right) and the columns (from top to bottom) of matrices by the numbers $1,\ldots,n,-n,\ldots,-1$ (respectively, by the numbers $1,\ldots,n,0,-n,\ldots,-1$ and $1,\ldots,n,-n,\ldots,-1$). Note that ${\mathfrak{g}}={\mathfrak{h}}\oplus{\mathfrak{n}}\oplus{\mathfrak{n}}_-$, where ${\mathfrak{n}}_-=\langle e_{-\alpha},~\alpha\in\Phi^+\rangle_{{\mathbb{C}}}$, and, by definition, $e_{-\alpha}=e_{\alpha}^T$. (The superscript $T$ always indicates matrix transposition.) The set $\{e_{\alpha},~\alpha\in\Phi\}$ can be extended to a unique Chevalley basis of ${\mathfrak{g}}$.

Let $G$ be one of the following classical Lie groups: ${\mathrm{SL}}_n({\mathbb{C}})$, ${\mathrm{SO}}_{2n}({\mathbb{C}})$, ${\mathrm{SO}}_{2n+1}({\mathbb{C}})$ or ${\mathrm{Sp}}_{2n}({\mathbb{C}})$. The group ${\mathrm{SO}}_{2n}({\mathbb{C}})$ (respectively, ${\mathrm{SO}}_{2n+1}$ and ${\mathrm{Sp}}_{2n}({\mathbb{C}})$) is realized as the subgroup of ${\mathrm{SL}}_{2n}({\mathbb{C}})$ (respectively, of ${\mathrm{SL}}_{2n+1}({\mathbb{C}})$ and ${\mathrm{SL}}_{2n}({\mathbb{C}})$) which preserves the form $\beta$. Let $H$ (respectively, $B$ and $N$) be the set of all diagonal (respectively, upper-triangular and upper-triangular with 1 on the diagonal) matrices from $G$. Then $H$ is a maximal torus of $G$, $B$ is a Borel subgroup of $G$ containing $H$, $N$ is the unipotent radical of $B$, and ${\mathfrak{g}}$ (respectively, ${\mathfrak{h}}$, ${\mathfrak{b}}$ and ${\mathfrak{n}}$) is the Lie algebra of $G$ (respectively, of $H$, $B$ and $N$).

Denote by $U({\mathfrak{n}})$ the enveloping algebra of ${\mathfrak{n}}$, and by $S({\mathfrak{n}})$ the symmetric algebra of ${\mathfrak{n}}$. Then ${\mathfrak{n}}$ and $S({\mathfrak{n}})$ are $B$-modules as $B$ normalizes $N$. Denote by $Z({\mathfrak{n}})$ the center of $U({\mathfrak{n}})$. It is well-known 