The displayed operators satisfy the bracket relations of $\mathfrak{sp}_{2\ell}$

For every $c\in\mathbb C$ and every polynomial $\varphi\in\mathbb C[T_{i,j}]$ there is a representation of $\mathfrak{sp}_{2\ell}(\mathbb C)$ on $\mathbb C[T_{i,j}]$ in which the generators act by the displayed formulas:

together with the displayed action of the diagonal negative root vectors $Y_{i,i}$, where $\widetilde X_{i,j}$ and $\widetilde Y_{i,j}$ are the first-order operators built from the symmetric variables and their formal partial derivatives.

Role. This is the entire construction of the family $\tau(c,\varphi)$, and after the reductions already in place it is the only part of that construction with content: choice supplies the dependence on the parameters, and rank-one freeness over the abelian nilradical is a consequence of the generator actions alone, since $T_{i,j}$ acts by multiplication by its variable.

What has to be checked is that the assignment respects the Lie bracket. The generators $X_{i,j}$, $T_{i,j}$ and $S_{i,j}$ span $\mathfrak{sp}_{2\ell}$, so it suffices to verify the bracket identities on pairs of them, and each such identity is a computation with first-order differential operators on a polynomial algebra. The nilradical relations are immediate --- multiplications by variables commute --- and the mixed relations are what force the specific shape of the displayed operators, in particular the appearance of $\widetilde X_{i,j}\varphi$ as a multiplication term and of the parameter $c$ only on the diagonal.

Every other headline result of the classification --- exhaustiveness, parameter uniqueness, the weight criterion, simplicity, finite length at the exceptional parameters, and the Hamiltonian application --- is stated for an arbitrary member of the family, so all of them are independent of this statement; but nothing in the mission is non-vacuous until it is available.