Every $\mathfrak{sp}_{2l}$-module free of rank one over $U(\mathfrak{n})$ is some $\tau(c,\varphi)$

Throughout, $\mathfrak{sp}_{2l}(\mathbb{C})$ is the symplectic Lie algebra with $l \ge 2$, realized concretely as $2l\times 2l$ matrices. Inside a maximal parabolic subalgebra sits the abelian nilradical $\mathfrak{n}$ spanned by the symmetric matrix units $T_{i,j}=T_{j,i}$, and $A = \mathbb{C}[T_{i,j} : i \le j]$ is the polynomial algebra on the corresponding variables, which is a copy of $U(\mathfrak{n})$.

The paper builds a two-parameter family of $\mathfrak{sp}_{2l}(\mathbb{C})$-module structures on $A$, indexed by a scalar $c \in \mathbb{C}$ and a polynomial $\varphi \in A$, written $\tau(c,\varphi)$. Its defining property is that the generators act by the displayed explicit differential operators and that each $T_{i,j}$ acts by multiplication by the variable $T_{i,j}$, so that $\tau(c,\varphi)$ is free of rank one over $U(\mathfrak{n})$.

This theorem is the exhaustiveness half of the classification: the family $\tau(c,\varphi)$ already contains every example. Precisely, if $M$ is any $\mathfrak{sp}_{2l}(\mathbb{C})$-module which is free of rank one over $U(\mathfrak{n})$, then there are a scalar $c$ and a polynomial $\varphi$ and an intertwining linear isomorphism

Together with the uniqueness statement for the parameters, this is what turns a construction into a classification: the modules free of rank one over the abelian nilradical of a maximal parabolic in type $C_l$ are exactly the $\tau(c,\varphi)$, with no further examples.