$\tau(c_1,\varphi_1)\cong\tau(c_2,\varphi_2)$ iff $c_1=c_2$ and $\varphi_1-\varphi_2$ is constant

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 determines exactly when two members of the family are isomorphic. For all scalars $c_1,c_2$ and polynomials $\varphi_1,\varphi_2$,

So the scalar $c$ is a genuine invariant of the isomorphism class, while the polynomial parameter matters only modulo constants: shifting $\varphi$ by a scalar changes the operators but not the module. Combined with exhaustiveness, this says that the isomorphism classes of modules free of rank one over $U(\mathfrak{n})$ are parametrized precisely by the set of pairs $(c, \varphi \bmod \mathbb{C})$.