$\tau(c,\varphi)$ is a weight module iff $\varphi$ 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 identifies which members of the family are weight modules and pins down the highest-weight-type structure in that case. Two assertions are made.

First, for every $c$ and $\varphi$,

A nonconstant polynomial parameter therefore produces a module that is genuinely not a weight module, which is the reason such families fall outside the classical highest-weight theory.

Second, when $\varphi$ is constant the constant polynomial $1 \in A$ is a lowest-weight cyclic generator of weight $c\Lambda_l$: it is annihilated by the diagonal negative-root vector $Y_{l,l}$ and by the strictly lower generators $X_{i,j}$ with $j+1=i$, the successive differences $X_{i,i}-X_{j,j}$ act on it by zero, the top Cartan generator acts by the scalar $c$, and no proper invariant subspace contains it.