The module is a weight module exactly when the parameter is constant

Let $\tau(c,\varphi)$ be the family of $\mathfrak{sp}_{2\ell}(\mathbb C)$-modules on $\mathbb C[T_{i,j}]$ determined by the displayed generator actions. Then $\tau(c,\varphi)$ is a weight module — spanned by simultaneous eigenvectors of the diagonal generators $X_{i,i}$ — if and only if the parameter $\varphi$ is a constant polynomial.

Role. This is the dichotomy that makes the family genuinely new. The classical modules parabolically induced from a character are weight modules, and they are exactly the members of the family with constant parameter; every nonconstant $\varphi$ produces a module that is free of rank one over the nilradical yet carries no basis of weight vectors, so it lies outside the weight-module theory entirely. That the same two-parameter family contains both is what allows a single classification to cover the weight and non-weight cases at once.

The easy direction is a computation: for constant $\varphi$ the diagonal operator $X_{i,i}$ acts on a monomial $\prod T_{a,b}^{n_{a,b}}$ by the scalar $c+\sum_k \epsilon_k n_{k,i}$, so the monomials are weight vectors and they span. The converse is the substantive one: a nonconstant $\varphi$ contributes a multiplication operator $\widetilde X_{i,i}\varphi$ that is not scalar on any monomial basis, and one has to rule out every other spanning set of eigenvectors, not merely the obvious one.