At exceptional $c$, $\tau(c,\varphi)$ has finite length and Jordan--H\ddot{o}lder uniqueness

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 describes what happens at the exceptional parameters, where simplicity fails. Let

and let $\varphi$ be arbitrary. Then $\tau(c,\varphi)$ is still as well behaved as a nonsimple module can be: ascending and descending chains of invariant subspaces both stabilize, so the module is Noetherian and Artinian; it admits a finite composition series, that is a finite chain of invariant subspaces with no invariant subspace strictly between consecutive terms; and any two such composition series have the same length and isomorphic composition factors up to permutation, which is the Jordan–Hölder property.

The point is that the exceptional locus does not produce pathological infinite-length modules. Each $\tau(c,\varphi)$ decomposes into finitely many well-defined simple pieces, so the classification of the family extends to a description of the exceptional members in terms of their composition factors.