Representations of Hamiltonian vector fields on a torus

In this paper, we study the structure of Shen-Larsson modules over the Hamiltonian Lie algebra, also known as the Lie algebra of Hamiltonian vector fields on a torus. We establish necessary and sufficient conditions for the irreducibility of these modules and explicitly describe the Jordan-H\"older series of all reducible ones, which we call exceptional modules. We further prove that the submodules appearing in these composition series' exhaust all possible submodules of the exceptional modules (up to trivial modules), which in turn shows that all these modules are indecomposable, except in a solitary case. The submodules of these exceptional modules can be naturally realized as kernels and images of certain differential maps acting on tensor field modules over a symplectic torus. In particular, we provide complete answers to the questions recently posed by Pei-Sheng-Tang-Zhao [J. Inst. Math. Jussieu 2023]. Additionally, our methods yield simpler, conceptual proofs of known irreducibility results for Shen-Larsson modules over the Lie algebra of polynomial vector fields on a torus and its subalgebra of divergence-free vector fields, in a uniform manner. We conclude by classifying the submodules of Shen-Larsson modules over the polynomial vector fields on a torus, which shows that these Shen-Larsson modules are always indecomposable.