Rank-one geometry and mixed complexes in representations of Cartan type Lie algebras on a torus

In this paper, we develop a unified theory of reducibility and indecomposability for Shen-Larsson modules over the Witt, special and Hamiltonian type Lie algebras on a torus. Our approach is based on a rank-one mechanism governing irreducible submodules, Loewy filtrations, rank reduction, uniseriality and mixed complex structures. We first provide a uniform intrinsic characterization of the trivial and fundamental representations of $gl_N, sl_N, sp_{2n}$ in terms of quadratic relations satisfied by rank-one elements of these matrix Lie algebras and utilize it to determine the irreducibility of Shen-Larsson modules over $W_N, S_N, H_{2n}$. Using the rank-one operators arising from these relations, we then construct rank-reducing operators corresponding to distinguished lattice directions and apply them to show that the submodule structure of the reducible Shen-Larsson modules over $W_N, S_N, H_{2n}$ attached to the fundamental representations of $gl_N, sl_N, sp_{2n}$ respectively are generically uniserial. In the Hamiltonian case, we show that the submodules of these reducible Shen-Larsson modules come from kernels and images of differentials of the de Rham and Koszul-type complexes. These differentials anti-commute and thus endow the tensor field modules with a mixed complex structure, which also admit a natural interpretation formally analogous to the de Rham differential and co-differential type operator appearing in symplectic Hodge theory. In particular, we provide complete answers to the questions recently posed by Pei-Sheng-Tang-Zhao [J. Inst. Math. Jussieu 2023] concerning the structure of Shen-Larsson modules over $H_{2n}$.