Partial classification of cuspidal simple modules for Virasoro-like algebra

Let $\mathfrak{P}$ be the Lie algebra of Hamiltonian vector fields on the torus, which is also known as the Virasoro-like algebra, a special kind of the so-called Block type Lie algebra. And let $\mathfrak{A}$ be the Laurent polynomial algebra in two variables. In this paper, by following S.E. Rao’s strategy of “backward induction”, we prove that any quasi-finite simple $(\mathfrak{A},\mathfrak{P})$-module has to come from Larsson-Shen’s construction.