Simple Weak Modules for Some Subalgebras of the Heisenberg Vertex Algebra and Whittaker Vectors

Let $\mathcal {M}(p)$ (p = 2,3,…) be the singlet vertex operator algebra and ω its conformal vector. We classify the simple weak $\mathcal {M}(p)$-modules with a non-zero element u such that for some integer s ≥ 2, $\omega _{i} u\in \mathbb {C} u$ (i = ⌊s/2⌋ + 1,⌊s/2⌋ + 2,…,s − 1), $\omega _{s} u\in \mathbb {C}^{\times } u$, and ωiu = 0 for all i > s.