Simple weak modules for the fixed point subalgebra of the Heisenberg vertex operator algebra of rank 1 by an automorphism of order 2 and Whittaker vectors

Let M (1) M (1) be the vertex operator algebra with the Virasoro element ω associated to the Heisenberg algebra of rank 1 1 and let M (1) + M (1) ^+ be the subalgebra of M (1) M (1) consisting of the fixed points of an automorphism of M (1) M (1) of order 2 2. We classify the simple weak M (1) + M (1) ^+ -modules with a non-zero element w w such that for some integer s ≥ 2 s 2, ω i w ∈ C w ᵢw Cw (i = ⌊ s / 2 ⌋ + 1, ⌊ <mml: m