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Let $M(1)$ be the vertex operator algebra with the Virasoro element $\omega$associated to the Heisenberg algebra of rank $1$ and let $M(1)^{+}$ be thesubalgebra of $M(1)$ consisting of the fixed points of an automorphism of$M(1)$ of order $2$. We classify the simple weak $M(1)^{+}$-modules with anon-zero element $w$ such that for some integer $s\geq 2$, $\omega_iw\in{\mathbb C}w$ ($i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s-1$),$\omega_{s}w\in{\mathbb C}^{\times}w$, and $\omega_i w=0$ for all $i>s$. Theresult says that any such simple weak $M(1)^{+}$-module is isomorphic to somesimple weak $M(1)$-module or to some $\theta$-twisted simple weak$M(1)$-module.

proceedings of the american mathematical society

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