On irreducibility of modules of Whittaker type for cyclic orbifold vertex algebra

Abstract

We extend the Dong-Mason theorem on the irreducibility of modules for orbifold vertex algebras from [C. Dong, G. Mason, Duke Math. J. 86 (1997)] 305-321] for the category of weak modules. Let $V$ be a vertex operator algebra, $g$ an automorphism of order $p$. Let $W$ be an irreducible weak $V$--module such that $W,W\circ g,\dots ,W\circ g^{p-1}$ are inequivalent irreducible modules. We prove that $W$ is an irreducible weak $V^{⟨g⟩}$-module. This result can be applied on irreducible modules of certain Lie algebra $\mathfrak{L}$ such that $W,W\circ g,\dots ,W\circ g^{p-1}$ are Whittaker modules having different Whittaker functions. We present certain applications in the cases of the Heisenberg and Weyl vertex operator algebras.

18 pages, a few changes, new result on complete reducibility added