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

We extend the Dong-Mason theorem on irreducibility of modules for orbifold vertex algebras (cf. [18]) to 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 circle g, ...,W circle 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 L such that W, W circle g, ..., W circle 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. (C) 2019 Elsevier Inc. All rights reserved.