Relaxed highest-weight modules III: Character formulae

Abstract

This is the third of a series of articles devoted to the study of relaxed highest weight modules over vertex operator algebras. Relaxed highest weight modules over affine vertex algebras associated to higher rank Lie algebras Aℓ and arbitrary non-critical levels whose top spaces are cuspidal representations of Aℓ are extensively studied. In particular, the string functions of irreducible relaxed highest weight modules whose top spaces are cuspidal Aℓ-modules are shown to be the quotients by a power of the Dedekind eta series of the q-characters of irreducible ordinary modules over affine W-algebras associated with the minimal nilpotent elements of Aℓ.