Relaxed highest-weight modules III: Character formulae

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 $A_{\mathrm{\ell }}$ are extensively studied. In particular, the string functions of simple relaxed highest weight modules whose top spaces are simple cuspidal $A_{\mathrm{\ell }}$-modules are shown to be the quotients by a power of the Dedekind eta series of the $q$-characters of simple ordinary modules over affine W-algebras associated with the minimal nilpotent elements of $A_{\mathrm{\ell }}$.