Admissible representations of simple affine vertex algebras

We provide an explicit combinatorial description of highest weights of simple highest weight modules over the simple affine vertex algebra L kappa(s[n+1) with n is an element of N of admissible level kappa. For admissible simple highest weight modules correspond-ing to the principal, subregular and maximal parabolic nilpo-tent orbits we give a realization using the Gelfand-Tsetlin theory, which also allows us to obtain a realization of certain classes of simple admissible s[2-induced modules in these or-bits. In particular, simple admissible s[2-induced modules are fully classified for the principal nilpotent orbit. (c) 2023 Elsevier Inc. All rights reserved.