ar X iv : 1 60 5 . 07 58 0 v 1 [ m at h . R T ] 2 4 M ay 2 01 6 WEIGHT REPRESENTATIONS OF ADMISSIBLE AFFINE VERTEX ALGEBRAS

For an admissible affine vertex algebra Vk(g) of type A, we describe a new family of relaxed highest weight representations of Vk(g). They are simple quotients of representations of the affine Kac-Moody algebra ĝ induced from the following g-modules: 1) generic Gelfand-Tsetlin modules in the principal nilpotent orbit, in particular all such modules induced from sl2; 2) all Gelfand-Tsetlin modules in the principal nilpotent orbit which are induced from sl3; 3) all simple Gelfand-Tsetlin modules over sl3. This in particular gives the classification of all simple relaxed highest weight representations of Vk(g) with finite dimensional weight spaces for g = sl3.