Weight representations of admissible affine vertex algebras

Abstract

For an admissible affine vertex algebra $V_k(\mathfrak{g})$ of type $A$, we describe a new family of relaxed highest weight representations of $V_k(\mathfrak{g})$. They are simple quotients of representations of the affine Kac-Moody algebra $\hat{\mathfrak{g}}$ induced from the following $\mathfrak{g}$-modules: 1) generic Gelfand-Tsetlin modules in the principal nilpotent orbit, in particular all such modules induced from ${\mathfrak{sl}}_2$; 2) all Gelfand-Tsetlin modules in the principal nilpotent orbit which are induced from ${\mathfrak{sl}}_3$; 3) all simple Gelfand-Tsetlin modules over ${\mathfrak{sl}}_3$. This in particular gives the classification of all simple positive energy weight representations of $V_k(\mathfrak{g})$ with finite dimensional weight spaces for $\mathfrak{g}={\mathfrak{sl}}_3$.

revised version, to appear in Comm. Math. Phys

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