Weight representations of admissible affine vertex algebras

TL;DR

This paper introduces a new family of relaxed highest weight representations for admissible affine vertex algebras of type A, expanding the classification of simple weight representations with finite-dimensional weight spaces.

Contribution

It describes a novel class of relaxed highest weight modules for admissible affine vertex algebras, constructed from Gelfand-Tsetlin modules, and classifies all such modules for sl_3.

Findings

Constructed new relaxed highest weight modules from Gelfand-Tsetlin modules.

Classified all simple positive energy weight representations with finite-dimensional weight spaces for sl_3.

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 $\widehat{\mathfrak{g}}$ 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 $\mathfrak{g}=\mathfrak{sl}_3$.