A note on representations of some affine vertex algebras of type \(D\) (Q2843831)

If \(\mathfrak{g}\) is a simple Lie algebra of type \(D_{\ell}\), the author provides explicit formulas for singular vectors for the corresponding universal affine vertex algebra \(\mathcal{N}_{D_{\ell}}(n-\ell+1,0)\), \(n\) a positive integer. In particular, for \(n=1\), he considers the vertex algebra \(\mathcal{V}_{D_{\ell}}(-\ell+2,0)\) which is a quotient of \(\mathcal{N}_{D_{\ell}}(-\ell+2,0)\), describes the corresponding Zhu's algebra, and gives a complete list of irreducible weak \(\mathcal{V}_{D_{\ell}}(-\ell+2,0)\)-modules from the category \(\mathcal{O}\). As a consequence, he also obtains a complete list of irreducible \(\mathcal{V}_{D_{\ell}}(-\ell+2,0)\)-modules. In the special case when \(\ell=4\) he provides a complete list of irreducible weak \(L_{D_{\ell}}(-2,0)\)-modules from the category \(\mathcal{O}\), he shows that \(L_{D_{\ell}}(-2,0)\) is the unique irreducible ordinary module for \(L_{D_{\ell}}(-2,0)\), and any ordinary \(L_{D_{\ell}}(-2,0)\)-module is completely reducible.