On the relevance of some constructions of highest weight modules over (super-) Kac-Moody algebras (In connection with a paper by Wakimoto) (Q1103039)

In [Commun. Math. Phys. 104, 605--609 (1986; Zbl 0587.17009)] \textit{M. Wakimoto} constructed a family of highest weight modules \(V(\mu,\nu)\) over the affine Lie algebra \(A_1^{(1)}\) parametrized by (\(\mu,\nu)\in \mathbb{C}^2\). Using the Kac-Kazhdan criterion [Adv. Math. 34, 97--108 (1979; Zbl 0427.17011)] Wakimoto made a statement on the irreducibility of his modules, \(V(\mu,\nu) = L(\mu,\nu)\), which contains a mistake. The present author corrects the statement and further discusses the (ir)reducibility of the highest weight modules over a Kac-Moody Lie (super)algebra in a certain class.