Decomposably-generated modules of simple Lie algebras (Q2885384)

Let \(\mathfrak{g}\) be a complex simple Lie algebra and \(V\) an irreducible orthogonal \(\mathfrak{g}\)-module. Then the spinor representation of \(\mathfrak{so}(V)\), when restricted to \(\mathfrak{g}\) via the inclusion \(\mathfrak{g}\rightarrow \mathfrak{so}(V)\), gives rise to a \(\mathfrak{g}\)-module \(\text{Spin}(V)\). This turns out to be a multiple of a \(\mathfrak{g}\)-module \(\text{Spin}_0(V)\), the ``reduced Spin'' of \(V\).NEWLINENEWLINE\textit{D. I. Panyushev} in [Transform. Groups 6, No. 4, 371--396 (2001; Zbl 0994.17004)] classified the pairs \((\mathfrak{g},V)\) for which \(\text{Spin}_0(V)\) is irreducible and defined when \(\text{Spin}_0(V)\) is \textit{decomposably-generated}. In the paper under review, the authors show that there are only finitely many isoclasses of irreducible \(\mathfrak{g}\)-modules \(V\) such that \(\text{Spin}_0(V)\) is decomposably generated. In the case of algebras of type \(\mathsf A\) they also provide some additional classification of such modules.