Structure of certain Verma modules over affine Lie algebras
Verma modules over affine Lie algebras are studied. Let \(\overset \circ {\mathfrak g}\) be a finite dimensional simple Lie algebra over \({\underset{\tilde{}} {\mathbb{C}}}\), \({\mathfrak g}=\overset \circ {\mathfrak g}\otimes {\mathbb{C}}[t,t^{-1}]\oplus {\mathbb{C}}\cdot c\) the corresponding affine Lie algebra, \(\overset \circ {\mathfrak h}\) a fixed Cartan subalgebra of \(\overset \circ {\mathfrak g}\), Q the lattice of roots associated with \(\overset \circ {\mathfrak h}\) and W be the corresponding Weyl group. For any \(\lambda\in (\overset \circ {\mathfrak h})^*\) set \(W_{\lambda}=\{w\in W | w\cdot \lambda -\lambda \in Q\}\). Then \(W_{\lambda}\) is isomorphic to the Weyl group of a root system \(\Delta_{\lambda}\). For a ``regular'' \(\lambda\), identify \(W_{\lambda}\) with the orbit \(W_{\lambda}\cdot \lambda\) with the aid of the mapping \(w\mapsto w\cdot \lambda.\) In the paper under rewiew the following statement is announced: If \(\lambda\) is regular and \(rk\Delta_{\lambda}=2\), then for any \(u,w\in W_{\lambda}\) there exists the unique, up to a multiple, morphism \(V(u)\to V(w)\). Any submodule of V(w) can be presented as a sum of images of these morphisms. For the case \(\overset \circ {\mathfrak g}=sl_ 3({\mathbb{C}})\) more precise results are obtained.
scientific article; zbMATH DE number 4198279