Homomorphisms between Verma modules in characteristic p
Let \({\mathfrak g}\) be a complex semisimple Lie algebra, U its universal enveloping algebra and K an algebraically closed field of characteristic p. Then U has a \({\mathbb{Z}}\)-form \(U_{{\mathbb{Z}}}\) (that is, a subring such that \(U_{{\mathbb{Z}}}\otimes {\mathbb{C}}=U)\). Let \(U_ K=U_{{\mathbb{Z}}}\otimes K\) and \(V_{\lambda}\) be the Verma-module of \({\mathfrak g}\) with highest weight \(\lambda\). If \(V_{\mu}\) is a simple composition factor of \(V_{\lambda}\) then there are \({\mathbb{Z}}\)-forms (i.e. \({\mathbb{Z}}\)-lattices invariant under \(U_{{\mathbb{Z}}})\) of \(V_{\mu}\) and \(V_{\lambda}\) which can be tensored to yield \(U_ K\)-modules \(V_{\mu,K}\) and \(V_{\lambda,K}\), respectively. The author studies homomorphisms between \(V_{\mu,K}\) and \(V_{\lambda,K}\). The main result is a construction of a nonzero homomorphism from \(V_{\mu,K}\) to \(V_{\lambda,K}\) for \(\mu =\lambda -dr\), where r is a root and d is a positive integer which is small enough and satisfies a certain integrality condition with respect to \(\lambda\).
scientific article; zbMATH DE number 4049834
scientific article