On submodules of a Verma module. The case of \(\mathfrak{sl}(4, \mathbb C)\) (Q1241795)

Let \(L\) be a complex semisimple Lie algebra with Cartan subalgebra \(H\). Each \(\chi\in\Hom(H,\mathbb C)\) generates a certain factor space of \(U(L)\), the universal enveloping algebra of \(L\). This is a \(U(L)\)-module, denoted by \(M(\chi)\) and called the Verma module induced by \(\chi\). \textit{D. N. Verma} [Dissertation, Yale University, 1966] proved that a submodule of \(M(\chi)\) generated by an extreme vector is isomorphic to another Verma module which is called a Verma submodule. Subsequently examples of submodules not of Verma type have been constructed. For this, \(\mathfrak{sl}(4, \mathbb C)\) has been used for \(L\) and \(\chi\) is a certain weight \(\omega\). In the present note the submodules not of Verma type are constructed when \(L=\mathfrak{sl}(4, \mathbb C)\) and \(\chi=n\omega\) where \(n\) is any positive integer.