On the structure of an \(\alpha\)-stratified generalized Verma module over Lie algebra \(sl(n, \mathbb{C})\)
Let \({\mathfrak g}\) be a finite-dimensional simple complex Lie algebra. A \({\mathfrak g}\)-module \(V\) is said to be \(\alpha\)-stratified for some root \(\alpha\) if \(X_\alpha\) and \(X_{-\alpha}\) act injectively on \(V\), where \(X_{\pm \alpha}\) are root vectors for the one-dimensional root spaces \({\mathfrak g}_{\pm \alpha}\). Fixing \({\mathfrak g}\) to be \(sl(n, \mathbb{C})\), the author continues his investigation into the structure of \(\alpha\)-stratified generalized Verma modules \(M(\lambda, p)\) [Ukr. Math. J. 45, No. 9, 1360-1371 (1993); translation from Ukr. Mat. Zh. 45, No. 9, 1215-1224 (1993; Zbl 0827.17009)]. The Verma module \(M(\lambda, p)\) is generated by a weight vector \(v_\lambda\) of weight \(\lambda\), which is killed by all positive root vectors except \(X_\alpha\), and such that the `small Casimir operator' of the \(sl(2, \mathbb{C})\) subalgebra associated to the root \(\alpha\) acts on \(v_\lambda\) as multiplication by \(p^2\). The main result of the paper is to give a description of the submodules of an \(\alpha\)-stratified Verma module similar to that in the classic Bernstein-Gelfand-Gelfand paper [Funct. Anal. Appl. 5, 1-8 (1971); translation from Funkts. Anal. Prilozh. 5, No. 1, 1-9 (1971; Zbl 0246.17008)]. Once the \(\alpha\)-stratified technical machinery is constructed in a series of lemmas, the outline of the proof of the main theorem follows that of BGG.
\(\alpha\)-stratified generalized Verma modules
scientific article; zbMATH DE number 866150