Rigidity of generalized Verma modules (Q2773356)

This paper continues the study of certain parabolic generalizations, \({\mathcal O}({\mathfrak p}, \Lambda)\), of the category \(\mathcal O\) of semisimple complex finite-dimensional Lie algebras, where \(\mathfrak p\) is a parabolic subalgebra of the Lie algebra with a fixed triangular decomposition and \(\Lambda\) is a certain admissible category of \(\mathfrak a\)-modules. These categories \({\mathcal O}({\mathfrak p}, \Lambda)\) were introduced by V. Futorny, S. König and V. Mazorchuk. It is shown that the blocks of the category \({\mathcal O}({\mathfrak p}, \Lambda)\) contain generalized Verma modules. The main interest in the present paper is the question whether these modules are rigid, that is, whether the socle filtration and the radical filtration of a generalized Verma module coincide with each other. This is an analogous question to that for the Verma modules in the category \(\mathcal O\) introduced by I. N. Bernstein, I. M. Gelfand and S. I. Gelfand. The nice result in the paper under review shows that generalized Verma modules induced from the generic Gelfand-Zetlin modules or associated with the Enright-complete modules are rigid.