Further generalization of generalized Verma modules
The purpose of this paper is a generalization of Verma modules. Let \(G\) be a complex semisimple Lie group, \(B\) a Borel subgroup and \(P\) a parabolic subgroup containing \(B\). The Lie algebras corresponding to \(G\), \(B\) and \(P\) are denoted by \(\mathfrak g\), \(\mathfrak b\) and \(\mathfrak p\), respectively. For a Lie subalgebra \(\mathfrak l\), we denote by \(U({\mathfrak l})\) the enveloping algebra of \(\mathfrak l\). Let \(E\) be a finite dimensional irreducible \(U({\mathfrak p})\)-module. A \(U({\mathfrak g})\)-module of the form \(U({\mathfrak g}) \otimes_{U({\mathfrak p})} E\) is called a generalized Verma module and, in the special case where \({\mathfrak p} = {\mathfrak b}\), it is called a Verma module. It had been proved that Verma modules correspond to the local cohomologies at the \(B\)-orbits on \(G/B\) via the localization functor. In this paper the author proves that a further generalization of the generalized Verma module appears in order to find a corresponding object to the cohomologies at the \(B\)-orbits on \(G/P\).
scientific article; zbMATH DE number 487011
scientific article