Further generalization of generalized Verma modules
type: Item
instance of: scholarly article
title: Further generalization of generalized Verma modules
zbMATH Open document ID: 0810.17002
author: Akihiko Gyoja
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1995-04-17T00:00:00Z
review text: 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 \(G/P\).
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 487011
zbMATH Keywords: algebraic group
zbMATH Keywords: orbits
zbMATH Keywords: generalized Verma module
zbMATH Keywords: cohomologies
reviewed by: Masakazu Muro
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195167048
OpenAlex ID: W2032589272
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DOI: 10.2977/PRIMS/1195167048
description: scientific article; zbMATH DE number 487011