Jacquet modules for semisimple Lie groups having Verma module filtrations
type: Item
instance of: scholarly article
title: Jacquet modules for semisimple Lie groups having Verma module filtrations
zbMATH Open document ID: 0716.22005
DOI: 10.1016/0021-8693(91)90051-9
author: David H. Collingwood
published in: Journal of Algebra
publication date: 1991-01-01T00:00:00Z
review text: Let G be a connected semisimple real matrix Lie group, F be a finite dimensional representation of G. Let \({\mathfrak g}\) be the complexification of Lie G and \({\mathfrak b}\) be an Iwasawa Borel subalgebra of \({\mathfrak g}\). We have two categories of representations: the category \({\mathcal H}{\mathcal C}_ F\) of Harish-Chandra modules with the same infinitesimal character as F and the category \({\mathcal O}'_ F\) of finitely generated U(\({\mathfrak g})\)-modules which are \({\mathfrak b}\)-locally finite and have the same infinitesimal character as F. There is a Jacquet functor J from the first category to the second one. The category \({\mathcal O}'_ F\) admits several basic families of indecomposable modules. For instance, the indecomposable projective modules and the indecomposable self-dual Verma flag modules. The paper reviewed determines when J(\(\pi\)) for irreducible \(\pi\) is in these families.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E47
zbMATH DE Number: 4179681
zbMATH Keywords: connected semisimple real matrix Lie group
zbMATH Keywords: finite dimensional representation
zbMATH Keywords: categories of representations
zbMATH Keywords: Harish-Chandra modules
zbMATH Keywords: infinitesimal character
zbMATH Keywords: Jacquet functor
zbMATH Keywords: indecomposable modules
zbMATH Keywords: indecomposable self-dual Verma flag modules
reviewed by: V. F. Molchanov
MaRDI profile type: Publication
full work available at URL: 0021-8693(91)90051-9
OpenAlex ID: W2089548641
Wikidata QID: Q115365819
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description: scientific article; zbMATH DE number 4179681