Prehomogeneous spaces for parabolic group actions in classical groups.
type: Item
instance of: scholarly article
title: Prehomogeneous spaces for parabolic group actions in classical groups.
zbMATH Open document ID: 1064.20048
author: Gerhard Röhrle
author: Simon M. Goodwin
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: Let \(G\) be a reductive linear algebraic group, \(P\) a parabolic subgroup of \(G\), and \({\mathfrak p} _u\) the Lie algebra of the unipotent radical for \(P\). The authors are mainly concerned with the question of when terms \({\mathfrak p }_u^{(l)}\) of the descending central series for \({\mathfrak p }_u\) contain a dense \(P\)-orbit under the adjoint action. In the classical case, the authors prove that if \(B\) is a Borel subgroup of \(G\) and the characteristic of the base field is either zero or good for \(G\), then each \({\mathfrak b}_u^{(l)}\) contains a dense \(B\)-orbit. In the case \(G=\text{GL}_n\), the authors also address more general parabolics. Specifically, the authors prove that if \(P=P(d_1,\dots,d_t)\) with \(d_1\leq\cdots\leq d_s\geq d_{s+1}\geq\cdots\geq d_t\) for some \(s\), then each \({\mathfrak p}_u^{(l)}\) contains a dense \(P\)-orbit.
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 17B45
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 11S90
zbMATH DE Number: 2105420
zbMATH Keywords: parabolic subgroups
zbMATH Keywords: dense orbits
zbMATH Keywords: prehomogeneous spaces
zbMATH Keywords: adjoint actions
reviewed by: John Lorch
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.11.005
OpenAlex ID: W2044189975
cites work: Q3999894
cites work: Finite, tame, and wild actions of parabolic subgroups in \(\text{GL}(V)\) on certain unipotent subgroups
cites work: The \(\Delta\)-filtered modules without self-extensions for the Auslander algebra of \(k[T]/\langle T^n\rangle\)
cites work: Finiteness for parabolic group actions in classical groups.
cites work: Finite orbit modules for parabolic subgroups of exceptional groups.
cites work: On parabolic subgroups of classical groups with a finite number of orbits on the unipotent radial
cites work: A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical
cites work: Variation on a theme of Richardson.
cites work: Q4842690
cites work: Conjugacy Classes in Parabolic Subgroups of Semisimple Algebraic Groups
cites work: On the modality of parabolic subgroups of linear algebraic groups
cites work: Linear algebraic groups
cites work: Q5662587
cites work: Q4930353
DOI: 10.1016/J.JALGEBRA.2003.11.005
description: scientific article; zbMATH DE number 2105420