Finite orbit modules for parabolic subgroups of exceptional groups.
type: Item
instance of: scholarly article
title: Finite orbit modules for parabolic subgroups of exceptional groups.
zbMATH Open document ID: 1069.20034
DOI: 10.1016/S0019-3577(04)90014-6
author: Gerhard Röhrle
author: Simon M. Goodwin
published in: Indagationes Mathematicae. New Series
publication date: 2005-01-03T00:00:00Z
review text: Let \(G\) be a reductive linear algebraic group defined over an algebraically closed field, \(P\) a parabolic subgroup of \(G\) and \(P_u\) be its unipotent radical. Considering the adjoint action of \(P\) on the Lie algebra \(L_u\) of \(P_u\), each term \(L^{(n)}_u\) of the descending central series of \(L_u\) is stable under this action. All instances when \(P\) acts on \(L^{(n)}_u\) for \(n\geq 0\) with a finite number of orbits were determined for classical \(G\) and for \(G\) of exceptional type. The authors extend these results to groups of type \(F_4\) and \(E_6\); for each parabolic subgroup \(P\) of \(G\) of these types, \(n\geq 1\), they determine whether \(P\) acts on \(L^{(n)}_u\) with a finite number of orbits or, otherwise, they determine whether \(P\) still acts with a dense orbit. For groups \(G\) of type \(E_7\) or \(E_8\) they investigate the case of a Borel subgroup.
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 17B45
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 2124638
zbMATH Keywords: parabolic subgroups
zbMATH Keywords: Borel subgroups
zbMATH Keywords: reductive linear algebraic groups
zbMATH Keywords: numbers of orbits
zbMATH Keywords: dense orbits
reviewed by: Melih Boral
describes a project that uses: GAP
MaRDI profile type: Publication
cites work: Q5580321
cites work: Q4095018
cites work: Finite, tame, and wild actions of parabolic subgroups in \(\text{GL}(V)\) on certain unipotent subgroups
cites work: Finiteness for parabolic group actions in classical groups.
cites work: Q3752549
cites work: Prehomogeneous spaces for parabolic group actions in classical 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: MOP—Algorithmic Modality Analysis for Parabolic Group Actions
cites work: A finiteness theorem for parabolic subgroups of fixed modality
cites work: Q4338611
cites work: Invariant Theory
cites work: Conjugacy Classes in Parabolic Subgroups of Semisimple Algebraic Groups
cites work: Parabolic subgroups of positive modality
cites work: A note on the modality of parabolic subgroups
cites work: On normal Abelian subgroups in parabolic groups
cites work: On the modality of parabolic subgroups of linear algebraic groups
full work available at URL: s0019-3577(04)90014-6
OpenAlex ID: W2137348712
description: scientific article; zbMATH DE number 2124638