Note on rational points in nilpotent orbits of semisimple groups
type: Item
instance of: scholarly article
title: Note on rational points in nilpotent orbits of semisimple groups
zbMATH Open document ID: 0918.22008
DOI: 10.1016/S0019-3577(97)87565-9
published in: Indagationes Mathematicae. New Series
publication date: 1999-02-17T00:00:00Z
review text: The author considers the problem of the existence of rational points on nilpotent orbits (1985, Carter; 1992, Collingwood, McGovern) of a connected semisimple linear algebraic \(k\)-group \(G\) (1991, Borel; 1994, Springer) defined over a field \(k\) with characteristic 0 and identified here with the group \(G(\overline k)\) of \(\overline k\)-points of \(G\) (\(\overline k\) being an algebraic closure of \(k\)). The group \(G_k\) of \(k\)-rational points of \(G\) and the related Lie algebra \(g_k\) are also introduced. Necessary and sufficient conditions are determined and proved for the nonzero nilpotent \(G(\overline k)\)-orbit \(O\) belonging to the Lie algebra \(g(\overline k)\) of \(G\) to be defined over the field \(k\) (Theorem 1) as well as to have a \(k\)-rational point, i.e., \(O\cap g_k\neq 0\) (Theorem 2).
reviewed by: A. A. Bogush
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 17B45
Mathematics Subject Classification ID: 20G99
zbMATH DE Number: 1251249
zbMATH Keywords: nilpotent orbits
zbMATH Keywords: semisimple linear algebraic \(k\)-group
zbMATH Keywords: \(k\)-rational points
zbMATH Keywords: Lie algebra
author: Dragomir Ž. Đoković
MaRDI profile type: Publication
cites work: Q3999894
cites work: Q3682663
cites work: Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers
cites work: Classification of nilpotent elements in simple real Lie algebras \(E_{6(6)}\) and \(E_{6(-26)}\) and description of their centralizers
cites work: Rational conjugacy classes in reductive groups
cites work: Q3998356
cites work: Remarks on real nilpotent orbits of a symmetric pair
cites work: Q5442209
cites work: Q5648549
full work available at URL: s0019-3577(97)87565-9
OpenAlex ID: W2151218546
description: scientific article; zbMATH DE number 1251249