Polar representations
type: Item
instance of: scholarly article
title: Polar representations
zbMATH Open document ID: 0611.22009
DOI: 10.1016/0021-8693(85)90136-X
author: Jiri Dadok
author: Victor G. Kac
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let G be a complex affine reductive algebraic group acting on a complex vector space V by a rational representation. Let \(v\in V\) be a vector with a closed G-orbit. Define a subspace \(c_ v=\{x\in V|\) \({\mathfrak g}\cdot x\subset {\mathfrak g}\cdot v\}\), \({\mathfrak g}\) the Lie algebra of V. If dim \(c_ v=\dim {\mathbb{C}}[V]^ G\) the action is called polar and \(c=c_ v\) a Cartan subspace. Assume this. The authors show: (1) Cartan subspaces are G-conjugate; (2) all closed orbits intersect c; (3) \(W=N_ G(c)/Z_ G(c)\) is finite and V/G\(\approx c/W\); (4) \({\mathbb{C}}[V]^ G\approx {\mathbb{C}}[c]^ W\); (5) W is a pseudo-reflection group. A list of all polar representations for G connected, simple is given.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 3987583
zbMATH Keywords: reductive algebraic group
zbMATH Keywords: Cartan subspace
zbMATH Keywords: closed orbits
zbMATH Keywords: pseudo- reflection group
zbMATH Keywords: polar representations
MaRDI profile type: Publication
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full work available at URL: 0021-8693(85)90136-x
OpenAlex ID: W4210374490
description: scientific article; zbMATH DE number 3987583