On distinguished orbits of reductive representations
type: Item
instance of: scholarly article
title: On distinguished orbits of reductive representations
zbMATH Open document ID: 1297.22009
author: Edison Alberto Fernández-Culma
published in: Journal of Algebra
publication date: 2014-09-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1301.4949
review text: Let \(G\) be a real reductive Lie group and let \(\tau :G\to\text{GL}(V)\) be a real reductive representation of \(G\) with (restricted) moment map \(m_{\mathfrak{g}} :V\setminus\{0\}\to\mathfrak{g}\). The author introduces the notion of nice space of a real reductive representation to study the problem of how to determine if a \(G\)-orbit is distinguished (i.e. it contains a critical point of the norm squared of \(m_{\mathfrak{g}}\)). He gives an elementary proof of the well-known convexity theorem of Atiyah-Guillemin-Sternberg in the present context and uses it to give an easy-to-check sufficient condition for a \(G\)-orbit of an element in a nice space to be distinguished. In the case where \(G\) is algebraic and \(\tau\) is a rational representation, the above condition is also necessary (making heavy use of recent results of Michael Jablonski), obtaining a generalization of Nikolayevsky's nice basis criterion. The author also provides useful characterizations of nice spaces in terms of the weights of \(\tau\). Finally, some applications to ternary forms are presented.
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 11E20
zbMATH DE Number: 6340912
zbMATH Keywords: real reductive representations
zbMATH Keywords: real reductive Lie groups
zbMATH Keywords: distinguished orbits
zbMATH Keywords: moment map
zbMATH Keywords: ternary form
reviewed by: Andreas Arvanitoyeorgos
MaRDI profile type: Publication
OpenAlex ID: W2061987164
arXiv ID: 1301.4949
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DOI: 10.1016/J.JALGEBRA.2013.07.031
description: scientific article; zbMATH DE number 6340912