Closed orbits for actions of maximal tori on homogeneous spaces
type: Item
instance of: scholarly article
title: Closed orbits for actions of maximal tori on homogeneous spaces
zbMATH Open document ID: 1040.22005
DOI: 10.1215/S0012-7094-03-11926-2
author: George Tomanov
author: Barak Weiss
published in: Duke Mathematical Journal
publication date: 2004-05-27T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.dmj/1082744736
review text: The action of a torus \(T\) (containing a maximal \(R\)-split torus) on a homogeneous space is studied. One considers a group \(G\) defined over the set \(Q\) of rational numbers and then the homogeneous space \(G/\Gamma \) is constructed. It is proven that the closed orbits for the action of \(T\) on \(G/\Gamma \) admit a simple algebraic description. When \(G\) is reductive, it is shown that an orbit \(Tx\Gamma \) is closed if and only if \(x^{-1}Tx\) is a product of a compact torus and a torus defined over \(Q.\) In addition, it is established that such an orbit is divergent if and only if the maximal \(R\)-split subtorus of \(x^{-1}Tx\) is defined over \(Q\) and \(Q\)-split. It is concluded that all closed orbits are ``standard'', that is they correspond to \(Q\)-subtori in \(G\). Some illustrative examples are also given and open questions for further researches are formulated.
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 2068476
zbMATH Keywords: homogeneous space
zbMATH Keywords: closed orbit
zbMATH Keywords: maximal torus
zbMATH Keywords: divergent orbit
reviewed by: Gheorghe Zet
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2068476