Linear maps preserving orbits

Schwarz, Gerald W.

Linear maps preserving orbits
[Applications linéaires qui préservent des orbites]

Annales de l'Institut Fourier, Tome 62 (2012), p. 667-706 / Harvested from Numdam

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Abstract

Soit $H \subset GL (V)$ un groupe complexe réductif connexe où $V$ est un espace vectoriel complexe de dimension finie. Soient $v \in V$ et $G = {g \in GL (V) ∣ g H v = H v}$ . D’aprés Raïs nous disons que l’orbite $H v$ est caractéristique pour $H$ si la composante connexe de l’identité de $G$ est $H$ . Si $H$ est semi-simple, nous disons que $H v$ est semi-caractéristique pour $H$ si la composante connexe de l’identité de $G$ est une extension de $H$ par un tore. Nous classifions les orbites de $H$ qui ne sont pas (semi)-caractéristiques dans plusieurs cas.

Let $H \subset GL (V)$ be a connected complex reductive group where $V$ is a finite-dimensional complex vector space. Let $v \in V$ and let $G = {g \in GL (V) ∣ g H v = H v}$ . Following Raïs we say that the orbit $H v$ is characteristic for $H$ if the identity component of $G$ is $H$ . If $H$ is semisimple, we say that $H v$ is semi-characteristic for $H$ if the identity component of $G$ is an extension of $H$ by a torus. We classify the $H$ -orbits which are not (semi)-characteristic in many cases.

Publié le : 2012-01-01

MR 2985513 | Zbl 1255.14040

DOI : https://doi.org/10.5802/aif.2691
Classification: 20G20, 22E46
Mots clés: Orbites caractéristiques, problèmes de préservation linéaires

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     author = {Schwarz, Gerald W.},
     title = {Linear maps preserving orbits},
     journal = {Annales de l'Institut Fourier},
     volume = {62},
     year = {2012},
     pages = {667-706},
     doi = {10.5802/aif.2691},
     zbl = {1255.14040},
     mrnumber = {2985513},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2012__62_2_667_0}
}

Schwarz, Gerald W. Linear maps preserving orbits. Annales de l'Institut Fourier, Tome 62 (2012) pp. 667-706. doi : 10.5802/aif.2691. http://gdmltest.u-ga.fr/item/AIF_2012__62_2_667_0/

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