Linear maps preserving orbits

Schwarz, Gerald W.

Annales de l'Institut Fourier, Tome 62 (2012) no. 2, pp. 667-706.

Soit $H\subset GL\left(V\right)$ un groupe complexe réductif connexe où $V$ est un espace vectoriel complexe de dimension finie. Soient $v\in V$ et $G=\{g\in \mathrm{GL}(V)\mid gHv=Hv\}$. D’aprés Raïs nous disons que l’orbite $Hv$ est caractéristique pour $H$ si la composante connexe de l’identité de $G$ est $H$. Si $H$ est semi-simple, nous disons que $Hv$ 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 $H\subset GL\left(V\right)$ be a connected complex reductive group where $V$ is a finite-dimensional complex vector space. Let $v\in V$ and let $G=\{g\in \mathrm{GL}(V)\mid gHv=Hv\}$. Following Raïs we say that the orbit $Hv$ is characteristic for $H$ if the identity component of $G$ is $H$. If $H$ is semisimple, we say that $Hv$ 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.

Classification : 20G20, 22E46
Keywords: Characteristic orbits, linear preserver problems
Mot clés : Orbites caractéristiques, problèmes de préservation linéaires

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