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- W3104185469 abstract "We prove a generalization of a theorem of Borel-Harish-Chandra on closed orbits of linear actions of reductive groups. Consider a real reductive algebraic group $G$ acting linearly and rationally on a real vector space $V$. $G$ can be viewed as the real points of a complex reductive group $G^mathbb C$ which acts on $V^mathbb C := V otimes mathbb C$. Borel-Harish-Chandra show that $G^mathbb C cdot v cap V$ is a finite union of $G$-orbits; moreover, $G^mathbb C cdot v$ is closed if and only if $Gcdot v$ is closed. We show that the same result holds not just for closed orbits but for the so-called distinguished orbits. An orbit is called distinguished if it contains a critical point of the norm squared of the moment map on projective space. Our main result compares the complex and real settings to show $Gcdot v$ is distinguished if and only if $G^mathbb C cdot v$ is distinguished. In addition, we show that if an orbit is distinguished, then under the negative gradient flow of the norm squared of the moment map the entire $G$-orbit collapses to a single $K$-orbit. This result holds in both the complex and real settings. We finish with some applications to the study of the left-invariant geometry of Lie groups." @default.
- W3104185469 created "2020-11-23" @default.
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- W3104185469 date "2012-10-01" @default.
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- W3104185469 title "Distinguished orbits of reductive groups" @default.
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- W3104185469 doi "https://doi.org/10.1216/rmj-2012-42-5-1521" @default.
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