On considère une opération d’un groupe de Lie compact connexe sur une variété de Stein par des transformations holomorphes. On démontre que la variété est sphérique si, et seulement si, il existe une involution antiholomorphe conservant toute orbite. De plus, pour une variété de Stein sphérique, on construit une involution antiholomorphe et équivariante par rapport à l’involution de Weyl du groupe opérant. On en déduit que cette involution laisse stable toute orbite. La construction utilise quelques propriétés des sous-groupes sphériques invariantes par certains automorphismes réels des groupes réductifs complexes.
We consider an action of a connected compact Lie group on a Stein manifold by holomorphic transformations. We prove that the manifold is spherical if and only if there exists an antiholomorphic involution preserving each orbit. Moreover, for a spherical Stein manifold, we construct an antiholomorphic involution, which is equivariant with respect to the Weyl involution of the acting group, and show that this involution stabilizes each orbit. The construction uses some properties of spherical subgroups invariant under certain real automorphisms of complex reductive groups.
@article{AIF_2009__59_3_1029_0,
author = {Akhiezer, Dmitri},
title = {Spherical Stein manifolds and the Weyl involution},
journal = {Annales de l'Institut Fourier},
volume = {59},
year = {2009},
pages = {1029-1041},
doi = {10.5802/aif.2456},
zbl = {pre05580733},
mrnumber = {2543661},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2009__59_3_1029_0}
}
Akhiezer, Dmitri. Spherical Stein manifolds and the Weyl involution. Annales de l'Institut Fourier, Tome 59 (2009) pp. 1029-1041. doi : 10.5802/aif.2456. http://gdmltest.u-ga.fr/item/AIF_2009__59_3_1029_0/
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