Geodesic
Duality of Symmetric Spaces and Polar Actions
Journal of Lie theory, Tome 21 (2011) no. 4, pp. 961-986.

Voir la notice de l'article provenant de la source Heldermann Verlag

We study isometric actions on Riemannian symmetric spaces of noncompact type which are induced by reductive algebraic subgroups of the isometry group. We show that for such an action there exists a corresponding isometric action on a dual compact symmetric space, which reflects many properties of the original action. For example, the principal isotropy subgroups of both actions are locally isomorphic and the dual action is (hyper)polar if and only if the original action is (hyper)polar. This fact provides many new examples for polar actions on symmetric spaces of noncompact type and we use duality as a method to study polar actions by reductive algebraic subgroups in the isometry group of an irreducible symmetric space. Among other applications, we show that they are hyperpolar if the space is of type III and of higher rank; we prove that such actions are orbit equivalent to Hermann actions if they are hyperpolar and of cohomogeneity greater than one. Furthermore, we classify polar actions by reductive algebraic subgroups of the isometry group on noncompact symmetric spaces of rank one.
Classification : 53C35, 57S20
Mots-clés : Symmetric spaces, duality, polar actions 
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     author = {A. Kollross },
     title = {Duality of {Symmetric} {Spaces} and {Polar} {Actions}},
     journal = {Journal of Lie theory},
     pages = {961--986},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2011},
     url = {http://geodesic.mathdoc.fr/item/JLT_2011_21_4_JLT_2011_21_4_a10/}
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A. Kollross . Duality of Symmetric Spaces and Polar Actions. Journal of Lie theory, Tome 21 (2011) no. 4, pp. 961-986. http://geodesic.mathdoc.fr/item/JLT_2011_21_4_JLT_2011_21_4_a10/