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Duality of symmetric spaces and polar actions. (English) Zbl 1234.53016
A proper isometric action of a Lie group on a Riemannian manifold is called polar if it admits a section, i.e., a complete immersed submanifold which intersects all orbits perpendicularly at each point of intersection. If the section is flat in the induced metric, then the action is called hyperpolar.
In this paper, the author uses duality between symmetric spaces of compact and of noncompact type to transfer results about polar actions on symmetric spaces of compact type to the noncompact case. More precisely, if \({\mathfrak g} = {\mathfrak k} \oplus {\mathfrak p}\) is the Cartan decomposition of a symmetric space \(M=G/K\) of noncompact type, and \(H\subset G\) is a canonically embedded Lie subgroup, i.e., one whose Lie algebra decomposes as \({\mathfrak h}=({\mathfrak h}\cap {\mathfrak k})\oplus ({\mathfrak h}\cap {\mathfrak p})\), then we obtain an action on a compact dual \(M^*\) induced by the Lie algebra \({\mathfrak h}^*=({\mathfrak h}\cap {\mathfrak k})\oplus i({\mathfrak h}\cap {\mathfrak p})\). It is shown that the \(H\)-action is (hyper)polar if and only if the thus constructed dual action is (hyper)polar.
Not all Lie subgroups of \(G\) are canonically embedded with respect to a Cartan decomposition of \(G\), but this is the case for reductive algebraic subgroups of \(G\), which is the statement of a generalization of the Karpelevich-Mostow theorem, see [A. L. Onishchik, E. B. Vinberg and V. V. Gorbatsevich, Lie groups and Lie algebras III. Structure of Lie groups and Lie algebras. Berlin: Springer-Verlag (1994; Zbl 0797.22001)], originally proven for semisimple subgroups by F. I. Karpelevich [“Surfaces of transitivity of a semisimple subgroup of the group of motions of a symmetric space” (Russian), Dokl. Akad. Nauk SSSR, n. Ser. 93, 401–404 (1953; Zbl 0052.38901)] and G. D. Mostow [“Some new decomposition theorems for semi-simple groups”, Mem. Am. Math. Soc. 14, 31–54 (1955; Zbl 0064.25901)].
Under the additional assumption that the acting subgroup is reductive algebraic, this duality between (hyper)polar actions directly implies that several results for (hyper)polar actions on irreducible symmetric spaces of compact type obtained by the author in [“A classification of hyperpolar and cohomogeneity one actions”, Trans. Am. Math. Soc. 354, No. 2, 571–612 (2002; Zbl 1042.53034); “Polar actions on symmetric spaces”, J. Differ. Geom. 77, No. 3, 425–482 (2007; Zbl 1139.53025); “Low cohomogeneity and polar actions on exceptional compact Lie groups”, Transform. Groups 14, No. 2, 387–415 (2009; Zbl 1179.22014)] carry over to the noncompact setting, such as the statement that a polar action on a space of rank greater than one is automatically hyperpolar. It should be noted that for arbitrary subgroups (not necessarily reductive algebraic) this statement is not true, as shown by J. Berndt, J. C. Díaz-Ramos and H. Tamaru [“Hyperpolar homogeneous foliations on symmetric spaces of noncompact type”, J. Differ. Geom. 86, No. 2, 191–235 (2010; Zbl 1218.53030)].
Finally, the author obtains a classification of polar actions of reductive algebraic subgroups on noncompact rank-one symmetric spaces, using duality and the classification in the compact case by F. Podestà and G. Thorbergsson [“Polar actions on rank-one symmetric spaces”, J. Differ. Geom. 53, No. 1, 131–175 (1999; Zbl 1040.53071)].

MSC:
53C35 Differential geometry of symmetric spaces
57S20 Noncompact Lie groups of transformations
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