[Paper Review] Duality of symmetric spaces and polar actions

Andreas Kollross|arXiv (Cornell University)|Jan 9, 2011

Advanced Algebra and Geometry33 references15 citations

TL;DR

This paper establishes a duality between isometric actions on Riemannian symmetric spaces of noncompact type and their compact duals, showing that an action is (hyper)polar if and only if its dual action is. The method enables classification of polar and hyperpolar actions on noncompact symmetric spaces by reducing them to compact duals, yielding new examples and generalizing known results.

ABSTRACT

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.

Motivation & Objective

• To establish a duality between isometric actions on symmetric spaces of noncompact type and their compact duals.
• To investigate whether polar and hyperpolar actions on noncompact symmetric spaces can be studied via their dual actions on compact symmetric spaces.
• To generalize classification results from compact to noncompact symmetric spaces using duality.
• To determine conditions under which polar actions on noncompact symmetric spaces are hyperpolar.
• To classify polar actions by reductive algebraic subgroups on noncompact symmetric spaces of rank one.

Proposed method

• Utilizes the duality between symmetric spaces of noncompact and compact type, particularly via the Lie algebra decomposition 𝔤 = 𝔨 ⊕ 𝔭 and its complexification.
• Applies the correspondence between reductive algebraic subgroups of the isometry group of a noncompact symmetric space and their duals in the compact dual space.
• Relies on the fact that principal isotropy subgroups of dual actions are locally isomorphic, preserving geometric properties like polarity and hyperpolarity.
• Employs the criterion that an action is (hyper)polar if and only if its dual action is (hyper)polar, enabling transfer of classification results.
• Uses classification results on compact symmetric spaces—especially for rank one and higher rank spaces—to infer properties of noncompact actions.
• Applies known results on Hermann actions, isotropy actions, and symmetric pairs to construct dual actions and verify their geometric properties.

Experimental results

Research questions

• RQ1 Does a polar action on a noncompact symmetric space of noncompact type correspond to a polar action on its compact dual?
• RQ2 Under what conditions is a polar action on a noncompact symmetric space necessarily hyperpolar?
• RQ3 Can classification results for hyperpolar actions on compact symmetric spaces be extended to noncompact symmetric spaces via duality?
• RQ4 Are there nontrivial polar actions on irreducible noncompact symmetric spaces of higher rank, and if so, under what conditions?
• RQ5 What is the structure of polar actions by reductive algebraic subgroups on noncompact symmetric spaces of rank one?

Key findings

• An isometric action on a symmetric space of noncompact type is (hyper)polar if and only if its dual action on the compact dual space is (hyper)polar.
• The duality construction provides many new examples of polar and hyperpolar actions on noncompact symmetric spaces of noncompact type.
• All polar actions on the octonionic projective plane 𝕆ℙ² are shown to have a totally geodesic orbit, as a consequence of their dual actions on the compact dual.
• For irreducible symmetric spaces of type III and higher rank, any polar action by a reductive algebraic subgroup of the isometry group is hyperpolar.
• Hyperpolar actions of cohomogeneity greater than one on noncompact symmetric spaces are orbit equivalent to Hermann actions.
• The classification of polar actions on noncompact symmetric spaces of rank one is completed via duality, showing all such actions arise from dual compact actions.