Polar actions on symmetric spaces of higher rank

TL;DR

This paper proves that polar actions of cohomogeneity two on higher rank symmetric spaces are hyperpolar, confirming a conjecture that all polar actions on such spaces are hyperpolar, thus advancing understanding of symmetry and group actions.

Contribution

It establishes that polar actions of cohomogeneity two on higher rank symmetric spaces are hyperpolar, extending previous results and confirming a key conjecture in the field.

Findings

Polar actions of cohomogeneity two are hyperpolar on higher rank symmetric spaces.

Polar actions induced by reductive algebraic subgroups are hyperpolar.

The conjecture that all polar actions on higher rank symmetric spaces are hyperpolar is confirmed.

Abstract

We show that polar actions of cohomogeneity two on simple compact Lie groups of higher rank, endowed with a biinvariant Riemannian metric, are hyperpolar. Combining this with a recent result of the second-named author, we are able to prove that polar actions induced by reductive algebraic subgroups in the isometry group of an irreducible Riemannian symmetric space of higher rank are hyperpolar. In particular, this result affirmatively settles the conjecture that polar actions on irreducible compact symmetric spaces of higher rank are hyperpolar.