Non-existence of exceptional orbits under polar actions on Hilbert spaces

TL;DR

This paper proves that polar actions on Hilbert spaces by connected Lie groups lack exceptional orbits, extending finite-dimensional results and providing geometric insights into symmetric spaces.

Contribution

It generalizes the non-existence of exceptional orbits for polar actions from finite-dimensional Euclidean spaces to infinite-dimensional Hilbert spaces.

Findings

Polar actions on Hilbert spaces have no exceptional orbits.

Finite-dimensional results are extended to infinite-dimensional settings.

Provides a geometric proof for symmetric spaces case.

Abstract

We prove that any polar action on a separable Hilbert space by a connected Hilbert Lie group does not have exceptional orbits. This generalizes a result of Berndt, Console and Olmos in the finite dimensional Euclidean case. As an application, we give a simple geometric proof of the fact that any hyperpolar action on a simply connected compact Riemannian symmetric space by a connected Lie group does not have exceptional orbits.