Representations of low copolarity

TL;DR

This paper classifies certain irreducible representations of compact Lie groups based on the geometric structure of their orbit spaces, revealing close relations to symmetric spaces with one notable exception.

Contribution

It provides a classification of irreducible representations with low copolarity, linking their orbit space geometry to symmetric spaces and identifying a unique exception.

Findings

Orbit spaces are isometric to those of 7, 8, or 9-dimensional Lie group representations.

Most such representations are related to symmetric spaces.

One exceptional case deviates from the symmetric space pattern.

Abstract

We classify irreducible representations of compact connected Lie groups whose orbit space is isometric to the orbit space of a representation of a compact Lie group of dimension~$7$, $8$ or $9$. They turn out to be closely related to symmetric spaces, with one exception only.