On orbit spaces of representations of compact Lie groups

TL;DR

This paper explores the structure of orbit spaces resulting from orthogonal representations of compact Lie groups, classifying certain irreducible representations and analyzing their quotient space properties.

Contribution

It provides new structural insights into orbit spaces and classifies irreducible representations with specific cohomogeneity, advancing understanding of representation quotients.

Findings

Classified all irreducible representations with cohomogeneity four or five.

Identified properties of quotient spaces that are shared by different representations.

Provided structural results linking representation properties to their orbit space metrics.

Abstract

We investigate orthogonal representations of compact Lie groups from the point of view of their quotient spaces, considered as metric spaces. We study metric spaces which are simultaneously quotients of different representations and investigate properties of the corresponding representations. We obtain some structural results and apply them to study irreducible representations of small copolarity. As an important tool, we classify all irreducible representations of connected groups with cohomogeneity four or five.