A Quotient Restriction Theorem for actions of real reductive groups

TL;DR

This paper extends the Chevalley Restriction Theorem to real reductive group actions on topological spaces, providing a stratification-based description of the quotient space in terms of orbit types.

Contribution

It introduces a new stratification for real reductive group actions and describes the quotient in terms of normalizer subgroup quotients, generalizing classical results.

Findings

Established a stratification based on orbit types of closed orbits.

Provided a description of the quotient space using normalizer subgroup quotients.

Extended the Chevalley Restriction Theorem to a broader class of group actions.

Abstract

We prove a version of the Chevalley Restriction Theorem for the action of a real reductive group G on a topological space X which locally embeds into a holomorphic representation. Assuming that there exists an appropriate quotient X//G for the G-action, we introduce a stratification which is defined with respect to orbit types of closed orbits. Our main result is a description of the quotient X//G in terms of quotients by normalizer subgroups associated to the stratification.