Reduced invariant sets

TL;DR

This paper investigates conditions under which the ideal of a K-stable algebraic set equals the ideal generated by its invariant part, extending the analysis to complex reductive group actions.

Contribution

It provides necessary and sufficient conditions for the equality of ideals and their radicals in the context of compact Lie groups and complex reductive groups.

Findings

Criteria for I(X)= I_K(X) in real algebraic sets

Criteria for (X)=_K(X) in real algebraic sets

Extension of results to complex reductive group actions

Abstract

Let K be a compact Lie group and W a finite-dimensional real K-module. Let X be a K-stable real algebraic subset of W. Let I(X) denote the ideal of X in R[W] and let I_K(X) be the ideal generated by I(X)^K. We find necessary conditions and sufficient conditions for I(X)= I_K(X) and for \sqrt{I_K(X)}=I(X). We consider analogous questions for actions of complex reductive groups.