Reduced invariant sets (Q1761409)

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 \(I(X)\cap R[W]^K\). The author finds necessary conditions and sufficient conditions for \(I(X) = I_K(X)\) and for \(\sqrt{I_K(X)} = I(X)\). These conditions are stated utilising the slice representations. To prove such conditions the author consider analogous questions for actions of complex reductive groups.