Assume that $G$ is a finite group acting by automorphisms on a topological space $X$ (For concreteness, assume that $X$ is a complex manifold). Let $H$ be a parabolic subgroup of $G$. Denote by $X_H$ the isotropy subset of elements $x$ in $X$ with stabilizer $Stab(x)=H$ and let $X^H$ be the set of $H$-fixed points in $X$. The isotropy subset $X_H$ is a subset in $X^H$. Is it true that $X_H$ is open and dense in $X^H$?
On page 20 of the paper https://arxiv.org/pdf/math/0212279.pdf it is stated that in the case when $X$ is a (symplectic) vector space, $X_H$ is Zariski-open, hence dense in $X^H$. Does anyone know a proof of that statement and more importantly if it generalizes to arbitary topological spaces $X$ with a finite group action.