Let $G$ be an infinite compact separable Hausdorff metric group, and $H\subset G$ a closed subgroup, such that the left $G$-action on $G/H$ is effective (i.e., $H$ doesn't contain a non-trivial closed normal subgroup of $G$). For every closed subgroup $K\subset G$ let $G/H^K\subset G/H$ be the fixed point set of $K$ in $G/H$, which is a closed subset.
Question:
Under which circumstances can $G/H^K$ have a non-empty interior for $K\neq\{1\}$? We may assume $G/H$ to be connected, although I am not sure that this is critical.
Discussion:
If $G$ is Lie then its action on $G/H$ is by analytic diffeomorphisms. An analytic map that is identity in a neighbourhood would be such on the entire connected component, I think. Thus $G/H^K$ either contains no neighbourhoods or occupies an entire connected component.
In the general case above, $G/H^K$ with non-empty interior still sounds pathological to me, but I haven't found a good description of that pathology so far.
Thank you.
