Suppose $G$ is a p-adic algebraic group, $P=MN$ a parabolic subgroup of $G$ with its Levi decomposition, $\sigma$ be a irreducible representation of $M$, we use $I(\sigma)$ to denote the unique quotient of the normalized parabolic induction from $P$ to $G$. Now let $H$ be a closed subgroup of $G$.
The question I want to ask is that what's the relations between the existence of open-$H$ orbits in $P\G$ and $H$-invariant distributions on $I(\sigma)$, especially when we take $P$ to be the minimal parabolic subgroup?
Also I'm also wondering what's happening in the archimedean case.
Any answer or reference is welcome. Thanks.