Double Coset Space Hausdorff?

Question

Let $G$ be a locally compact, totally disconnected Hausdorff group, and let $H,N \subseteq G$ be closed subgroups. It is known that $G/N$ is Hausdorff (in the quotient topology, and has the other two properties as well). What can be said about when, or if, the double coset space $H\backslash G/N$ is Hausdorff? (Context: $G$ is a $p$-adic connected reductive group, and $H,N$ are the unipotent radicals of parabolic subgroups.)

Answer 1

In general, no. For instance if $N$ is normal, then the quotient is Hausdorff iff $NH$ is closed. For a counterexample, choose $G=\mathbf{Z}_p\times\mathbf{Z}$, $H$ cyclic generated by $(0,1)$, and $N$ cyclic generated by $(z,1)$ with $z\neq 0$. Then $N+H$ (it's an additive group, so it corresponds to $NH$) is the direct product $\langle z\rangle\times\mathbf{Z}$, which is not closed.

In your specific context, it's another question. Indeed, for algebraic actions of unipotent groups on affine varieties, orbits are (Zariski) closed. This is not exactly what's needed (one needs $G/N$ affine, which is not always the case although it's quasi-affine in the cases I can think of, and one has to check closedness of orbits at the level of points, in the $p$-adic topology), but at least it gives some clue to hope a positive answer.