Let $G$ be a Lie group and $H\subseteq G$ a closed subgroup. Then, $G/H$ is a smooth manifold and inherits a smooth action of $G$: $$G\times G/H\longrightarrow G/H,\quad g_1\cdot(g_2H)=g_1g_2H.$$
Question: Is this action proper?
If not, are there reasonable conditions so that it is proper? If the group $G$ is compact then of course it will be proper. But I am more interested in the case where $G$ is non-compact. For example, what happens if $G$ is the complexification of a compact Lie group $K$ and $H=T_{\Bbb C}$ is the complexification of a maximal torus $T$ in $K$?