This is the question that I actually meant to ask in Which groups $H$ act transitively on a noncompact symmetric space $G/K$? I got confused about the definition of parabolic subgroups, so the answer to that question is something that I'm actually not interested in. What I'm really interested in is the following:
All Lie groups here are assumed to be real. Let $M=G/K$ be a symmetric space of noncompact type and $H \subset G$. $H$ acts on $G/K$ by left-multiplication. Assume that action of $H$ on $M$ is transitive. Let $G=KAN$ be the Iwasawa decomposition of $G$.
Question 1: Does $H$ contain a copy of $AN$?
That is: Is there a subgroup of $H$ isomorphic to $AN$?
Question 2: Does $H$ contain a subgroup $g AN g^{-1}$, where $g \in G$?
Here is what I've thought so far:
For convenience we write $G=NAK$. $H$ acting on $M$ transitively means that in particular for all $na \in NAK$ we find $\hat{h} \in H$, $\hat{k} \in K$ such that $\hat{h}=na \hat{k}$. Let $\hat{h}=n_h a_h k_h$ be the Iwasawa decomposition of $\hat{h}$. From the uniqueness of this representation follows that $n_h=n$, $a_h=a$, $k_h=\hat{k}$. Thus, we get an injective map $$ AN \rightarrow H \\ an \mapsto ank_h. $$ Now, this map need not be continuous, but it suggests that the answer to question 1 will be "yes".
