Let $G$ be the $k$-points of a connected, reductive group $\mathbf G$ over a local field $k$. I have heard several statements about compact subgroups and Iwasawa decomposition, mostly in the context of semisimple real lie groups. I wanted to know what was true for reductive groups. So my question is, which of the following six statements are true? (one is a definition)
1 . There exist finitely many conjugacy classes of maximal compact subgroups of $G$. If $k = \mathbf R$, then all maximal compact subgroups are conjugate.
2 . If $k$ is nonarchimedean, then maximal compact subgroups are always open.
Let $K$ be a compact subgroup of $G$, and $P$ a parabolic subgroup of $G$. Suppose that the product mapping $P \times K \rightarrow G$ is a homeomorphism.
3 . If $P, K$ are as above, $G = PK$ is called an Iwasawa decomposition.
If my definition of Iwasawa decomposition is incorrect, please keep in mind my meaning of the term in the following questions.
4 . If $P$ is a parabolic subgroup of $G$, then there exists a compact subgroup $K$ such that $PK$ is an Iwasawa decomposition.
5 . If $P_0$ is a minimal parabolic, there exists a maximal compact subgroup $K$ such that $P_0K$ is an Iwasawa decomposition.
Let $P_0$ be a minimal parabolic of $G$, let $\mathbf A_0$ be a maximal $k$-split torus of $\mathbf G$ with $\mathbf A_0(k) \subseteq P_0$, and let $\Phi = \Phi(\mathbf A_0,\mathbf G)$ be the roots of $\mathbf A_0$ in $G$. Let $\Delta$ be the base of $\Phi$ corresponding to $P_0$, so that the subsets of $\Delta$ correspond to $P_0$-standard parabolics of $G$. Let $\theta \subseteq \Delta$, and let $w \in G(k)$ be an element in the Weyl group $N_G(A_0)/Z_G(A_0)$ such that $w(\theta) \subseteq \Delta$. Let $P, P'$ be the standard parabolics of $G$ corresponding to $\theta, w(\theta)$.
6 . If $K$ is a compact subgroup such that $PK$ is an Iwasawa decomposition, then $P'K$ is also one.