## Maximal compact subgroups for locally profinite groups and generalization

Let $G$ a locally profinite not compact group and we suppose that $G$ have maximal compact subgroups, for example $G$ reductive group over p-adic field.

We call projective Tower of $G$ any strictly decreasing sequence $\mathcal{K}=(K_{n})_{n\in\mathbb{N}}$ of open compact subgroup of $G$. We define an ordering on the set $\mathcal{Pt}(G)$ of projective tower of $G$ as follows , $\mathcal{K}\leqslant\mathcal{L}$ if and only if $\mathcal{K}$ is a sub-sequence of $\mathcal{L}$, it is not difficut to verify that is an ordering.

We call maximal projective tower of $G$ any $\mathcal{K}\in\mathcal{Pt}(G)$ maximal for this ordering. It is not difficut to verify that there exists such elements and for such element $\mathcal{K}=(K_{n})_{n\in\mathbb{N}}$ the first term of the sequence is maximal compact subgroup of $G$.

The group $G$ acts on the set $\mathcal{Pt}_{max}(G)$ of maximal projective tower of $G$ by conjugation.

My question is : Is this action are transitive ? I think is not transitive.

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No, not even if $G=\mathbf{G}(k)$ for a split connected semisimple group $\mathbf{G}$ over a $p$-adic field $k$. If $\mathbf{G}^{\rm{ad}}$ denotes the adjoint central quotient of $\mathbf{G}$ then $\mathbf{G}^{\rm{ad}}(k)$ acts on $G$ and typically $G \rightarrow \mathbf{G}^{\rm{ad}}(k)/\mathbf{G}^{\rm{ad}}(O_k)$ is not surjective; its "cokernel" set indexes $G$-conjugacy classes in the $\mathbf{G}^{\rm{ad}}(k)$-orbit of $\mathbf{G}(O_k)$ (and that's just the hyperspecials!); e.g., the ${\rm{PGL}}_n(k)$-orbit of ${\rm{SL}}_n(O_k)$ consists of $n$ distinct ${\rm{SL}}_n(k)$-conjugacy classes. – kreck Dec 24 at 2:14