First, I'd like to understand what the compact open subgroups of $H(\mathbb{Q}\_p)$ are, where $H$ is an inner form of $GL\_n$ over $\mathbb{Q}\_p$.


Second, I'd like to know where I can read about this for other reductive groups.


Any pointers would be greatly appreciated. Thanks!



Reference Request: Compact subgroups of p-adic Reductive Groups
You get compact subgroups by taking compact-open subgroups of algebraic subgroups. My understanding is that they are more-or-less all compact subs, e.g. any compact subgroup $H$ should have a finite index subgroup of this form.


Uri Bader's reference to Pink's 1998 paper is a good start. Pink proves this sort of rigidity theorem over a local field of positive characteristic. I believe he should explain why it is "eazzzy" over a local field of zero characteristic but I have no intention of checking it: no pink till Xmas! 


