Let $F$ be a local field of characteristic zero, and $G$ a linear algebraic group with finite connected components over $F$. We will consider the $G(F)$, and give it $p$-adic topology. Let $C$ be some compact subset in $G(F)$, I'm wondering if we have a description/characterization of $C$?
For example, for $GL_n(F)$, if $C$ is a compact subset, then the components of elements in $C$ is bounded from above. Of course, this is not enough to ensure $C$ is compact, and I would like to know if there is any (partial) description/characterization for general $G$.
In particular, if $\omega$ is a representative of some Weyl group element, $a$ is an element in a maximal split torus, intersect $C$ with $a\omega U$, where $U$ is the unipotents of a Borel subgroup of $G(F)$. Can we conclude that $C\cap a\omega U$ is contained in $a\omega Y$ for some open compact subgroup $Y\subset U$?
This is true for $GL_n(F)$, and I'd like to know if it holds for general $G$. If not, for which class of groups (classical?) this is OK?
I checked some books on algebraic groups,but didn't find an answer. Sorry if this question is not appropriate here, and much appreciated for any comments.