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.

  • $\begingroup$ does not the question amount to calssifying all closed subsets of a fixed compact open subgroup? And closed subsets are pretty much arbitrary (of course, they are closed). $\endgroup$ – Venkataramana Jul 20 '13 at 13:00
  • $\begingroup$ Yes. Thought I formulate the question in general, I'm particularly interested in what dose the intersection of a compact subset $C$ with $a\omega U$ look like? Does this have a simple description? $\endgroup$ – user1832 Jul 21 '13 at 1:18

R. Pink's article "Compact subgroups of linear algebraic groups" is about a qualitative classification of all compact subgroups $Γ ⊂ GL_n(F)$, where $F$ is a local field and $n$ is arbitrary. This is not a complete answer to your question (it is about compact subgroups indeed), but the introduction alone gives a good idea, what the nature of this question is, and will be helpful, I think. http://www.math.ethz.ch/~pink/ftp/LastVersion.pdf.


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