Abstract :
The general problem underlying this article is to give a qualitative classification of all compact subgroups Γ GLn(F), whereFis a local field andnis arbitrary. It is natural to ask whether Γ is an open compact subgroup ofH(E), whereHis a linear algebraic group over a closed subfieldE F. We show that Γ indeed has this form, up to finite index and a finite number of abelian subquotients. When Γ is Zariski dense in a connected semisimple group, we give a precise openness result for the closure of the commutator group of Γ. In the case char(F) = 0 the answers have long been known by results of Chevalley and Weyl. The motivation for this work comes from the positive characteristic case, where such results are needed to study Galois representations associated to function fields. We also derive openness results over a finite number of local fields.
