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You mean restriction to $G(o)$ has single multiplicity? Then yes for $n=2$. This follows from the theory of types, but should be already in Silberger's LNM. I don't know a more conceptual proof. You need classiifcation of all irreducible smooth admissible representations and then restricting, though. It's annoying. For $n>3$, we don't actually know the representation theory of $GL_n(Z_p)$, so I thing it's pretty much open. The corresponding question for $GL(n,R)$ or $GL(n, C)$ seem to be wrong for $n>3$, but are right for $n=2$. I would only care about the types necessary to classify smooth admissible representations. There I think you have a possitive answer meaning they occur with single multiplicity in irreducible admissible represenations.