Is $(G,K)$ a strong Gelfand pair? Let $F$ be a $p$-adic field with ring of integers $\mathcal{O}$. When $G={\rm GL}_n$, it is a classical result that $(G(F),G(\mathcal{O}))$ is a Gelfand pair. Is it actually a strong Gelfand pair? I am particularly interested in the case $n=2$.
 A: $\DeclareMathOperator\Res{Res}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\GL{GL}$You mean restriction of irreducible smooth, admissible representations of $G(F)$ to $G(o)$ decomposes with multiplicity one? Then yes for $n=2$.
Here are some more exact references:

*

*One-dimensional representations are obvious.


*For supercuspidal representations: Kristina Hansen. "Restriction to $\GL_2({\scr O})$ of supercuspidal representations of $\GL_2(F)$", Pacific J. Math. 130 (2) 327–349, 1987. https://projecteuclid.org/euclid.pjm/1102690181
-For principal series representations and Steinberg: Casselman - Restriction to $GL_2(o)$: https://doi.org/10.1007/BF01355984
Note here that $\Res_{G(o)} \Ind_{B(F)}^{G(F)} \mu = \Ind_{B(o)}^{G(o)} \mu$, of which Casselman gives an explicit decomposition in the first lemma. The Steinberg as a quotient/submodule of some $\Ind_{B(F)}^G(F) \mu$ for some $\mu$ has then also the property.
I don't know a more conceptual proof. You need classification of all irreducible smooth admissible representations and then you need to look at the restriction, though. It's annoying. For $n>3$, we don't actually know the representation theory of $\GL_n(Z_p)$, so I think it's pretty much open. The corresponding questions 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 positive answer meaning they occur with single multiplicity in irreducible admissible representations.
