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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 ${\rm GL}_2({\scr O})$ of supercuspidal representations of ${\rm GL}_2(F)$", 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 $Res_{G(o)} Ind(B(F))^{G(F)} \mu = Ind_{B(o)}^{G(o)} \mu$, 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 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.

$\DeclareMathOperator\Res{Res}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\GL{GL}$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.

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: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=imagefirstpage_1&handle=euclid.pjm/1102690181

-For principal series representations and Steinberg: Casselman - Restriction to $GL_2(o)$: http://link.springer.com/article/10.1007%2FBF01355984#page-1 Note here that $Res_{G(o)} Ind(B(F))^{G(F)} \mu = Ind_{B(o)}^{G(o)} \mu$, 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 classifcation 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 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.

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 ${\rm GL}_2({\scr O})$ of supercuspidal representations of ${\rm GL}_2(F)$", 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 $Res_{G(o)} Ind(B(F))^{G(F)} \mu = Ind_{B(o)}^{G(o)} \mu$, 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 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.

You mean restriction of irreducibele smooth, admissible representations of $G(F)$ to $G(o)$ decomposes single multiplicity? Then yes for $n=2$. This follows from the theory of types, but should be already in Silberger's LNM.

Here are some more exact references:

One-dimensional representations are obvious.

For supercuspidal representations: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=imagefirstpage_1&handle=euclid.pjm/1102690181

For principal series representations and Steinberg: Casselman - Restriction to $GL_2(o)$: http://link.springer.com/article/10.1007%2FBF01355984#page-1

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.

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: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=imagefirstpage_1&handle=euclid.pjm/1102690181

-For principal series representations and Steinberg: Casselman - Restriction to $GL_2(o)$: http://link.springer.com/article/10.1007%2FBF01355984#page-1 Note here that $Res_{G(o)} Ind(B(F))^{G(F)} \mu = Ind_{B(o)}^{G(o)} \mu$, 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 classifcation 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 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.