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There are many papers studying branching laws of irreducible admissible complex representations of classical groups over local fields, are there some analogues for $p$-adic representations?

For example, consider an irreducible admissible $p$-adic representation of $GL_2(\mathbb Q_p)$, what are the multiplicities of its restriction to $GL_1(\mathbb Q_p)$ ? How about $GL_n(\mathbb Q_{p^2})$ to $GL_n(\mathbb Q_p)$ ?

More precisely, $p$-adic representations of $G$ mean admissible unitary $L$-Banach representations of $G(\mathbb Q_p)$ where $G$ is a reductive group over $\mathbb Q_p$, and $L$ is a finite extension of $\mathbb Q_p$. They are natural objects in the $p$-adic Langlands program. And we care about dimension of $Hom_H(\pi|_{H}, \sigma)$ as in the classical branching law where $H$ is a reductive subgroup of $G$, and the representations $\pi$ and $\sigma$ are both irreducible.

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  • $\begingroup$ Can you clarify whether you are asking about representations defined over $p$-adic fields or over $\mathbb C$? Paul Broussous' answer assumes the latter, though this falls under the heading of "irreducible admissible complex representations of classical groups over local fields" which you said you knew about. $\endgroup$ – Kimball Oct 25 at 9:23
  • $\begingroup$ @Kimball You're right I did not understand the question correctly ... $\endgroup$ – Paul Broussous Oct 25 at 11:31
  • $\begingroup$ @PaulBroussous Well, I'm not sure---I find the question a bit vague as the term "$p$-adic representation" is. Also, previous questions of the OP make me wonder which is actually meant. $\endgroup$ – Kimball Oct 25 at 14:29
  • $\begingroup$ @Kimball Sorry I shall be more specific, thank you! $\endgroup$ – sawdada Oct 25 at 17:19
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If you're asking about admissible p-adic Banach space representations in the sense of Schneider--Teitelbaum, then I think virtually nothing is known in this setting about branching laws, even in the simplest cases like branching from $GL_2(\mathbb{Q}_p)$ to the diagonal maximal torus (a case which we understand extremely well for smooth representations, thanks to Waldspurger's theorem).

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  • $\begingroup$ Are there any $p$-adic multiplicity 1 results known, like a (non-reductive) analogue of Whittaker models? $\endgroup$ – Kimball Oct 25 at 18:19
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    $\begingroup$ Colmez's work on p-adic Langlands uses something he calls a Kirillov model; but I don't know if it has the same uniqueness properties as the Whittaker/Kirillov models in the smooth theory. $\endgroup$ – David Loeffler Oct 26 at 9:13

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