# branching laws for $p$-adic representations of reductive groups

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.

• 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. – Kimball Oct 25 '19 at 9:23
• @Kimball You're right I did not understand the question correctly ... – Paul Broussous Oct 25 '19 at 11:31
• @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. – Kimball Oct 25 '19 at 14:29
• @Kimball Sorry I shall be more specific, thank you! – sawdada Oct 25 '19 at 17:19

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).
• Are there any $p$-adic multiplicity 1 results known, like a (non-reductive) analogue of Whittaker models? – Kimball Oct 25 '19 at 18:19