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

Question

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.

Answer 1

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).