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Yes, this also follows from the same approximation theorems. One very concrete way of stating the strong approximation theorem for $Sp_{2g}$ is that $Sp_{2g}(\mathbf{Z})$ surjects onto $Sp_{2g}(\math …

In general, all we can say from "general abstract nonsense" is that if $\sigma$ is a subrepresentation of $Ind_P^G(\pi)$, then $\pi$ is a quotient of $J_N(\sigma)$; but you don't immediately get any f …

Let $\pi$ be the irreducible generic unramified representation of $Sp(W) $ that is a subquotient of $Ind(\chi_1, \dots, \chi_n)$. I think the key here is to realise that this does not exist for all …

(Expanded version of my earlier comment, reposted as an answer): If $F^2(a) = a$, $F^2(b) = b$, and $a F(a) + \pi b F(b) = 1$, then it is not too difficult to show that $a$ and $b$ have to be in $\ma …

There's an explicit description of maximal compact subgroups of all unitary groups over local fields (not necessarily quasi-split) in section 3 of this paper: Gan, Hanke, and Yu, "On an exact mass fo …

Let's look at the case of representations associated to modular forms. I'm going to switch the roles of $\ell$ and $p$, because I find $\ell$-adic Hodge theory disturbing; so I'm going to look at $\rh …

This came up in a paper of mine not so long ago, and my coauthors and I were surprised that it wasn't made explicit in the standard references, so we wrote it out ourselves: Dembélé, Lassina; Loeffler …

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 s …