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Let $G = SO_5(\mathbf{Q}_p)$ be the split special orthogonal group over $\mathbf{Q}_p$, and $H \subset G$ the group $SO_4(\mathbf{Q}_p)$, embedded in the usual way as the stabiliser of an anisotropic vector.

Let $\pi$ be an irreducible unramified principal series representation of $G$. Is $Hom_H( \pi \mid_H, \mathbf{C})$ non-zero? (That is, is $\pi$ necessarily $H$-distinguished?)

(NB: I'm aware that the Gan--Gross--Prasad conjectures give all sorts of insights into the spaces $Hom_H( \pi \mid_H, \tau)$ where $\tau$ is an irreducible rep of $H$. But they always seem to assume $\tau$ is generic, which the trivial representation is not.)

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    $\begingroup$ This is a comment, but I don't have enough reputation points to comment. I think the representations of $\mathrm{SO}(n)$ that have nonzero periods over some embedded $\mathrm{SO}(n-1)$ are the ones that are lifts from $\widetilde{SL}_2$ via the theta correspondence. (So to be clear: I think the answer to your question is "Yes, precisely when $\pi$ is a theta lift from $\widetilde{SL}_2(\mathbf{Q}_p)$".) $\endgroup$
    – user104335
    Commented Feb 24, 2017 at 18:47
  • $\begingroup$ That sounds very interesting, but how do I make it more concrete? If I know the Satake parameter of $\pi$, how do I tell if it's a theta lift? $\endgroup$
    – David Loeffler
    Commented Feb 24, 2017 at 19:24
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    $\begingroup$ Since the sum of the dimension of the $H=SO(n-1)$ and the dimension of the minimal parabolic $P$ in the (maximally split) $G=SO(n)$ is greater than or equal to the dimension of $G$, it is likely that the double coset space $P\backslash G/H$ is finite. So a Mackey-Bruhat computation should determine conditions for that space of intertwinings to be non-trivial. $\endgroup$
    – paul garrett
    Commented Feb 24, 2017 at 20:47

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