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Let $F$ be a $p$-adic field, $G=SO_{2n}(F)$ the split special orthogonal group and $H=SO_{2n-1}(F)$, taken as a subgroup of $G$. Assume that we have an irreducible, admissible representation $\pi$ of $G$ such that its restriction $\pi^\prime$ to $H$ is also irreducible (and admissible).

Now let $K$ be a compact open subgroup of $G$. It's obvious that $\pi^K\neq 0$ implies $\pi^{\prime K\cap H}\neq 0$. However, is the other implication true?

Does $\pi^{\prime K\cap H}\neq 0$ imply $\pi^K\neq 0$?

Might the question be easier to answer for some special $K$, e.g. congruence subgroups?

I got interested in this by looking at what happens to $K$-fixed vectors under the local descent construction of Jiang and Soudry. In that situation, we can also assume $\pi^\prime$ to be supercuspidal and generic.

Studying the irreducibility of $\pi^\prime$ is the setting of the Gross-Prasad Conjecture. If I understand it correctly, it would tell us for which representations $\pi$ we have that $\pi^\prime$ is irreducible. However, I don't see if it could be useful in looking at the $K$-fixed vectors.

Or Frobenius reciprocity might be useful, relating $\pi^\prime$ to $Ind_H^G \pi^\prime$. It would for example suffice to show that $Ind_H^G \pi^\prime\simeq\pi$ - although I don't think this can be true in general.

(This is my first MO question, so I'll appreciate any comments or suggestions. Thanks!)

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