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Let $E/F$ be a quadratic field extension of p-adic fields. Let $V$ be a (skew-)Hermitian space and $U(V)$ be the unitary group. Let $GU(V)$ be the similitude unitary group. Given an irreducible smooth representation $\pi$ of $GU(V)$, do we know that the restriction $\pi|_{U(V)}$ has multiplicity one?

For the pair $(GL(n),SL(n))$ similar results are proved by Tadic. For the pair $(GSp, Sp)$, similar results are proved by Adler-Prasad. I am wondering if the unitary group version is true or not.

Thanks.

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In case you didn't already see it, this question has now been answered in the affirmative by Adler and Prasad in Theorem 12a here: $[$1$]$

$[$1$]$ Jeffrey D. Adler, Dipendra Prasad. Multiplicity upon restriction to the derived subgroup, 2018. (arXiv link)

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