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.



multiplicity one for restriction of representations from $GU$ to unitary group
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](https://arxiv.org/abs/1806.03635))


