Let $F$ be a local field of characteristic 0.
I know that $\pi=\text{Ind}_{B_k(F)}^{GL_k(F)}(\chi_1 \boxtimes \cdots \chi_k)$ for some unramified characters $\chi_i$'s is irrducible if there is no $\chi_i,\chi_j$ such that $\chi_i \chi_j^{-1} = |\cdot|$ or $|\cdot|^{-1}$.
I am wondering that something similar also holds in classical group.
That is, let $U(n)$ be the quasi-split unitary group and $B$ its Borel subgroup.
Let $\chi=(\chi_1 \boxtimes \cdots \boxtimes \chi_{[\frac{n}{2}]})$ character of $B$. Then $\pi=\text{Ind}_{B}^{U(n)} \chi$ is irreducible if and only if there is no $\chi_i,\chi_j$ such that $\chi_i \chi_j^{-1} = |\cdot|$ or $|\cdot|^{-1}$?
Thank you very much!