$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\GL{GL} $Let $F$ be local field of characteristic zero and $(W,\langle,\rangle)$ be a $2n$-dimensional symplectic space over $F$.

Let $B$ be a Borel subgroup of $\Sp(W)$ and $\{\chi_1,\ldots,\chi_n\}$, $\{\chi_1',\ldots,\chi_n'\}$ be two set of characters of $F^{\times}$.

Then I am wondering that $\operatorname{Hom}_{\Sp(W)}(\operatorname{Ind}_{B}^{\Sp(W)} (\chi_1\otimes\cdots \otimes \chi_n),\operatorname{Ind}_{B}^{\Sp(W)} (\chi_1'\otimes\cdots \otimes \chi_n')) \ne 0$ is equivalent to $\{\chi_1,\ldots,\chi_n,\chi_1^{-1},\ldots,\chi_n^{-1}\}=\{\chi_1',\ldots,\chi_n',\chi_1'^{-1},\ldots,\chi_n'^{-1}\}$ as a set?

Any comments are welcome!