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

Let $B$ be a Borel subgroup of $\Sp(W)$ and $\{\chi_1,\chi_2\}$ be some characters of $F^{\times}$.

Then I am wondering if $\operatorname{Hom}_{\Sp(W)}(\operatorname{Ind}_{B}^{\Sp(W)} \chi_1,\operatorname{Ind}_{B}^{\Sp(W)} \chi_2) \ne 0$, then $\chi_1=\chi_2$ or $\chi_1=\chi_2^{-1}$?

Any comments are welcome!