Let $G=Sp(2)$ be a symplectic group over a $p$-adic field and $P$ be the Siegel parabolic subgroup of $G$. Let $\text{Ind}_P^G$ be the normalized induction functor from smoothe representations of $P$ to that of $G$. Let $J_P$ be the unnormalized Jacquet functor from smooth representations of $G$ to that of $P$.
Let $M \simeq GL_1$ be the Levi subgroup of $P$ and $\chi$ is a character of $M$. Then what is $J_P(Ind_P^G(\chi))$? Is it just $\chi$ or $\chi\cdot \delta_{P}^{\frac{1}{2}}$? (Here, $\delta_{P}$ is the modulus character of $P$.)