To check whether my understaning on the geometric lemma is right, I would like to ask some specific question related to it.

Let $F$ be a $p$-adic local field of characteristic zero. Let $W_n$ be a symplectic space over $F$ of dimension $2n$. Let $Sp_{n}(W_n)$ be a symplectic group and $B_n$ its standard Borel subgroup. Let $Q_{t,n-t}$ be a standard parabolic subgroup of $G$ preserving a totally isotropic subspace of $W_n$ of dimension $t$.

Let ${chi_i}_{1 le i le n}$ be unramified characters of $GL_1(F)$ such that $chi_i(omega)=1$ and let $pi$ be the normalized parabolic induced representation $text{Ind}_{Q_{3,n-3}}^{Sp_{n}}(chi_1 circ text{det}_{GL_3},chi_2,cdots,chi_{n-2})$.

I am considering $J_{Q_{2,n-2}}(pi)$, the normalized Jacquet module of $pi$ to $Q_{2,n-2}$. By the geometric lemma, there is some filtration of $J_{Q_{2,n-2}}(pi)$

$$0=tau_0 subset tau_1 subset cdots subset tau_m=J_{Q_2}(pi)$$ such that $tau_{i}backslashtau_{i+1}$ is some induced representation of the Jacquet module of the inducing data of $pi$.

I guess that such subquotient appearing in this filtration has the form $|cdot|^{-frac{1}{2}}cdot(chi_1 circ text{det}_{GL_2}) boxtimes text{Ind}_{B_{n-2}}^{Sp_{n-2}}(chi_1′,chi_2′,cdots,chi_{n-2}’)$ or

$text{Ind}_{B_2}^{GL_2}(chi_1′,chi_2′) boxtimes text{Ind}_{Q_{3,n-5}}^{Sp_{n-2}}(chi_1 circ text{det}_{GL_3},chi_4′,cdots,chi_{n-2}’)$?

(Here, ${chi_1′,cdots,chi_{n-2}’} subset {chi_1,cdots,chi_{n-2},chi_1^{-1},cdots,chi_{n-2}^{-1}}.)$

Any comments are highly appreaciated!