Question on the geometric lemma in $p$-adic classical groups

Question

$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}\DeclareMathOperator\B{B} $My question is about the geometric lemma of $p$-adic classical groups not $\GL_n$. For $\GL_n$, Kenta Suzuki gave a very nice answer! (See Jacquet module of irreducible principal series.)

Let $F$ be a $p$-adic field and $\Sp_{2n}$ the symplectic group over a $2n$-dimensional symplectic space over $F$. Let $B_n$ be $p$-adic group defined over $F$. Let $\B_n$ be the borel subgroup of $\GL_n$.

Let $\chi=\chi_1\otimes \dotsb \otimes \chi_n$ be a character of $B_n(F)$ and consider $\pi=\Ind_{B_n(F)}^{\Sp_{2n}(F)} \chi$, the normalized induced representation from $\chi$ to $\Sp_{2n}(F)$.

For $1\le a \le n$, let $P_a$ be the parabolic subgroup of $\Sp_{2n}(F)$ whose Levi subgroup is isomorphic to $\GL_a(F) \times \Sp_{2(n-a)}(F)$.

Let $J_{P_a}$ be the normalized Jacquet funtor. Let $$S_{n,a}=\{s\in S_n|s^{-1}(1)<\dotsb <s^{-1}(a),s^{-1}(a+1)<\cdots<s^{-1}(n)\}$$ be the subset of the symmetric group $S_n$ of $\{1,\cdots,n\}$

Then I am wondering whether the geometric lemma tells that $J_{P_a}(\pi)$ is equal to $\bigoplus_{s\in S_{n,a}}\Ind_{\B_a}^{\GL_a}(\chi_{s(1)}\otimes \cdots \otimes \chi_{s(a)}) \boxtimes \Ind_{B_{2n-2a}}^{\Sp_{2n-2a}}(\chi_{s(a+1)}\otimes \dotsb \otimes \chi_{s(n)})$, up to semisimplication.

As is well known, in the Weyl group of $\Sp(2n)$, there are involutive elements $e_i$ such that $e_i^2=1$. (i.e. it changes $t_i$ and $t_i^{-1}$ in the diagonal elements of the maximal split torus.) However, such $e_i$ seems not an element $W^{M,N}$ in the Berstein and Zelvinski’s paper Induced representations of reductive $\mathfrak p$-adic groups. I page 448. (Here, $M$ is the maximal split torus and $N$ is the Levi of $P_a$.)

Is this right?

Any comments are appreciated!