$\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 $X,X^*$ be maximal totally isotropic subspaces of $W$, which are dual with respect to $\langle,\rangle$.

Let $e_1,\dotsc,e_n$ be a basis of $X$ and $f_1,\dotsc,f_n$ be a basis of $X^*$. Let $B_{n}=M_nU_n$ be the parabolic subgroup of $\Sp(W)$ stabilizing the flag $$\langle e_1\rangle \subset \langle e_1, e_2\rangle \subset \dotsb \subset\langle e_1, e_2, \dotsc, e_{n}\rangle.$$

Let $\{ \chi_i \}$ be characters of $F^{\times}$ and $U_n$ the unipotent radical of $B_n$.

Let $\pi$ be the irreducible generic unramified representation of $\Sp(W)$ that is a subquotient of $\operatorname{Ind}_{B_{n}}^{\Sp(W)} (\chi_1 \otimes \dotsb \chi_{n} )$. (Here, the induction is normalized.)

Then for any $1 \le i \le n$, let $W_i=\left<e_i,f_i\right>$ and $B_{i}$ be the parabolic subgroup of $\Sp(W_i)$ stabilizing $\left<e_i\right>$. Let $\pi_i=\operatorname{Ind}_{B_{i}}^{\Sp(W_i)} \chi_i$.

Then I am wondering whether the irreducible unramified constituent of $\pi_i$ is also generic. Is it true? And I also guess $\pi_i$ is irreducible.

Any comments are welcome!