I would like to some question concering the induced representation of tensor product.

Let $F$ be a local fields of charateristic 0 and $0<a<m$ are two positive integers.

For $1 \le i \le m$, let $\lambda_i$ be an unramified character of $F^{\times}$ such that $|\chi_i|=q^{-s}$ for some $-\frac{1}{2}<s<\frac{1}{2}$.

Let $B_k$ denote the subgroup of upper triangualr matrices in $GL_k(F)$.

For some irreducible principal series representation $\sigma^{\prime}=Ind_{B_a}^{GL_a}(\chi_1 \boxtimes \chi_2 \cdots \boxtimes \chi_a)$, write $\sigma=\sigma^{\prime} \boxtimes (\sigma^{\prime})^{\vee}$, irreducible representation of $GL_a \times GL_a$.

Write $\pi=Ind_{B_m}^{GL_m}(\chi_1 \boxtimes \chi_2 \cdots \boxtimes \chi_m)$.

Write $n=m+2a$ and for $k$-tuple integers $\mathbb{a}=(a_1,\cdots,a_k)$ whose sum is $n$, we denote by $P_{\mathbb{a}}=M_{\mathbb{a}}U_{\mathbb{a}}$ the standard parabloic subgroup of $GL_n$ whose Levi subgroup $M_{\mathbb{a}}$ is isomorphic to $GL_{a_1} \times \cdots \times GL_{a_k}$.

If we set $v$ by a character of $GL_a \times GL_a$ defined by $v(g_1,g_2)=|\det(g_1)|\cdot |\det(g_2)|$, then why $Ind_{P_{3,\cdots,3,1,\cdots,1}}^{GL_{n}}(\chi_1 \circ \det_{GL_3} \boxtimes \cdots \boxtimes \chi_a \circ \det_{GL_3} \boxtimes \chi_{a+1} \boxtimes \cdots \boxtimes \chi_{m})$ is an irreducible subqutient of $Ind_{P_{(a,a,m)}}^{GL_{n}}((\sigma \cdot v) \boxtimes \pi)$?

Any comments will be highly appreciated!