Some question on the induced representation of tensor product

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!

• Without necessarily quite referring to the specifics of your question, the general context would seem (to me) to be about irreducibility of various degenerate, but unramified, principal series and their subs or quots. Is this correct? If so, certainly the easiest partial answer regarding the subs/quots with spherical vectors is Casselman's 1980 paper, which tracks spherical vectors in the intertwinings, although there have been some more recent sharper results, of course. Is this the sort of thing you're asking? – paul garrett Jul 10 '18 at 21:58
• @garrett, The fact that $I=Ind_{P_{3,\cdots,3,1,\cdots,1}}^{GL_{n}}(\chi_1 \circ \det_{GL_3} \boxtimes \cdots \boxtimes \chi_a \circ \det_{GL_3} \boxtimes \chi_{a+1} \boxtimes \cdots \boxtimes \chi_{m})$ is irreducible is from Zelevinsky's result. (Thm 4.2 in "Induced representations of reductive p-adic groups".) Instead of the irreducibility of $I$, I am much more curious about why $I$ appears as a subquotient of $Ind_{P_{(a,a,m)}^{GL_n}}((\sigma \cdot v)\boxtimes \pi)$. – Monty Jul 11 '18 at 11:09
• Could you give any idea where this question comes from? As things stand, it's technically clearly stated but seems to me rather a jumble of apparently ad hoc things that you can do with $\mathrm{GL}_n$ representations. – LSpice Jul 11 '18 at 11:30
• @LSpice, You are right. The original situation is concerning about global unitary group and this question comes from it when the relevant informations are unramified at the split places. My question arised in the proof of Lemma 5.5 of the following Ichino-Yamana's paper. degruyter.com/abstract/j/crll.ahead-of-print/crelle-2015-0107/… – Monty Jul 11 '18 at 11:46
• I have read some argument which seems to have somthing to do with this. I leave this for those who can answer my original question from this. That is if $\tau$ is the representation of $GL_2$ induced from the Borel subgroup and the character $\mu|\cdot|^{\frac{1}{2}}\otimes \mu|\cdot|^{-\frac{1}{2}}$. Then this representation admits the chatacter $\mu(\det_{GL_2})$. – Monty Jul 14 '18 at 7:18