Non-vanishing criterion of the Hom space of induced representation of p-adic groups?

Question

$\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 $B$ be a Borel subgroup of $\Sp(W)$ and $\{\chi_1,\ldots,\chi_n\}$, $\{\chi_1',\ldots,\chi_n'\}$ be two set of characters of $F^{\times}$.

Then I am wondering that $\operatorname{Hom}_{\Sp(W)}(\operatorname{Ind}_{B}^{\Sp(W)} (\chi_1\otimes\cdots \otimes \chi_n),\operatorname{Ind}_{B}^{\Sp(W)} (\chi_1'\otimes\cdots \otimes \chi_n')) \ne 0$ is equivalent to $\{\chi_1,\ldots,\chi_n,\chi_1^{-1},\ldots,\chi_n^{-1}\}=\{\chi_1',\ldots,\chi_n',\chi_1'^{-1},\ldots,\chi_n'^{-1}\}$ as a set?

Any comments are welcome!

Answer 1

The main theorem of section 2.9 of Bernstein, Zelevinsky "Induced representations of reductive $p$-adic groups - I" gives a criterion for the existence of a non-zero intertwining operator between two parabolically induced representations. In your case the characters of the maximal split torus have to be conjugate under an elementof the corresponding Weyl group.

You do not need to assume that $F$ has characteristic $0$.