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$\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!

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    $\begingroup$ What are the irreducibility criteria for these induced reps? $\endgroup$
    – Kimball
    Commented Nov 9, 2021 at 1:14
  • $\begingroup$ @Kimball, You can assume that two representations are irreducible. But is that important? $\endgroup$
    – Andrew
    Commented Nov 9, 2021 at 4:18
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    $\begingroup$ Yes, because then you are just asking an isomorphism question. $\endgroup$
    – Kimball
    Commented Nov 9, 2021 at 12:33
  • $\begingroup$ @Kimball, Right! But I guess my reasoning would be true for reducible cases also! $\endgroup$
    – Andrew
    Commented Nov 10, 2021 at 7:04
  • $\begingroup$ Frobenius Reciprocity/adjunction? $\endgroup$
    – paul garrett
    Commented May 8, 2022 at 2:40

1 Answer 1

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The main theorem of section 2.9 of Bernstein, Zelevinsky "Induced representations of reductive $\mathfrak p$-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 element of the corresponding Weyl group.

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

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    $\begingroup$ I believe it may add to this clear and concise answer to note that, yes, the proposed equivalence in the question is the same as the two characters being Weyl conjugate. $\endgroup$
    – LSpice
    Commented Sep 24 at 12:26

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