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edit approved May 8, 2022 at 5:13
F Zaldivar
edit approved May 8, 2022 at 5:13
F Zaldivar
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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,\chi_2,\chi_n\}$$\{\chi_1,\ldots,\chi_n\}$, $\{\chi_1',\cdots,\chi_n'\}$$\{\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,\cdots,\chi_n,\chi_1^{-1},\cdots,\chi_n^{-1}\}=\{\chi_1',\cdots,\chi_n',\chi_1'^{-1},\cdots,\chi_n'^{-1}\}$$\{\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!

$\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 $\{\chi_1,\chi_2,\chi_n\}$, $\{\chi_1',\cdots,\chi_n'\}$ 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 $\{\chi_1,\cdots,\chi_n,\chi_1^{-1},\cdots,\chi_n^{-1}\}=\{\chi_1',\cdots,\chi_n',\chi_1'^{-1},\cdots,\chi_n'^{-1}\}$ as a set?

Any comments are welcome!

$\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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Andrew
Andrew
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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 $2$$2n$-dimensional symplectic space over $F$.

Let $B$ be a Borel subgroup of $\Sp(W)$ and $\{\chi_1,\chi_2\}$$\{\chi_1,\chi_2,\chi_n\}$, $\{\chi_1',\cdots,\chi_n'\}$ be sometwo set of characters of $F^{\times}$.

Then I am wondering if $\operatorname{Hom}_{\Sp(W)}(\operatorname{Ind}_{B}^{\Sp(W)} \chi_1,\operatorname{Ind}_{B}^{\Sp(W)} \chi_2) \ne 0$, thenthat $\chi_1=\chi_2$ or$\operatorname{Hom}_{\Sp(W)}(\operatorname{Ind}_{B}^{\Sp(W)} (\chi_1\otimes\cdots \otimes \chi_n),\operatorname{Ind}_{B}^{\Sp(W)} (\chi_1'\otimes\cdots \otimes \chi_n')) \ne 0$ is equivalent to $\chi_1=\chi_2^{-1}$$\{\chi_1,\cdots,\chi_n,\chi_1^{-1},\cdots,\chi_n^{-1}\}=\{\chi_1',\cdots,\chi_n',\chi_1'^{-1},\cdots,\chi_n'^{-1}\}$ as a set?

Any comments are welcome!

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\GL{GL} $Let $F$ be local field of characteristic zero and $(W,\langle,\rangle)$ be a $2$-dimensional symplectic space over $F$.

Let $B$ be a Borel subgroup of $\Sp(W)$ and $\{\chi_1,\chi_2\}$ be some characters of $F^{\times}$.

Then I am wondering if $\operatorname{Hom}_{\Sp(W)}(\operatorname{Ind}_{B}^{\Sp(W)} \chi_1,\operatorname{Ind}_{B}^{\Sp(W)} \chi_2) \ne 0$, then $\chi_1=\chi_2$ or $\chi_1=\chi_2^{-1}$?

Any comments are welcome!

$\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 $\{\chi_1,\chi_2,\chi_n\}$, $\{\chi_1',\cdots,\chi_n'\}$ 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 $\{\chi_1,\cdots,\chi_n,\chi_1^{-1},\cdots,\chi_n^{-1}\}=\{\chi_1',\cdots,\chi_n',\chi_1'^{-1},\cdots,\chi_n'^{-1}\}$ as a set?

Any comments are welcome!

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

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\GL{GL} $Let $F$ be local field of characteristic zero and $(W,\langle,\rangle)$ be a $2$-dimensional symplectic space over $F$.

Let $B$ be a Borel subgroup of $\Sp(W)$ and $\{\chi_1,\chi_2\}$ be some characters of $F^{\times}$.

Then I am wondering if $\operatorname{Hom}_{\Sp(W)}(\operatorname{Ind}_{B}^{\Sp(W)} \chi_1,\operatorname{Ind}_{B}^{\Sp(W)} \chi_2) \ne 0$, then $\chi_1=\chi_2$ or $\chi_1=\chi_2^{-1}$?

Any comments are welcome!