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Let $F$ be a local field of characteristic 0.

I know that $\pi=\text{Ind}_{B_k(F)}^{GL_k(F)}(\chi_1 \boxtimes \cdots \chi_k)$ for some unramified characters $\chi_i$'s is irrducible if there is no $\chi_i,\chi_j$ such that $\chi_i \chi_j^{-1} = |\cdot|$ or $|\cdot|^{-1}$.

I am wondering that something similar also holds in classical group.

That is, let $U(n)$ be the quasi-split unitary group and $B$ its Borel subgroup.

Let $\chi=(\chi_1 \boxtimes \cdots \boxtimes \chi_{[\frac{n}{2}]})$ character of $B$. Then $\pi=\text{Ind}_{B}^{U(n)} \chi$ is irreducible if and only if there is no $\chi_i,\chi_j$ such that $\chi_i \chi_j^{-1} = |\cdot|$ or $|\cdot|^{-1}$?

Thank you very much!

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  • $\begingroup$ I doubt this. For $U(3)$, the condition "there is no $\chi_i,\chi_j$ such that $\chi_i\chi_j^{-1}=|~|,|~|^{-1}$" is empty. Does that mean every induced representation of $U(3)$ is irreducible? Keys had a paper on principal series on $SU(n)$. I think Shahidi might also have some papers on this. $\endgroup$
    – Q. Zhang
    Mar 23, 2020 at 2:49

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