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Let $F$ be a local field of characteristic zero and $G$ be a classical group over $F$.

Let $P=MN$ be a parabolic subgroup of $G$ and $\pi$ a irreducible smooth representation of $M$.

Let $\sigma$ be an irreducible constituent of normalized parabolic induction $\operatorname{Ind}_{M}^G(\pi)$. Then I am wondering whether the normalized Jacquet module $J_{N}(\sigma)$ has $\pi$ as a quotient? Some paper argues in this way without proof but I don't know the reason exactly. Why does it hold?

There is one more question. What is the difference of irreducible subquotient and irreducible consitituent of a module? I guess the people use the later when the given module is of finite length and the former is used in more general situations. Am I right?

Thank you very much!

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In general, all we can say from "general abstract nonsense" is that if $\sigma$ is a subrepresentation of $Ind_P^G(\pi)$, then $\pi$ is a quotient of $J_N(\sigma)$; but you don't immediately get any further information about other composition factors.

However, if the $M$-representation $\pi$ is supercuspidal (which is usually the most interesting case) -- in particular, when $P$ is a Borel subgroup -- then we have the implications

"$\pi$ occurs as a subquotient of $J_N(\sigma)$" $\iff$ "$\pi$ occurs as a quotient of $J_N(\sigma)$" $\iff$ "$\pi$ occurs as a sub of $J_N(\sigma)$".

This is because supercuspidal $M$-representations are projective objects once you fix the central character.

The question about "constituent" vs "subquotient" seems to have been answered in the comments.

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  • $\begingroup$ I think that the guess at the distinction between 'constituent' and 'subquotient' in the post is probably correct: a constituent is a subquotient, but perhaps it makes most sense to save 'constituent' for when we can think of our subquotient as a term in some composition series. $\endgroup$
    – LSpice
    Jun 18, 2021 at 17:06
  • $\begingroup$ @LSpice, thanks you very much for clarification. $\endgroup$
    – Andrew
    Jun 22, 2021 at 15:23
  • $\begingroup$ @David, Thank you for your reply. In your comment “it occurs as a quotient, or as a sub”, you mean $\sigma$ occurs as a quotient or a sub of $\operatorname{Ind}_P^G \pi$? $\endgroup$
    – Andrew
    Jun 22, 2021 at 15:25

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