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!


1 Answer 1


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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.