# Jacquet module and Frobenius reciprocity

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!

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.

• 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. Jun 18, 2021 at 17:06
• @LSpice, thanks you very much for clarification. Jun 22, 2021 at 15:23
• @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 $\operatorname{Ind}_P^G \pi$? Jun 22, 2021 at 15:25