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Let $F$ be a $p$-adic field. Let $G$ be a connected reductive group and $\rho$ an irreducible admissible representation of $G(F)$. Let $P$ be a parabolic subgroup of $G$ and suppose further that $\rho$ is essentially square integrable. It seems like the Jacquet module $\mathrm{Jac}_P(\rho)$ is essentially square integrable or at least a sum of essentially square integrable representations. Is this true/is there a reference for this fact?

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Yes, your claim is true.

It follows from proposition 43 of the Bernstein notes on representation of $p$-adic groups. That proposition gives a characterization of essentially square integrable representations (called square integrable modulo center, in this source) in terms of certain exponents of all the (dual) Levi subalgebras. Your claim follows then by the transitivity of the Jacquet functor.

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  • $\begingroup$ Great! I'm happy that my claim is true, and happier still that you pointed me to this fact about square integrable representations which I had missed. $\endgroup$ – Alexander Dec 5 '17 at 23:49

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