Let $G = GL_n(F)$ be the general linear group over a finite extension $F$ of $Q_p$. This question could be posed for a larger class of groups, but let us stay with $Gl_n$ for the moment.

Let $\pi$ be a smooth irreducible complex representation of $G$. Let $P \subset G$ be a parabolic subgroup and $P =MN$ a Levi-decomposition. The Jacquet module $\pi_N$ of $\pi$ is by definition the module of $N$-coinvariants in $\pi$.

Via the local Langlands correspondence $\pi$ corresponds to a Weil-Deligne representation $\sigma_\pi$. Furthermore, in the cases where $\pi_N$ is irreducible, the representation $\pi_N$ has a corresponding Weil-Deligne representation via local Langlands for $M$.

My question is, does the operation $\pi \to \pi_N$ from $G$-representations to $M$-representations has a "satisfactory" interpretation on the Galois side, via local Langlands for $G$ and $M$?

A point of caution is that $\pi_N$ need not be $M$-irreducible, so it does not go directly into local Langlands for $M$.