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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 $M$.

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1 Answer 1

Let us consider the simple case: $G=GL_2(F)$, $n=2$. (cf. ''The local langlands conjecture for $GL_2(F)$'' C.J. Bushnell and G.Henniart)

In order to tell the story, first we need to give some definitions. Clearly we only need to consider the non-cuspidal case. Let $\chi=\chi_1\otimes \chi_2$ be the character of $T$, we denote $ \chi^{\omega}=\chi_2\otimes \chi_1$, we define $\pi_{\chi}=Ind_B^G(\delta_B^{-\frac{1}{2}}\otimes \chi)$ where $\delta_B$ is the modular function of the group $B$ i,e $\delta_B(tn)=||t_2t_1^{-1}||$ for $t=diag(t_1,t_2)$, $n\in N$, we write $\phi\circ det$, $\phi \cdot St_G$ two other kind of principal series for $GL_2(F)$.

Now we arrive to write the Jacquet functor $J: Rep(G) \longrightarrow Rep(T); (\pi, V) \longrightarrow (\pi_N, V_N)$.

(1) For $\chi_1\chi_2^{-1}\neq ||.||^{\pm}$, $\pi=\pi_{\chi}$ is irreducible, then $\pi_N=\delta_B^{-\frac{1}{2}}\otimes \chi \oplus \delta_B^{-\frac{1}{2}}\otimes \chi^{\omega}$.

(2) $\pi=\phi\circ det$, then $\pi_N=\phi\otimes \phi$.

(3) $\pi=\phi \cdot St_G$, then $\pi_N=||.||\phi\otimes ||.||^{-1}\phi$.

We recall some result about local langlands correspondance for general linear group. We denote $\mathcal{G}_2(F)$ to be the set of equivalence classes of 2-dimensional Frobenius semisimple, Deligne representation of the Weil group $\mathcal{W}_F$; also $\mathcal{A}_2(F)$ to be the set of equivalence classes of irreducible smooth representations of $GL_2(F)$. The local langlands correspondance tell us that there is a natural bijective map $l_2$ between $\mathcal{A}_2(F)$ and $\mathcal{G}_2(F)$. The naturality often involves some compatibility conditions. ( For detail one should see the article of Borel in Corvallis).

Assume $\pi$ is irreducible, lying in $\mathcal{A}_2(F)$, we denote $l_2(\pi)=(\rho,W,\mathbf{n})$.

(1) if $\pi=\pi_{\chi}$, then $\rho=\chi_1 \oplus \chi_2$ and $\mathbf{n}=0$, here we regard $\chi_i$ as the representation of Weil group $\mathcal{W}_F$.

(2) if $\pi=\phi\circ det$, then $\rho=||.||^{-\frac{1}{2}}\phi \oplus ||.||^{\frac{1}{2}}\phi$ and $\mathbf{n}=0$.

(3) if $\pi=\phi \cdot St_G$, then $\rho=||.||^{-\frac{1}{2}}\phi \oplus ||.||^{\frac{1}{2}}\phi$, but in this case $\mathbf{n}\neq 0$.

Finally we comme to the question that Arno asks. We translate directly ''the Jacquet functor'' to the Galois side via the local langlands correspondence.

$J: \mathcal {G}_2(F) \longrightarrow \mathcal{G}_1(F)^{\otimes 2}$. More precisely, the result is outlined as follows:

(1) $J\big((\pi_{\chi}, \mathbf{n}=0)\big)=(\delta_B^{-\frac{1}{2}}\otimes \chi) \oplus (\delta_B^{-\frac{1}{2}}\otimes \chi^{\omega})$;

(2) $J\big((\phi\circ det, \mathbf{n}=0)\big)=\phi\otimes \phi$.

(3) $J\big((\phi \cdot St_G, \mathbf{n}=0)\big)= ||.||\phi\otimes ||.||^{-1}\phi$.

Remark: for general case, we take $\pi \in Irr_{\mathbb{C}}(G)$, one knows $\pi_N$ has finite length and is admissible as the representation over its Levi subgroup $M$, although we don't even know it is semi-simple or not.

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