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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 $\mathcal{G}_2(F)$, \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 {A}_2(F) G}_2(F) \longrightarrow \mathcal{A}_1(F)^{\otimes 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$, although we don't even know it is semi-simple or not. 6 added 1 characters in body Let us consider the simple case:$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$is the module modular function of the group$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{G}_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 {A}_2(F) \longrightarrow \mathcal{A}_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$, although we don't even know it is semi-simple or not. 5 added 2 characters in body Let us consider the simple case:$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$is the module function of the group$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{G}_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 {A}_2(F) \longrightarrow \mathcal{A}_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\circ detJ\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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