This is a natural question. For example, using Colmez's results, as completed by Paskunas (who shows that Colmez's p-adic local Langlands describes all topologically irreducible unitary admisisble Banach space representations of $GL_2(\mathbb Q_p)$) one can start to prove purely representation-theoretic facts about unitary admissible Banach space reps.
of $GL_2(\mathbb Q_p)$, using Colmez's description in terms of $(\phi,\Gamma)$-modules. Now while some of these might naturally be related to unitarity,
there are certainly results that now seem accessible in the unitary case, which I suspect
don't actually require unitarity in order to hold. However, if one is going to use Colmez's and Paskunas's results, one needs unitarity as a hypothesis.
One could imagine (and here I am talking at the vaguest level) working with some kind of Weil group representations rather than Galois representations in order to include the non-unitary representations. I think that Schneider and Teitelbaum may have pondered this at some point, but I don't know what came of it. And I don't know how reasonable it is to hope for such a correspondence. I am just making the most absolutely naive guess, which you've probably also made yourself!
(One thing that makes me nervous is that when one works with unitary reps., there is a natural way to go from locally analytic reps. to Banach ones, by passing to universal unitary completions, and this is sometimes sensibly behaved, e.g. in the case of locally analytic inductions attached to crystabelline reps., by Berger--Breuil. But if one starts
to imagine completions that are not unitary, then I could imagine that they are much more wild; but again, this is just speculation.)
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.
Best Answer
OK...I think I see how to do this now. In the end, I am seeing $(p-1)^2$ distinct $(\phi,\Gamma)$-modules which matches well with the Galois side.
To do this, let $D$ be any 1-dimensional etale $(\phi,\Gamma)$-module. Let $e$ be a basis, and set $\phi(e)=h(T)e$ with $h(T) \in F_p((T))^\times$. Write $h(T) = h_0 T^a f(T)$ with $h_0 \in F_p^\times$ and $f(T) \in F_p[[T]]$ with $f(0)=1$.
Changing basis from $e$ to $u(T)e$ with $u(T) \in F_p((T))^\times$ gives $$ \phi(u(T)e) = u(T^p)h(T)e = \frac{u(T^p)}{u(T)} h(T) (u(T)e). $$ I claim one can find $u(T)$ such that $u(T)/u(T^p)$ equals any element of $1+TF_p[[T]]$. Indeed, for such an element $g(T)$, the infinite product $\prod_{j=1}^\infty \phi^j(g(T))$ (which hopefully converges since $g(0)=1$) works.
Thus, we can change basis so that $\phi$ has the form $\phi(e) = h_0 T^a e$ -- i.e. we can kill off the $f(T)$ term. Further, by making a change of basis of the form $e$ goes to $T^b e$, we may assume that $0 \leq a < p-1$.
Now, we use the fact that the $\phi$ and $\Gamma$ actions commute (which is a strong condition even in dimension 1). Namely, let $\gamma$ be a generator of $\Gamma$, and set $\gamma e = g(T) e$. Then $\gamma \phi e = \phi \gamma e$ implies $$ ((1+T)^{\chi(\gamma)}-1)^a g(T) = g(T^p) T^a. $$ Comparing leading coefficients, we see this is only possible if $a=0$ and $g(T)$ is a constant.
Thus, $D$ has a basis $e$ so that $\phi(e) = h_0 e$ and $\gamma(e) = g_0 e$ with $h_0,g_0 \in F_p^\times$ as desired.
Does this look okay? Any takers for the 2-dimensional case?