Dear Kevin,
Here are some things that you know.
(1) Every non-tempered representation is a Langlands quotient of an induction of a non-tempered twist of a tempered rep'n on some Levi, and this description is canonical.
(2) Every tempered rep'n is a summand of the induction of a discrete series on some Levi.
(3) The discrete series for all groups were classified by Harish-Chandra.
Now Langlands's correspondence is (as you wrote) completely canonical: discrete series
with fixed inf. char. lie in a single packet, and the parameter is determined from the
inf. char. in a precise way.
All the summands of an induction of a discrete series rep'n are also declared to lie
in a single packet. So all packet structure comes from steps (1) and (2).
The correspondence is compatible in a standard way with twisting, and with parabolic induction.
So:
If we give ourselves the axioms that discrete series correspond to irred. parameters,
that the correspondence is compatible with twisting, that the correspondence is compatible
with parabolic induction, and that the correspondence is compatible with formation of
inf. chars., then putting it all together, it seems that we can determine step 1, then
2, then 3.
I don't know if this is what you would like, but it seems reasonable to me.
Why no need for epsilon-factor style complications: because there are no supercuspidals,
so everything reduces to discrete series, which from the point of view of packets are described by their inf. chars. In the p-adic world this is just false: all the supercuspidals are disc. series, they have nothing analogous (at least in any simple way) to an inf. char., and one has to somehow identify them --- hence epsilon factors to the rescue.
[Added: A colleague pointed out to me that the claim above (and also discussed below
in the exchange of comments with Victor Protsak) that the inf. char. serves to determine
a discrete series L-packet is not true in general. It is true if the group $G$ is semi-simple, or if the fundamental Cartan subgroups (those which are compact mod their centre) are connected. But in general one also needs a compatible choice of central character to determine the $L$-packet. In Langlands's general description of a discrete series parameter, their are two pieces of data: $\mu$ and $\lambda_0$. The former is giving the inf. char., and the latter the central char.]
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
In the case of $GL_{N}$, the $L$-packets are a non-issue, and the surjective map in the local Langlands correspondence becomes a bijection. At that point, we can think of allowing the information to flow the other way. Here's a simple application.
Let $f(z)$ be a classical modular form of weight $4k+2$ for the group $\Gamma_{0}(4)$ (that is also a cusp form, in the new subspace, and is an eigenform of all the Hecke operators). If $L(f,s)$ is the $L$-function for $f(z)$, what is the sign of the functional equation for $L(f,s)$?
The sign of the functional equation is always $1$, for the following reason. It is determined by the local components of the automorphic representation $\pi$ attached to $f$, and we only have to worry about the local components at $\infty$ (which is a discrete series representation that contributes a factor of $1$ to the sign because the weight is $\equiv 2 \pmod{4}$), and the local representation $\pi_{2}$ at $2$. The fact that the level of the modular form is $4$ shows that $\pi_{2}$ corresponds (under local Langlands) to a representation $\rho : W_{\mathbb{Q}_{2}} \to GL_{2}(\mathbb{C})$ that comes from a character $\chi$ of $W_{K}$, where $K = \mathbb{Q}_{2}(\omega)$ is the unramified quadratic extension of $\mathbb{Q}_{2}$, and that this character has order $6$. It follows that $\rho$ comes from an $S_{3}$ extension of $\mathbb{Q}_{2}$, and it turns out that there is a unique $S_{3}$ extension of $\mathbb{Q}_{2}$. From this, $\rho$ and hence $\pi_{2}$ is uniquely determined, and it turns out that the local root number of $\pi_{2}$ is also $1$.
(This fact was also observed by Atkin and Lehner in 1970, but the explanation above gives a more conceptual reason for it to be true, in my opinion.)