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.]
The Langlands correspondence for higher local fields is still at an early stage of development. I haven't really kept up with it, but here's some key points.
As the question stated, and Loren commented, the starting point is the $GL_1$ case, which is class field theory for higher local fields. Local class field theory relates the abelianized Galois group $Gal_F^{ab}$ of a local field $F$ to the multiplicative group $F^\times = K_1(F)$. For a higher local fields $E$, Kato's class field theory relates the abelianized Galois group $Gal_E^{ab}$ to the Milnor K-group $K_n(E)$.
For example, let $E = {\mathbb Q}_p((t))$. Then there's a canonical homomorphism $\Phi \colon K_2(E) \rightarrow Gal_E^{ab}$ such that for all finite abelian $L/E$, $\Phi$ induces an isomorphism from $K_2(E) / N_{L/E} K_2(L)$ to $Gal(L/E)$. This gives a bijection between finite abelian extensions of $E$ (in a fixed algebraic closure) and open, finite-index subgroups of $K_2(E)$. This is the main theorem described in
Kato, Kazuya, A generalization of local class field theory by using K-groups. I, Proc. Japan Acad., Ser. A 53, 140-143 (1977). ZBL0436.12011.
You can look at this paper to see the topology on $K_2(E)$ and more details. In particular, this suggests a possible Weil group for $E$. Namely, Kato reciprocity gives an isomorphism from a completion of $K_2(E)$ to $Gal_E^{ab}$. One might let the abelianized Weil group be the subgroup $Weil_E^{ab}$ of $Gal_E^{ab}$ corresponding to the uncompleted $K_2(E)$. And perhaps the (nonabelian) Weil group should be defined by pulling back. I.e., look at the map $\pi \colon Gal_E \rightarrow Gal_E^{ab}$, and define $Weil_E = \pi^{-1}(Weil_E^{ab})$. I haven't explored if this is the right idea though.
Kato goes beyond this, from 2-dimensional to n-dimensional local fields, and from $K_2$ to $K_n$ accordingly. These aren't hard to find, and there are surveys floating around. See the Invitation to Higher Local Fields volume, for example. Even $K_2$ is interesting, I think!
Note that Kato's paper was from 1977... so what about the Langlands program for fields like $E$? A natural first step is figuring out a suitable version of the Satake isomorphism, and the Iwahori-Hecke algebra. There's a series of papers by Kazhdan, Gaitsgory, Braverman, Patnaik, Rousseau, Gaussent (and certainly others) on the subject.
Recent landmark papers are
- Braverman, Alexander; Kazhdan, David, The spherical Hecke algebra for affine Kac-Moody groups. I, Ann. Math. (2) 174, No. 3, 1603-1642 (2011). ZBL1235.22027.
- Gaussent, Stéphane; Rousseau, Guy, Spherical Hecke algebras for Kac-Moody groups over local fields., Ann. Math. (2) 180, No. 3, 1051-1087 (2014). ZBL1315.20046.
- Braverman, Alexander; Kazhdan, David; Patnaik, Manish M., Iwahori-Hecke algebras for $p$-adic loop groups, Invent. Math. 204, No. 2, 347-442 (2016). ZBL1345.22011..
Note that a group like $SL_2(E)$ can be seen as a loop group over ${\mathbb Q}_p$. Hence the appearance of words like "loop group" and "Kac-Moody group".
The Langlands dual group certainly arises in these studies, but I haven't seen something quite as straightforward as a parameters from the Weil group (described above) to the dual group. I haven't looked too hard either, so maybe it's in there somewhere. There seems to be a fancier, more categorical, parameterization involved. I'd be tempted to bring it down to earth a bit, following Kato.
The other direction that I haven't seen -- and one that I think is worth pursuing -- is the case of (nonsplit) tori. That's important for any putative Langlands program, and should require an interesting mix of Milnor K-theory and Galois cohomology.
Best Answer
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.)