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.)
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.]
Best Answer
You seem to be expecting that mod $p$ local Langlands should satisfy the same compatibilities as "conventional" local Langlands (for smooth representations of $GL_2(\mathbf{Q}_p)$ and $WD(\mathbf{Q}_p)$ with coefficients in $\mathbf{C}$).
However, before you can even talk about reduction mod $p$, you need to check that the coefficients can be descended from $\mathbf{C}$ to a number field. I.e., if $\pi$ is a smooth irred rep of $GL_n(\mathbf{Q}_p)$ on an $L$-vector space, where $L$ is some subfield of $\mathbf{C}$, then is its Langlands parameter $\phi_{\pi}$ an $L$-valued Weil-Deligne rep? The answer, annoyingly, is "no": if $n$ is even, you have to add $\sqrt{p}$ to $L$ in order to get this to work. So it's common to re-normalise by twisting the correspondence for $GL_n$ by $|\det|^{(n-1)/2}$; this makes it compatible with coefficient fields, and also works better for local-global compatibility.
It's this same shift which you are seeing in the mod $p$ theory, and here it's completely impossible to get rid of, even if you extend the coefficient fields as much as you like: the character $|\cdot|^{1/2}$ of $\mathbf{Q}_p^\times$ is not $p$-adically unitary, so there is no way you can reduce it mod $p$.