# P-adic local Langlands for non-unitary representations?

In Colmez's work on the p-adic local Langlands correspondence for ${\rm GL}_2(\mathbb{Q}_p)$, he works with ${\rm GL}_2(\mathbb{Q}_p)$-representations on $p$-adic Banach spaces which admit an invariant norm, so the reduction modulo $p$ makes sense. To each irreducible admissible representation of this kind (let's call these "unitary" Banach representations), he attaches a rank 2 $(\varphi, \Gamma)$-module, and hence a 2-dimensional p-adic representation of ${\rm Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$.

The unitary condition is quite strict -- it rules out all nontrivial finite-dimensional algebraic representations of ${\rm GL}_2$, for instance. Is there any natural way to extend the correspondence to non-unitary admissible Banach space representations of ${\rm GL}_2(\mathbb{Q}_p)$, and what sort of Galois-theoretic objects would these match up with?

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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.
@Matt: My equally naive guess was that maybe the non-unitary Banach reps should correspond to non-etale $(\varphi, \Gamma)$-modules over Fontaine's ring $\mathscr{E}$ (Berger's $\mathbb{B}_{\mathbb{Q}_p}$) and non-unitarisable locally analytic reps to non-slope-zero $(\varphi, \Gamma)$'s over the Robba ring. But as far as I know there's no reason why a non-etale $(\varphi, \Gamma)$-module over $\mathscr{E}$ should be overconvergent, but any Banach rep will have locally analytic vectors, so maybe my guess is the wrong one. – David Loeffler Apr 26 '10 at 21:48