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.

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.)

  • $\begingroup$ Of course in the 1-d case then Weil reps are exactly what you get. Here's a perhaps related question. If we modify local Langlands for GL_2 so that the 1-d pi's don't show up and are replaced by reducible principal series, and if we stay away from p=2, then there's a nice map from (2-d complex reps of the inertia subgroup of the Galois group which extend to reps of the Weil group) to (reps of GL_2(Z_p)) (Henniart's appendix to Breuil-Mezard). Is there an analogue of that story in the p-adic case? $\endgroup$ Commented Apr 26, 2010 at 19:55
  • $\begingroup$ @Kevin: There is a mod p story along those lines (I remember Michael Schein telling me about this once). I don't know if there's a p-adic version. $\endgroup$ Commented Apr 26, 2010 at 21:42
  • $\begingroup$ @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. $\endgroup$ Commented Apr 26, 2010 at 21:48

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