Let $E/\mathbb{Q}_p$ be a finite extension. Let $\rho$ be a continuous irreducible representation of $Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ on a two dimensional $E$-vector space. To $\rho$, via the p-adic Langlands correspondence, there is an associated $E$-Banach vector space representation of $GL_2(\mathbb{Q}_p)$, which we denote by $\Pi(\rho)$.

Is it known if this construction behaves well in $p$-adic families? More precisely, if $S$ is a reduced affinoid algebra and $\rho_S$ is a continuous $S$-linear representation of $Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ on a free rank two $S$-module is there an associated $S$-Banach module, $\Pi(\rho_S)$, equipped with a contiuous $S$-linear action of $GL_2(\mathbb{Q}_p)$, such that for all $x \in Sp(S)$, $S/x \otimes\Pi(\rho_S) \cong \Pi(\rho_{S,x})$?

• I will remark that the classical local Langlands correspondence does not behave well in families. For if you consider a reducible representation $\chi_1\oplus\chi_2$ and then let $\chi_1$ and $\chi_2$ vary in a complex family, then occasionally $\chi_1/\chi_2$ can be $|.|^{\pm1}$ and in these cases the associated $\pi$ is 1-dimensional, so even the dimension of $\pi$ jumps. – Kevin Buzzard Apr 3 '11 at 22:56
• Dear Kevin, When considering the behaviour of the (classical) local Langlands correspondence in families, I think it is better to replace the correspondence as usually considered by the Breuil--Schneider correspondence, in which non-generic representations are replaced by the parabolic inductions of which they are the Langlands quotient. (This is discussed in my recent preprint with David Helm, for example.) I should add that the behaviour of this modified correspondence in families is still quite subtle (and this is the subject of my preprint with David). Best wishes, Matt – Emerton Apr 4 '11 at 2:46

There is a discussion of the behaviour of $\Pi(\rho_S)$ in Colmez's Asterisque paper which says that the answer is essentially yes. (Probably it deals with the case of a formal family rather than a rigid analytic one, though.) See also the discussion in section 3 of my paper on local-global compatibility, which presents the deformation-theoretic formulation of $p$-adic local Langlands as originally suggested by Kisin. (Again, this will handle formal families rather than rigid ones.)