TITLE: p-adic Langlands in Families
QUESTION [6 upvotes]: 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 $x \in Sp(S)$,  $S/x \otimes\Pi(\rho_S) \cong \Pi(\rho_{S,x})$?

REPLY [8 votes]: 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.)