## The image of Colmez’s Montreal functor.(English)Zbl 1297.22021

The subject of the paper is the study of $$p$$-adic representations of $$G =\mathrm{GL}_2(\mathbb Q_p)$$. Let $$L$$ be a finite extension of $$(\mathbb Q_p)$$, let $$\mathcal O$$ be the ring of integers of $$L$$ and $$k$$ its residue field. Considered are representations of $$G$$ on $$L$$-Banach spaces or on $$\mathcal O$$ -modules. In a unitary Banach space representation of $$G$$ one may choose a $$G$$-invariant $$\mathcal O$$-lattice $$\Theta$$ and reduction gives a $$k$$-representation of $$G$$. Let $$\mathcal G$$ denote the Galois group of $$\overline{\mathbb Q}_p /\mathbb Q_p$$.
Colmez’s functor is an exact covariant functor $$\mathbf V$$ from the category of smooth $$G$$-representations of finite length with a central character on torsion $$\mathcal O$$-modules to the category of continuous $$\mathcal G$$-representations of finite length on $$\mathcal O$$ -modules. The author extends this functor to a functor on $$L$$-Banach space representations. Let $$\Pi$$ be an absolutely irreducible admissible unitary $$L$$-Banach representation of $$G$$ with a central character. A theorem on the reduction $$\mathrm {mod}-p$$ of $$\Pi$$, proved here for $$p > 3$$, implies that $$\mathrm{dim}_L \mathbf V(\Pi)\leq 2$$ and determines for which $$\Pi$$ this dimension equals 2. For those $$\Pi$$ it is proved that their isomorphism classes correspond bijectively, via the functor $$\mathbf V$$, with the absolutely irreducible 2-dimensional continuous $$L$$-representations of $$\mathcal G$$ with determinant $$\varepsilon \xi$$ ($$\xi$$ is the central character of $$\Pi, \varepsilon$$ is the cyclotomic character). The final result is an anti-equivalence of categories between the category of admissible unitary $$L$$-Banach representations of $$G$$ of finite length, with central character $$\varepsilon$$ and all irreducible subquotients isomorphic to $$\Pi$$, and the category of modules of finite length over the ring representing the deformation functor of $$V = \mathbf V(\Pi)$$ to artinian local $$L$$-algebras.
An essential tool for the proof of these results is a deformation theory for $$k$$-representations of $$G$$, which is presented in such a way that it can be used for an arbitrary $$p$$-adic analytic group, when certain hypotheses (on Ext groups) are satisfied.
In the case of $$G =\mathrm{GL}_2(\mathbb Q_p)$$ the category of Banach space (resp. $$\mathcal O$$-module) representations of $$G$$ can be decomposed into a product of subcategories, indexed by blocks of irreducible smooth $$k$$-representations of $$G$$. There are four types of blocks to be considered and for which the application of the theory mentioned above is different. Also results of deformation theory for 2-dimensional representations of $$\mathrm{Gal}(\mathbb Q_p) / (\mathbb Q_p)$$ are needed. They are deduced from work of Böckle.

### MSC:

 22E50 Representations of Lie and linear algebraic groups over local fields 11S37 Langlands-Weil conjectures, nonabelian class field theory
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