Sur certains compl\'et\'es unitaires universels explicites pour GL_2(F)

TL;DR

This paper provides an explicit description of the universal unitary completion of specific locally Q_p-analytic representations of GL_2(F), extending previous results to more general fields using Banach spaces of C^r functions.

Contribution

It generalizes Berger-Breuil's results from Q_p to finite extensions F of Q_p by explicitly describing the universal unitary completion.

Findings

Explicit description of universal unitary completions for GL_2(F)

Extension of previous results from Q_p to finite extensions

Use of Banach spaces of C^r functions on O_F

Abstract

In this paper we give an explicit description of the universal unitary completion of certain locally Q_p-analytic representations of GL_2(F), where F is a finite extension of Q_p (this generalizes some results of Berger-Breuil for F=Q_p). To this aim, we make use of certain Banach spaces of C^r functions on O_F (for r a positive real number) introduced by the author.