Compl\'et\'es universels de repr\'esentations de GL_2(Q_p)

TL;DR

This paper characterizes the locally analytic vectors of certain unitary representations of GL_2(Q_p) using p-adic Langlands correspondence, showing their universal completion recovers the original representation.

Contribution

It provides a detailed description of the locally analytic vectors and their filtration for unitary GL_2(Q_p) representations via phi-Gamma modules, linking to p-adic Langlands.

Findings

Description of Pi^{an} in terms of phi-Gamma modules

Filtration of Pi^{an} by radius of analyticity

Universal completion of Pi^{an} equals Pi

Abstract

Let Pi be a unitary representation of GL_2(Q_p), topologically of finite length. We describe the sub-representation Pi^{an} made of its locally analytic vectors, and its filtration by radius of analyticity, in terms of the phi-Gamma module attached to Pi via the p-adic local Langlands correspondence, and we deduce that the universal completion of Pi^{an} is Pi itself.