Locally analytic distributions and p-adic representation theory, with applications to GL_2

TL;DR

This paper develops a new framework for locally analytic representations of p-adic groups, establishing their algebraic structure and applying it to prove irreducibility results for GL_2 representations.

Contribution

It introduces a category of admissible locally analytic representations and shows their algebraization via distribution algebras, with applications to GL_2.

Findings

Category of admissible representations can be embedded into modules over distribution algebra

Proves topological irreducibility of generic p-adic principal series of GL(2,Qp)

Establishes foundational properties of locally analytic representations

Abstract

Let L be a finite extension of Qp, and let K be a spherically complete non-archimedean extension field of L. In this paper we introduce a restricted category of continuous representations of locally L-analytic groups G in locally convex K-vector spaces. We call the objects of this category "admissible" representations and we establish some of their basic properties. Most importantly we show that (at least when G is compact) the category of admissible representations in our sense can be algebraized; it is faithfully full (anti)-embedded into the category of modules over the locally analytic distribution algebra D(G,K) of G over K. As an application of our theory, we prove the topological irreducibility of generic members of the p-adic principal series of GL(2,Qp).