On locally analytic Beilinson-Bernstein localization and the canonical dimension

TL;DR

This paper extends Beilinson-Bernstein localization to admissible locally analytic representations of p-adic groups and proves a locally analytic version of Smith's theorem on canonical dimension, advancing the understanding of p-adic representation theory.

Contribution

It develops a localization theorem for locally analytic representations of p-adic groups and extends methods of Ardakov-Wadsley to this setting, also proving a new dimension theorem.

Findings

Localization theorem for admissible locally analytic representations

Extension of Ardakov-Wadsley's methods to p-adic setting

Proof of a locally analytic version of Smith's dimension theorem

Abstract

Let G be a connected split reductive group over a p-adic field. In the first part of the paper we prove, under certain assumptions on G and the prime p, a localization theorem of Beilinson-Bernstein type for admissible locally analytic representations of principal congruence subgroups in the rational points of G. In doing so we take up and extend some recent methods and results of Ardakov-Wadsley on completed universal enveloping algebras to a locally analytic setting. As an application we prove, in the second part of the paper, a locally analytic version of Smith's theorem on the canonical dimension. This paper is in final form, is an expanded version of the former preprint 'On the dimension of locally analytic representations of semisimple p-adic groups' and has appeared in Mathematische Zeitschrift.