On locally analytic Beilinson-Bernstein localization and the canonical dimension

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.