Locally analytic representations and sheaves on the Bruhat-Tits building

TL;DR

This paper develops a functor linking admissible locally analytic representations of p-adic groups to equivariant sheaves on the Bruhat-Tits building, extending the geometric representation theory framework.

Contribution

It introduces a new functor connecting locally analytic representations with sheaves on the Bruhat-Tits building, generalizing existing smooth representation sheaf constructions.

Findings

Establishes a functor from admissible locally analytic G-representations to equivariant sheaves.

Shows compatibility with Beilinson-Bernstein localization for g-modules.

Relates sheaves for smooth representations to those by Schneider and Stuhler.

Abstract

Let L be a finite field extension of Q_p and let G be the group of L-rational points of a split connected reductive group over L. We view G as a locally L-analytic group with Lie algebra g. We define a functor from admissible locally analytic G-representations with prescribed infinitesimal character to a category of equivariant sheaves on the Bruhat-Tits building of G. For smooth representations, the corresponding sheaves are closely related to the sheaves constructed by S. Schneider and U. Stuhler. The functor is also compatible, in a certain sense, with the localization of g-modules on the flag variety by A. Beilinson and J. Bernstein.