Locally analytic representations and sheaves on the Bruhat–Tits building

Patel, Deepam; Schmidt, Tobias; Strauch, Matthias

doi:10.2140/ant.2014.8.1365

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Locally analytic representations and sheaves on the Bruhat–Tits building

Deepam Patel, Tobias Schmidt and Matthias Strauch

Vol. 8 (2014), No. 6, 1365–1445

DOI: 10.2140/ant.2014.8.1365

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 $g$ . The purpose of this work is to propose a construction which extends the localization of smooth $G$ -representations of P. Schneider and U. Stuhler to the case of locally analytic $G$ -representations. 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 of Schneider and 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.

Keywords

locally analytic representations, Bruhat–Tits, buildings, sheaves

Mathematical Subject Classification 2010

Primary: 22E50

Secondary: 20G25, 20G05, 32C38, 11S37, 13N10

Milestones

Received: 27 November 2012

Revised: 20 February 2014

Accepted: 23 May 2014

Published: 2 October 2014

Authors

Deepam Patel
Department of Mathematics Purdue University 150 North University Street West Lafayette, IN 47907 United States

Tobias Schmidt
Institut fuer Mathematik Humboldt-Universität zu Berlin Rudower Chaussee 25 D-12489 Berlin Germany

Matthias Strauch
Department of Mathematics Indiana University Rawles Hall Bloomington, IN 47405 United States

Vol. 8, No. 6, 2014