Vol. 8, No. 6, 2014

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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

Let L be a finite field extension of 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.

locally analytic representations, Bruhat–Tits, buildings, sheaves
Mathematical Subject Classification 2010
Primary: 22E50
Secondary: 20G25, 20G05, 32C38, 11S37, 13N10
Received: 27 November 2012
Revised: 20 February 2014
Accepted: 23 May 2014
Published: 2 October 2014
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
Matthias Strauch
Department of Mathematics
Indiana University
Rawles Hall
Bloomington, IN 47405
United States