Algebra & Number Theory

Locally analytic representations and sheaves on the Bruhat–Tits building

Deepam Patel, Tobias Schmidt, and Matthias Strauch

Full-text: Open access

Abstract

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.

Article information

Source
Algebra Number Theory, Volume 8, Number 6 (2014), 1365-1445.

Dates
Received: 27 November 2012
Revised: 20 February 2014
Accepted: 23 May 2014
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730251

Digital Object Identifier
doi:10.2140/ant.2014.8.1365

Mathematical Reviews number (MathSciNet)
MR3267141

Zentralblatt MATH identifier
1298.22021

Subjects
Primary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]
Secondary: 20G25: Linear algebraic groups over local fields and their integers 20G05: Representation theory 32C38: Sheaves of differential operators and their modules, D-modules [See also 14F10, 16S32, 35A27, 58J15] 11S37: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E50] 13N10: Rings of differential operators and their modules [See also 16S32, 32C38]

Keywords
locally analytic representations Bruhat–Tits buildings sheaves

Citation

Patel, Deepam; Schmidt, Tobias; Strauch, Matthias. Locally analytic representations and sheaves on the Bruhat–Tits building. Algebra Number Theory 8 (2014), no. 6, 1365--1445. doi:10.2140/ant.2014.8.1365. https://projecteuclid.org/euclid.ant/1513730251


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