Uniform analysis on local fields and applications to orbital integrals

Authors:
Raf Cluckers, Julia Gordon and Immanuel Halupczok

Journal:
Trans. Amer. Math. Soc. Ser. B **5** (2018), 125-166

MSC (2010):
Primary 14E18; Secondary 22E50, 40J99

DOI:
https://doi.org/10.1090/btran/25

Published electronically:
October 2, 2018

MathSciNet review:
3859937

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Abstract | References | Similar Articles | Additional Information

Abstract: We study upper bounds, approximations, and limits for functions of motivic exponential class uniformly in non-Archimedean local fields whose characteristic is $0$ or sufficiently large. Our results together form a flexible framework for doing analysis over local fields in a field-independent way. As corollaries, we obtain many new transfer principles, for example, for local constancy, continuity, and existence of various kinds of limits. Moreover, we show that the Fourier transform of an $L^2$-function of motivic exponential class is again of motivic exponential class. As an application in representation theory, we prove uniform bounds for the Fourier transforms of orbital integrals on connected reductive $p$-adic groups.

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Additional Information

**Raf Cluckers**

Affiliation:
Université de Lille, Laboratoire Painlevé, CNRS - UMR 8524, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France —and— Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium

Email:
Raf.Cluckers@univ-lille.fr

**Julia Gordon**

Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada

Email:
gor@math.ubc.ca

**Immanuel Halupczok**

Affiliation:
Mathematisches Institut, HHU Düsseldorf, Universitätsstrasse 1, 40225 Düsseldorf, Germany

Email:
math@karimmi.de

Keywords:
Transfer principles for motivic integrals,
uniform bounds,
motivic integration,
motivic constructible exponential functions,
loci of motivic exponential class,
orbital integrals,
admissible representations of reductive groups,
Harish-Chandra characters

Received by editor(s):
September 11, 2017

Received by editor(s) in revised form:
March 12, 2018

Published electronically:
October 2, 2018

Additional Notes:
The first author was supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement no. 615722 MOTMELSUM, by the Labex CEMPI (ANR-11-LABX-0007-01), and would like to thank both the Forschungsinstitut für Mathematik (FIM) at ETH Zürich and the IHÉS for the hospitality during part of the writing of this paper.

The second author was supported by NSERC

The third author was partially supported by the SFB 878 of the Deutsche Forschungsgemeinschaft. Part of the work was done while he was affiliated with the University of Leeds.

Article copyright:
© Copyright 2018
by the authors under
Creative Commons Attribution 3.0 License
(CC BY 3.0)