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Geometric stabilisation via $ p$-adic integration


Authors: Michael Groechenig, Dimitri Wyss and Paul Ziegler
Journal: J. Amer. Math. Soc. 33 (2020), 807-873
MSC (2010): Primary 11S37, 11S80, 14H60, 20G40, 14D24
DOI: https://doi.org/10.1090/jams/948
Published electronically: June 15, 2020
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Abstract: In this article we give a new proof of Ngô's geometric stabilisation theorem, which implies the fundamental lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme $ G$ to the cohomology of Hitchin fibres for the endoscopy groups $ H_{\kappa }$. Our proof avoids the decomposition and support theorem, instead the argument is based on results for $ p$-adic integration on coarse moduli spaces of Deligne-Mumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of $ G$-Higgs bundles in terms of endoscopic data, and extend duality for generic Hitchin fibres of Langlands dual group schemes to the quasi-split case.


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Additional Information

Michael Groechenig
Affiliation: Department of Mathematical and Computational Sciences, University of Toronta at Mississauga, 3359 Mississauga Rd N., Ontario Canada
Address at time of publication: Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4
Email: michael.groechenig@utoronto.ca

Dimitri Wyss
Affiliation: Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
Email: dimitri.wyss@epfl.ch

Paul Ziegler
Affiliation: Mathematical Institute, University of Oxford, Oxford, United Kingdom
Address at time of publication: Department of Mathematics, Technische Universität München, Munich, Germany
Email: paul.ziegler@ma.tum.de

DOI: https://doi.org/10.1090/jams/948
Received by editor(s): November 8, 2018
Received by editor(s) in revised form: October 28, 2019
Published electronically: June 15, 2020
Additional Notes: The first author was funded by a Marie Skłodowska-Curie fellowship: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 701679. \includegraphics[height = .75cm]euflag.eps
The second author was supported by the Foundation Sciences Mathématiques de Paris, as well as a public grant overseen by the French National Research Agency (ANR) as part of the Investissements d’avenir program (reference: ANR-10-LABX-0098) and also by ANR-15-CE40-0008 (Défigéo).
The third author was supported by the Swiss National Science Foundation.
Article copyright: © Copyright 2020 American Mathematical Society