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FINE DELIGNE–LUSZTIG VARIETIES AND ARITHMETIC FUNDAMENTAL LEMMAS

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry Linear algebraic groups and related topics

Published online by Cambridge University Press:  10 December 2019

XUHUA HE ,
CHAO LI  and
YIHANG ZHU
XUHUA HE
Affiliation:
Lady Shaw Building, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong; xuhuahe@math.cuhk.edu.hk
CHAO LI
Affiliation:
Department of Mathematics, Columbia University, 2990 Broadway, New York, NY10027, USA; chaoli@math.columbia.edu, yihang@math.columbia.edu
YIHANG ZHU
Affiliation:
Department of Mathematics, Columbia University, 2990 Broadway, New York, NY10027, USA; chaoli@math.columbia.edu, yihang@math.columbia.edu

Abstract

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We prove a character formula for some closed fine Deligne–Lusztig varieties. We apply it to compute fixed points for fine Deligne–Lusztig varieties arising from the basic loci of Shimura varieties of Coxeter type. As an application, we prove an arithmetic intersection formula for certain diagonal cycles on unitary and GSpin Rapoport–Zink spaces arising from the arithmetic Gan–Gross–Prasad conjectures. In particular, we prove the arithmetic fundamental lemma in the minuscule case, without assumptions on the residual characteristic.

MSC classification

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties 14G17: Positive characteristic ground fields
Secondary: 20G40: Linear algebraic groups over finite fields
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e47
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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