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Bessel F-isocrystals for reductive groups

Xu, Daxin and Zhu, Xinwen (2022) Bessel F-isocrystals for reductive groups. Inventiones Mathematicae, 227 (3). pp. 997-1092. ISSN 0020-9910. doi:10.1007/s00222-021-01079-5.

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We construct the Frobenius structure on a rigid connection Be_Ğ on G_m for a split reductive group Ğ introduced by Frenkel–Gross. These data form a Ğ-valued overconvergent F-isocrystal Be^†_Ğ on G_(m,F_p), which is the p-adic companion of the Kloosterman Ğ-local system Kl_Ğ constructed by Heinloth–Ngô–Yun. By studying the structure of the underlying differential equation, we calculate the monodromy group of Be^†_Ğ when Ğ is almost simple (which recovers the calculation of monodromy group of Kl_Ğ due to Katz and Heinloth–Ngô–Yun), and prove a conjecture of Heinloth–Ngô–Yun on the functoriality between different Kloosterman Ğ-local systems. We show that the Frobenius Newton polygons of Kl_Ğ are generically ordinary for every Ğ and are everywhere ordinary on |G_(m,F_p)| when Ğ is classical or G₂.

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Additional Information:© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021. Received: 30 November 2020 / Accepted: 23 September 2021 / Published online: 16 January 2022. We would like to thank Benedict Gross, Shun Ohkubo, Daqing Wan, Liang Xiao and Zhiwei Yun for valuable discussions. We are also grateful to an anonymous referee for his/her careful reading and valuable comments. X. Z. is partially supported by the National Science Foundation under agreement Nos. DMS-1902239 and a Simons Fellowship.
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Simons FoundationUNSPECIFIED
Issue or Number:3
Record Number:CaltechAUTHORS:20220310-752269000
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Official Citation:Xu, D., Zhu, X. Bessel F-isocrystals for reductive groups. Invent. math. 227, 997–1092 (2022).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:113873
Deposited By: George Porter
Deposited On:15 Mar 2022 16:10
Last Modified:15 Mar 2022 16:10

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