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Geometric Langlands for hypergeometric sheaves

Kamgarpour, Masoud and Yi, Lingfei (2021) Geometric Langlands for hypergeometric sheaves. Transactions of the American Mathematical Society, 374 (12). pp. 8435-8481. ISSN 0002-9947. doi:10.1090/tran/8509.

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Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler–Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. In this paper, we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems, thus confirming a central conjecture of the geometric Langlands program for hypergeometrics. The key new concept is the notion of hypergeometric automorphic data. We prove that this automorphic data is generically rigid (in the sense of Zhiwei Yun) and identify the resulting Hecke eigenvalue with hypergeometric sheaves. The definition of hypergeometric automorphic data in the tame case involves the mirabolic subgroup, while in the wild case, semistable (but not necessarily stable) vectors coming from principal gradings intervene.

Item Type:Article
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URLURL TypeDescription Paper
Yi, Lingfei0000-0001-8517-5499
Additional Information:© 2021 American Mathematical Society. Received by the editors February 14, 2021. Received by the editors February 14, 2021. The first author was supported by two Australian Research Council Discovery Projects. The second author was supported by a CalTech Graduate Student Fellowship. This paper fulfils a part of the vision of Zhiwei Yun for the role of rigidity in the geometric Langlands program. Our intellectual debt to the work of Yun and collaborators is obvious [HNY13, Yun14a, Yun14b, Yun16]. As noted above, this project was initiated in response to a question by Thomas Lam. We would like to thank him for raising this penetrating question and for many subsequent illuminating discussions. We would also like to thank Dima Arinkin, David Ben-Zvi, Javier Fresan, Jochen Heinloth, Konstantin Jakob, Paul Levy, Beth Romano, Will Sawin, Ole Warnaar, Daxin Xu, Zhiwei Yun, Xinwen Zhu, and Wadim Zudilin for helpful discussions. The first author would like to especially thank Dan Sage for collaborations on rigid connections [KS19, KS21] and for teaching him about parahorics and Moy–Prasad subgroups.
Funding AgencyGrant Number
Australian Research CouncilUNSPECIFIED
Subject Keywords:Hypergeometric local systems, rigid automorphic data, Hecke eigensheaves, geometric Langlands
Issue or Number:12
Classification Code:MSC (2020): Primary 14D24, 20G25, 22E50, 22E67
Record Number:CaltechAUTHORS:20220309-966703000
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:113853
Deposited By: George Porter
Deposited On:10 Mar 2022 21:32
Last Modified:10 Mar 2022 21:32

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