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Published December 2021 | Submitted + Published
Journal Article Open

Geometric Langlands for hypergeometric sheaves

Abstract

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.

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.

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Published - S0002-9947-2021-08509-8.pdf

Submitted - 2006.10870.pdf

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August 22, 2023
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