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On the physical rigidity of Frenkel-Gross connection

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Abstract

We show that the Frenkel-Gross connection on \({\mathbb {G}}_m\) is physically rigid as \(\check{G}\)-connection, thus confirming the de Rham version of a conjecture of Heinloth-Ngô-Yun. The proof is based on the construction of the Hecke eigensheaf of a \(\check{G}\)-connection with only generic oper structure, using the localization of Weyl modules.

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Notes

  1. They work in the setting of \(\ell \)-adic sheaves.

  2. This proposition can be proved alternatively using Segal-Sugawara operators, see [23, Proposition 28.(i)].

  3. Note that we use \(d_i\)’s to denote fundamental degrees, while in loc. cit. \(d_i\)’s are exponents.

  4. This result is originally due to an unpublished work of Tsao-Hsien Chen.

  5. Although in [23] they use a potentially different set of generators \(S_{ij}\) of the center \({\mathfrak {Z}}\) and assumes G is a classical group, their argument in the case of \(d=h\) works for any set of generators \(S_{ij}\) and any simple algebraic group.

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Acknowledgements

The author thanks Alexander Beilinson, Tsao-Hsien Chen and Xinwen Zhu for very valuable discussions. The author particularly thanks Zhu for pointing out Lemma 4, and thanks Chen for suggesting the key reference [9].

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  1. School of Mathematics, University of Minnesota, Twin cities, Minneapolis, MN, 55455, USA

    Lingfei Yi

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Yi, L. On the physical rigidity of Frenkel-Gross connection. Sel. Math. New Ser. 30, 41 (2024). https://doi.org/10.1007/s00029-024-00931-9

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