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Constructible tame Hecke eigensheaves in positive characteristic

OpenAI · October 5, 2026

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Abstract

We construct nonzero locally constructible perverse Hecke eigensheaves with Borel level at one marked point on a smooth projective curve of genus at least two over an algebraic closure of a finite field. The parameter is a Zariski-dense geometric ℓ-adic local system with tame regular-unipotent monodromy. The group is simple and simply connected and satisfies four explicit Lie-theoretic characteristic hypotheses. The eigensheaves have parabolic nilpotent singular support, and their eigenisomorphisms are compatible with tensor products, permutations, and fusion.

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Possibly relevant literature (found by the agent, not checked)

  • Shiyu Shen, Tamely ramified geometric Langlands correspondence in positive characteristic (arXiv:1810.12491, v3 April 2024; DOI 10.1093/imrn/rnae005). Proves a tamely ramified $\mathrm{GL}_n$ correspondence on an open spectral locus in characteristic $p>n$ with parabolic level, using crystalline differential operators rather than $\ell$-adic sheaves; a different-coefficient comparison for the tame Borel-level eigensheaf construction here, cited by the several-marked-points companion but not by this paper.

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Cite

@misc{OAI:Constructible-tame-Hecke-eigensheaves-in-positive-characteristic-October-5-2026,
  author = {{OpenAI}},
  title = {{Constructible tame Hecke eigensheaves in positive characteristic}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Constructible-tame-Hecke-eigensheaves-in-positive-characteristic-October-5-2026/constructible-tame-hecke-eigensheaves-positive-characteristic.pdf}{OAI:Constructible-tame-Hecke-eigensheaves-in-positive-characteristic-October-5-2026}},
  year = {2026}
}

Family 014: Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan

Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over $\overline{\mathbb F}_q$ under the four stated Lie-theoretic characteristic hypotheses. Over arbitrary algebraically closed fields of characteristic p > 0, the same conclusion holds assuming additionally that p is very good for the group and $p\nmid |W_G|$. Over global function fields, proves Ramanujan at every place for globally generic cuspidal representations of split adjoint absolutely simple exceptional groups, without characteristic or ramification-depth restrictions, and at every unramified place for cuspidal representations of split adjoint absolutely simple groups with a generic unramified component. Assuming the finite-level Ramanujan–Arthur decomposition, constructs global Arthur enhancements of occurring cuspidal excursion parameters for split connected semisimple groups at full finite level, recovering the given parameters by diagonal specialization on the entire Weil group, including inertia.

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