The Restricted Geometric Langlands Equivalence in Positive Characteristic

OpenAI

For a connected reductive group on a smooth projective connected curve over an algebraically closed field of characteristic p>0, the restricted geometric Langlands functor reaches every spectral component under either finite-field descent or, with p very good and p not dividing the Weyl-group order, an arbitrary algebraically closed base. Both regimes require a nondegenerate invariant Lie-algebra form (also on Levi centers), Chevalley restriction for every Levi, semisimple centralizers that are Levi, and nilpotents lying in parabolic unipotent radicals over every field extension.

Sources for this record

What was already known

Gaitsgory and Raskin had an equivalence onto a possibly proper open-and-closed union of components, with full support for general linear groups.

AI involvement AI-led

OpenAI authors the manuscript, and its release says an internal model produced ordinary evaluation manuscripts. Human task selection and curation are not detailed for this result.

OpenAI authorship and the repository's model-production account support AI-led work; the record does not establish independence from human problem selection.

Importance Major

Under the stated hypotheses, upgrades the restricted equivalence to the whole spectral category.

Limitations

The theorem relies on the cited earlier equivalence and specialization results. A completed human or machine check of the new full-support argument is undocumented. The later tame eigensheaf theorem is separate.

Technical verification Reported

The released manuscript supplies an argument, but no completed human or machine check of its full-support theorem is documented. Reported is the evidence default, not a failed-check finding.

Human understanding Few

No named human explains the new support and nonvanishing mechanism; Few is the missing-evidence default.

Correction and publication history

2026-10-18 · publication

Under the stated hypotheses, upgrades the restricted equivalence to the whole spectral category.

Read this published revision