Full spectral support in restricted geometric Langlands
Follow the latest progress by email.
Does the restricted geometric Langlands equivalence in positive characteristic reach every spectral component, rather than a possibly proper open-and-closed union? The group is connected reductive and the curve smooth, projective and connected over an algebraically closed field of characteristic p>0.
The proposed regimes are finite-field descent, or an arbitrary algebraically closed base when p is very good and does not divide the Weyl-group order. Both require a nondegenerate invariant Lie-algebra form, including on Levi centers; Chevalley restriction for every Levi; semisimple centralizers that are Levi; and nilpotents lying in parabolic unipotent radicals over every field extension.
Why it matters and what counts as progress
Why it matters: Full spectral support removes a component restriction from the correspondence between the automorphic and spectral categories.
What counts as progress: A full-support theorem under these explicit characteristic and group hypotheses, or a spectral component outside the functor’s image.
How the problem developed
- 4 Aug 2025 · Partial result [1]
Dennis Gaitsgory and Sam Raskin Established an equivalence for automorphic sheaves with nilpotent singular support and ind-coherent sheaves on some connected components of the Langlands-parameter stack; the result did not cover every component.
- 24 Sept 2026 · Reported claim [2]
OpenAI The manuscript claims the restricted geometric Langlands equivalence with full spectral support under its stated group, curve, and characteristic hypotheses.
AI contribution
The Restricted Geometric Langlands Equivalence in Positive Characteristic
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
- Gaitsgory and Raskin, Geometric Langlands in positive characteristic from characteristic zero
- OpenAI, Restricted geometric Langlands (manuscript dated 24 September 2026)
- The Restricted Geometric Langlands Equivalence in Positive Characteristic
- OpenAI restricted geometric Langlands introduction
- OpenAI mathematics release account
- Separate tame eigensheaf theorem
Website timeline
Publication dates and editorial changes.
OpenAI reports full support for restricted geometric Langlands in positive characteristic