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Math-2801 · mathematics · Number theory

Restricted geometric Langlands in the stated positive-characteristic regimes

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let k be algebraically closed of characteristic $p>0$, ell different from p, X a smooth projective curve and G connected reductive. Assume that the Lie algebra admits an invariant nondegenerate symmetric form nondegenerate on every Levi center, Chevalley restriction is an isomorphism for every Levi, semisimple Lie-element scheme centralizers are Levi subgroups, and every nilpotent Lie element over every extension belongs to the Lie algebra of the unipotent radical of an extension-defined parabolic. Assume either descent of k,X,G from a finite field with k its algebraic closure, or that p is very good for G and does not divide its Weyl-group order. Must the existing primed restricted geometric Langlands functor have full spectral support, giving a QCoh-linear equivalence from ell-adic sheaves on $Bun_G$ with nilpotent singular support to IndCoh with nilpotent support on the entire restricted dual-group local-system prestack?

Short English statement

Let k be algebraically closed of characteristic $p>0$, ell different from p, X a smooth projective curve and G connected reductive. Assume that the Lie algebra admits an invariant nondegenerate symmetric form nondegenerate on every Levi center, Chevalley restriction is an isomorphism for every Levi, semisimple Lie-element scheme centralizers are Levi subgroups, and every nilpotent Lie element over every extension belongs to the Lie algebra of the unipotent radical of an extension-defined parabolic. Assume either descent of k,X,G from a finite field with k its algebraic closure, or that p is very good for G and does not divide its Weyl-group order. Must the existing primed restricted geometric Langlands functor have full spectral support, giving a QCoh-linear equivalence from ell-adic sheaves on $Bun_G$ with nilpotent singular support to IndCoh with nilpotent support on the entire restricted dual-group local-system prestack?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2801-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2801

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 014; 2026-09-24).

Restricted geometric Langlands in the stated positive-characteristic regimes

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 014; 2026-09-24), the asserted result is that restricted geometric Langlands has full spectral support in positive characteristic in the two stated characteristic regimes, with additional very-good and Weyl-order conditions over arbitrary algebraically closed fields.

Mathematical statement of the claimed result

Manuscript 1.
Let $k$ be algebraically closed of characteristic $p>0$, $\ell\ne p$, $E=\overline{\mathbb Q}_{\ell}$, $X/k$ smooth projective connected and $G/k$ connected reductive. Require the four common conditions: an invariant nondegenerate Lie-algebra form nondegenerate on every Levi center, Chevalley restriction for every Levi, semisimple Lie-algebra centralizers being Levi, and nilpotent elements over every extension lying in parabolic unipotent-radical Lie algebras. Regime A additionally requires $k=\overline{\mathbb F}_q$ and descent of $X$ and split $G$ to $\mathbb F_q$. Regime B instead permits arbitrary algebraically closed $k$, but requires $p$ very good for $G$ and $p\nmid|W_G|$. Very good excludes primes dividing $n+1$ for $A_n$, $2$ for $B,C,D$, $2,3$ for $G_2,F_4,E_6,E_7$, and $2,3,5$ for $E_8$.

Let $Y=\operatorname{LS}_{\widehat G}^{\rm restr}(X)$ be the restricted spectral prestack, parametrizing right-t-exact symmetric monoidal functors from dual-group representations to $\operatorname{QCoh}(T)\otimes_E\operatorname{QLisse}_E(X)$ on derived affine $T$. The claim says the previously occurring open-and-closed union $Y'$ equals all of $Y$, giving a $\operatorname{QCoh}(Y)$-linear equivalence \[ \mathbb L_G^{\rm restr}:\operatorname{Shv}_{\rm Nilp}(\operatorname{Bun}_G(X)) \xrightarrow{\sim}\operatorname{IndCoh}_{\rm Nilp}(Y). \] The automorphic and spectral categories each retain nilpotent singular support. This positive-characteristic result is restricted to the two stated regimes.

Outline of the argument

For manuscript 1 (family 014; 2026-09-24), a hypothetical missing component is transported along a trait to a geometric specialization. Levi reduction and transverse Hecke modifications provide cotangent tests whose nonvanishing prevents that spectral summand from being annihilated.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 014; 2026-09-24).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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