Component exhaustion and Hecke eigensheaves for the restricted geometric Langlands correspondence in positive characteristic

For a smooth complete curve $X$ over an algebraically closed field of characteristic $p>0$ and a reductive group $G$, Gaitsgory and Raskin (2025) proved that the category of $\ell$-adic sheaves on the moduli stack of $G$-bundles with nilpotent singular support is equivalent to ind-coherent sheaves on the union of some connected components of the stack of restricted Langlands parameters, and that this union is the whole stack in characteristic $0$ and for $G=GL_n$. The open problem is to prove, or refute, that this component exhaustion always holds (Conjecture 1.3.10 of the cited paper) — equivalently, to complete the restricted geometric Langlands conjecture for arbitrary reductive groups in positive characteristic — and with it the existence of a non-zero Hecke eigensheaf for every irreducible dual-group local system, the facet the authors call the most mysterious part of the Langlands conjecture. Both questions remain open outside the known cases.

Background

Let $k$ be an algebraically closed field, $X$ a smooth complete curve over $k$, and $G$ a reductive group over $k$; fix a prime $\ell\neq\operatorname{char}(k)$ and work with $\ell$-adic sheaves (constructible sheaves with $\mathbb{Q}_\ell$-coefficients). Denote by $\operatorname{Bun}_G$ the moduli stack of principal $G$-bundles on $X$. The Hecke functors modify a $G$-bundle at a varying point of $X$ using the affine Grassmannian $\operatorname{Gr}_G=G((t))/G[[t]]$ (where $G((t))$ and $G[[t]]$ denote the groups of algebraic maps into $G$ from the formal punctured disc and the formal disc around a point of $X$, respectively); a Hecke eigensheaf with eigenvalue a $\check G$-local system $\sigma$ (a principal bundle with connection on $X$ for the Langlands dual group $\check G$, the reductive group whose root datum is dual to that of $G$) is a sheaf $\mathcal{F}$ with $H_V(\mathcal{F})\simeq\mathcal{F}\boxtimes\sigma_V$ for every representation $V$ of $\check G$. The geometric Langlands conjecture predicts that Hecke eigensheaves organize the category of sheaves on $\operatorname{Bun}_G$ in terms of spectral data on the stack of $\check G$-local systems. Laumon (1987) observed that all automorphic sheaves must satisfy a microlocal restriction: their singular support (a closed conical subset of the cotangent bundle, defined for $\ell$-adic sheaves by Arinkin–Gaitsgory–Kazhdan–Raskin–Rozenblyum–Varshavsky) must be nilpotent, i.e. must project to the nilpotent cone in the Hitchin-type fibers. This led to the restricted form of the conjecture (Conjecture 21.2.7 of arXiv:2010.01906, the AGKRRV series): the category $\operatorname{Shv}_{\mathrm{Nilp}}(\operatorname{Bun}_G)$ of sheaves with nilpotent singular support should be equivalent, compatibly with the natural action of quasi-coherent sheaves, to the category $\operatorname{IndCoh}_{\mathrm{Nilp}}(\operatorname{LS}^{\mathrm{restr}}_{\check G})$ of ind-coherent sheaves (the sheaf theory adapted to singular derived stacks) with nilpotent singular support on the stack $\operatorname{LS}^{\mathrm{restr}}_{\check G}$ of $\check G$-local systems with restricted variation (a refinement of the naive stack of local systems introduced in that series precisely so that the categorical conjecture makes sense for any constructible sheaf theory, including $\ell$-adic sheaves in positive characteristic). The connected components of $\operatorname{LS}^{\mathrm{restr}}_{\check G}$ are indexed by the semisimplification of the local system, so every irreducible $\check G$-local system occupies its own connected component, which is however stacky and non-reduced. In characteristic $0$, the correspondence for D-modules is a theorem (the 2024 proof of the geometric Langlands conjecture), and eigensheaves for irreducible parameters exist by the Beilinson–Drinfeld construction via localization of affine Kac–Moody modules at the critical level; for $G=GL_n$, eigensheaves for irreducible local systems exist in every characteristic (Frenkel–Gaitsgory–Vilonen, 2002). In positive characteristic, D-module localization is unavailable, and Gaitsgory–Raskin (arXiv:2508.02237) instead transfer the characteristic-$0$ theory to characteristic $p$ by spreading out. They construct a canonical Langlands functor and prove that it is an equivalence onto $\operatorname{IndCoh}_{\mathrm{Nilp}}({}'\mathrm{LS}^{\mathrm{restr}}_{\check G})$, where ${}'\mathrm{LS}^{\mathrm{restr}}_{\check G}$ is the union of some of the connected components of $\operatorname{LS}^{\mathrm{restr}}_{\check G}$, and that the inclusion ${}'\mathrm{LS}^{\mathrm{restr}}_{\check G}\subset\mathrm{LS}^{\mathrm{restr}}_{\check G}$ is an equality in characteristic $0$ and for $G=GL_n$. Given this theorem, the full restricted conjecture in positive characteristic is equivalent to the statement that no component is missed; the surrounding proofs also currently carry auxiliary hypotheses on the characteristic (Assumption (*) in Sect. 7.1.3 of the cited paper, and the characteristic hypotheses inherited from the AGKRRV series), and the mismatch between the structure sheaf and the dualizing sheaf on the arithmetic parameter stack — ultimately responsible for the Arthur $SL_2$ in the classification of automorphic functions — is only conjecturally understood (Remark 0.2.4 of the cited paper).

Problem Statement

Let $k$ be an algebraically closed field of characteristic $p>0$ (with $\ell\neq p$ fixed), $X$ a smooth complete curve over $k$, and $G$ a reductive group over $k$, with $\check G$ its Langlands dual group. Let $\mathrm{LS}^{\mathrm{restr}}_{\check G}$ denote the stack of $\check G$-local systems on $X$ with restricted variation, and let ${}'\mathrm{LS}^{\mathrm{restr}}_{\check G}\subset\mathrm{LS}^{\mathrm{restr}}_{\check G}$ be the union of connected components characterized by the equivalence $\operatorname{Shv}_{\mathrm{Nilp}}(\operatorname{Bun}_G)\simeq\operatorname{IndCoh}_{\mathrm{Nilp}}({}'\mathrm{LS}^{\mathrm{restr}}_{\check G})$ of Gaitsgory–Raskin (arXiv:2508.02237, Theorem 1.3.9). Determine whether the inclusion ${}'\mathrm{LS}^{\mathrm{restr}}_{\check G}\subset\mathrm{LS}^{\mathrm{restr}}_{\check G}$ is an equality for every reductive $G$ over every such field $k$ — Conjecture 1.3.10 of the cited paper — or exhibit a reductive group and a positive-characteristic field for which the inclusion is strict. By the cited theorem, this is equivalent to establishing the full restricted geometric Langlands conjecture in positive characteristic, $\operatorname{Shv}_{\mathrm{Nilp}}(\operatorname{Bun}_G)\simeq\operatorname{IndCoh}_{\mathrm{Nilp}}(\mathrm{LS}^{\mathrm{restr}}_{\check G}),$ compatible with the action of $\operatorname{QCoh}(\mathrm{LS}^{\mathrm{restr}}_{\check G})$, without restricting the components on the spectral side. As its decisive consequence — since every irreducible $\check G$-local system occupies its own connected component — the exhaustion equality entails that, for every irreducible $\check G$-local system on $X$, there is a non-zero Hecke eigensheaf in $\operatorname{Shv}_{\mathrm{Nilp}}(\operatorname{Bun}_G)$ (equivalently, over function fields, a non-zero cuspidal eigenfunction for every irreducible Langlands parameter). The problem concerns an arbitrary reductive group $G$ over a field $k$ of arbitrary positive characteristic subject to the standing $\ell$-adic framework; the cases $G=GL_n$, tori, and characteristic $0$ are known.

References

1. Dennis Gaitsgory and Sam Raskin, 'Geometric Langlands in positive characteristic from characteristic zero', arXiv:2508.02237 (2025), DOI 10.48550/arXiv.2508.02237, https://arxiv.org/abs/2508.02237
2. D. Arinkin, D. Gaitsgory, D. Kazhdan, S. Raskin, N. Rozenblyum and Y. Varshavsky, 'The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support', arXiv:2010.01906 (2020), DOI 10.48550/arXiv.2010.01906, https://arxiv.org/abs/2010.01906
3. Edward Frenkel, Dennis Gaitsgory and Kari Vilonen, 'On the geometric Langlands conjecture', Journal of the American Mathematical Society 15, no. 2 (2002), 367-417, DOI 10.1090/S0894-0347-01-00388-5, preprint arXiv:math/0012255 (2000), https://arxiv.org/abs/math/0012255
4. Gérard Laumon, 'Correspondance de Langlands géométrique pour les corps de fonctions', Duke Mathematical Journal 54, no. 1 (1987), 309–359, DOI 10.1215/s0012-7094-87-05418-4, https://doi.org/10.1215/s0012-7094-87-05418-4
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