Beraldo's conjecture: the action of $\mathbb{H}(\mathrm{LocSys}_{\check{G}})$ on $D(\mathrm{Bun}_G)$

An open problem posed by D. Beraldo, explicitly left unresolved in Færgeman–Raskin's unconditional proof of the Arinkin–Gaitsgory temperedness conjecture (arXiv:2108.02719), asks to construct and prove a canonical action of the rigid monoidal DG category $\mathbb{H}(\mathrm{LocSys}_{\check{G}})$ (a differential-graded category with a tensor product in which every object admits a dual) — Beraldo's categorification of the ring of differential operators on the stack of $\check{G}$-local systems on a curve — on the DG category $D(\mathrm{Bun}_G)$ of D-modules on the moduli stack of $G$-bundles, extending Gaitsgory's spectral action of $\mathrm{QCoh}(\mathrm{LocSys}_{\check{G}})$ and of Hecke/factorizable origin. The factorizable derived geometric Satake theorem of Campbell–Raskin (2023–2024), the enabling technology absent in 2021, now exists.

Background

In the de Rham geometric Langlands program one fixes a smooth, projective, geometrically connected curve $X$ over a field $k$ of characteristic $0$ and a split reductive group $G$, with Langlands dual group $\check{G}$. The two sides of the correspondence are $D(\mathrm{Bun}_G)$, the DG category of D-modules (sheaves of modules over algebraic differential operators) on the moduli stack of algebraic $G$-bundles on $X$, and the spectral category $\mathrm{IndCoh}_{\mathrm{Nilp}}(\mathrm{LocSys}_{\check{G}})$, where $\mathrm{LocSys}_{\check{G}}$ is the derived stack of $\check{G}$-bundles on $X$ equipped with a flat connection and $\mathrm{IndCoh}_{\mathrm{Nilp}}$ is the ind-coherent enhancement of quasi-coherent sheaves restricted by the nilpotent singular support condition of Arinkin–Gaitsgory (a microlocal constraint on the direction of the cotangent fibers, cutting out exactly the coherent sheaves expected to correspond to automorphic D-modules). The categorical geometric Langlands conjecture, in the form given by Arinkin and Gaitsgory following Beilinson and Drinfeld, predicts an equivalence $D(\mathrm{Bun}_G)\simeq\mathrm{IndCoh}_{\mathrm{Nilp}}(\mathrm{LocSys}_{\check{G}})$.

Inside the spectral side sits the plain subcategory $\mathrm{QCoh}(\mathrm{LocSys}_{\check{G}})$ of quasi-coherent sheaves. Arinkin and Gaitsgory defined the parallel subcategory of tempered D-modules on $\mathrm{Bun}_G$: 'tempered' refers, by analogy with automorphic forms, to objects without non-tempered (exponentially growing) Hecke components. Their definition used derived geometric Satake — the equivalence of the spherical Hecke category, i.e. equivariant D-modules on the affine Grassmannian (the quotient of the loop group of $G$ by its arc subgroup, parametrizing modifications of a $G$-bundle at a point), with ind-coherent sheaves on the classifying stack of $\check{G}$ with nilpotent singular support — at an auxiliary point $x\in X(k)$, and they conjectured that the resulting subcategory $D(\mathrm{Bun}_G)^{x\text{-}\mathrm{temp}}$ is independent of $x$. This temperedness conjecture was proved unconditionally by Færgeman and Raskin (arXiv:2108.02719).

A second structural statement about the spectral side is the spectral action. The spherical Hecke category at the points of $X$ organizes into a factorization category over the Ran space (the prestack of finite nonempty subsets of $X$, with union as multiplication), and the naive Satake functor realizes the category $\mathrm{Rep}(\check{G})_{\mathrm{Ran}}$ of $\check{G}$-representations spread over the Ran space as acting on $D(\mathrm{Bun}_G)$ by Hecke functors. Gaitsgory proved that this action factors through the monoidal localization $\mathrm{Rep}(\check{G})_{\mathrm{Ran}}\to\mathrm{QCoh}(\mathrm{LocSys}_{\check{G}})$, so $\mathrm{QCoh}(\mathrm{LocSys}_{\check{G}})$ itself acts monoidally on $D(\mathrm{Bun}_G)$; this is the spectral action.

Beraldo then asked what larger monoidal category should act. For a suitable derived stack $\mathcal{Y}$ he constructed a rigid monoidal DG category (a differential-graded monoidal category in which every object admits a monoidal dual) $\mathbb{H}(\mathcal{Y})=\mathrm{IndCoh}_0((\mathcal{Y}\times\mathcal{Y})^{\wedge})$ — ind-coherent sheaves with nilpotent singular support on the formal completion of $\mathcal{Y}\times\mathcal{Y}$ along the diagonal, multiplied by convolution — which behaves as a categorification of the ring of differential operators on $\mathcal{Y}$: module categories over $\mathbb{H}(\mathcal{Y})$ play the role of categorified D-modules, there is a monoidal functor $\Delta_{*,0}:\mathrm{QCoh}(\mathcal{Y})\to\mathbb{H}(\mathcal{Y})$ analogous to the inclusion of functions into differential operators, and $\mathbb{H}(\mathcal{Y})$ acts naturally on $\mathrm{IndCoh}(\mathcal{Y})$ and its singular-support variants. Beraldo conjectured that for $\mathcal{Y}=\mathrm{LocSys}_{\check{G}}$ the spectral action of $\mathrm{QCoh}(\mathrm{LocSys}_{\check{G}})$ on $D(\mathrm{Bun}_G)$ extends to an action of the larger category $\mathbb{H}(\mathrm{LocSys}_{\check{G}})$, to be constructed directly from the Hecke action via a Ran-space (factorizable) version of derived geometric Satake; he observed that such an extension would imply the Arinkin–Gaitsgory temperedness conjecture, and he gave a precise Betti-setting (topological) analogue in his work on the topological chiral homology of the spherical category. Færgeman and Raskin, whose proof of temperedness deliberately sidesteps $\mathbb{H}$ and factorizable Satake, state explicitly that their argument does not resolve Beraldo's conjecture and that it remained open, blocked at the time by the absence of factorizable derived Satake (a theorem claimed for over a decade by Gaitsgory–Lurie and later in progress by Campbell–Raskin).

Problem Statement

Let $X$ be a smooth, projective, geometrically connected curve over a field of characteristic $0$ and let $G$ be a split reductive group with Langlands dual group $\check{G}$. Prove Beraldo's conjecture: construct a canonical monoidal action of $\mathbb{H}(\mathrm{LocSys}_{\check{G}})$ — Beraldo's categorification of the ring of differential operators on the stack of $\check{G}$-local systems, defined as $\mathrm{IndCoh}_0((\mathrm{LocSys}_{\check{G}}\times\mathrm{LocSys}_{\check{G}})^{\wedge})$, the ind-coherent sheaves with nilpotent singular support on the formal completion of $\mathrm{LocSys}_{\check{G}}\times\mathrm{LocSys}_{\check{G}}$ along the diagonal, equipped with its convolution product — on the DG category $D(\mathrm{Bun}_G)$ of D-modules on the moduli stack of $G$-bundles on $X$, such that the restriction of the action along the monoidal functor $\Delta_{*,0}:\mathrm{QCoh}(\mathrm{LocSys}_{\check{G}})\to\mathbb{H}(\mathrm{LocSys}_{\check{G}})$ recovers the Gaitsgory spectral action of $\mathrm{QCoh}(\mathrm{LocSys}_{\check{G}})$. This is the deeper structural extension left open in Færgeman–Raskin, Section 1.2.4 (arXiv:2108.02719): the temperedness conjecture itself is settled and must not be conflated with it.

References

1. Joakim Færgeman and Sam Raskin. The Arinkin–Gaitsgory temperedness conjecture. arXiv:2108.02719; Bulletin of the London Mathematical Society 55(3):1419–1446 (2023), DOI 10.1112/blms.12801. https://arxiv.org/abs/2108.02719
2. Dario Beraldo. The center of the categorified ring of differential operators. arXiv:1709.07867; Journal of the European Mathematical Society 23(6):1999–2049 (2021), DOI 10.4171/jems/1048. https://arxiv.org/abs/1709.07867
3. Justin Campbell and Sam Raskin. Langlands duality on the Beilinson–Drinfeld Grassmannian. arXiv:2310.19734. https://arxiv.org/abs/2310.19734
4. Dennis Gaitsgory and Sam Raskin. Proof of the geometric Langlands conjecture I: construction of the functor. arXiv:2405.03599. https://arxiv.org/abs/2405.03599
5. Dima Arinkin and Dennis Gaitsgory. Singular support of coherent sheaves, and the geometric Langlands conjecture. arXiv:1201.6343; Selecta Mathematica 21(1):1–199 (2015), DOI 10.1007/s00029-014-0167-5. https://arxiv.org/abs/1201.6343
6. Dennis Gaitsgory. Outline of the proof of the geometric Langlands conjecture for GL(2). arXiv:1302.2506; Astérisque 400 (2018), DOI 10.24033/ast.966. https://arxiv.org/abs/1302.2506
7. Dario Beraldo. The topological chiral homology of the spherical category. arXiv:1802.08118; Journal of Topology 12(3):685–704 (2019), DOI 10.1112/topo.12098. https://arxiv.org/abs/1802.08118
8. Dario Beraldo. On the geometric Ramanujan conjecture. arXiv:2103.17211. https://arxiv.org/abs/2103.17211
9. Dario Beraldo and Lin Chen. Automorphic Gluing. arXiv:2204.09141. https://arxiv.org/abs/2204.09141
10. Joakim Færgeman. Quasi-Tempered Automorphic D-modules. arXiv:2210.09193. https://arxiv.org/abs/2210.09193
11. Peter Dillery and Arnaud Eteve. The extended Fargues–Scholze spectral action. arXiv:2608.02708. https://arxiv.org/abs/2608.02708
12. Dennis Gaitsgory, Vincent Lafforgue, and Sam Raskin. Tempered vs generic automorphic functions and the canonical filtration on automorphic functions. arXiv:2603.26925. https://arxiv.org/abs/2603.26925
13. Sam Raskin. Conjectures of Arthur and Ramanujan for Unramified Automorphic Forms: Announcement. Proceedings of the International Congress of Mathematicians 2026, Volume 3:619–636, DOI 10.1137/25m1810763. https://doi.org/10.1137/25m1810763
14. Laurent Fargues and Peter Scholze. Geometrization of the local Langlands correspondence. arXiv:2102.13459. https://arxiv.org/abs/2102.13459