Categorical geometric Langlands theorem

Statement

Let $X$ be a smooth projective connected curve over an algebraically closed field of characteristic $0$ , let $G$ be a connected reductive group, and let $\widehat G$ be its Langlands dual group. The modern de Rham categorical geometric Langlands theorem gives an equivalence

D\text{-}\operatorname{mod}_{1/2}\bigl(\operatorname{Bun}_G(X)\bigr) \simeq \operatorname{IndCoh}_{\mathcal N} \bigl(\operatorname{LocSys}_{\widehat G}(X)\bigr).

The left side is the derived category of half-twisted $D$ -modules on the stack of $G$ -bundles. The right side is the category of ind-coherent sheaves with nilpotent singular support on the stack of $\widehat G$ -local systems. Here $\mathcal N$ is the global nilpotent cone.

Pointwise consequence

At a sufficiently regular spectral point $E$ , the correspondence predicts an automorphic Hecke eigensheaf $\mathcal F_E$ . For every representation $V$ of $\widehat G$ ,

H_V(\mathcal F_E)\simeq \mathcal F_E\boxtimes E_V.

This is a sheaf-theoretic analogue of simultaneous Hecke eigenfunctions.

Scope and status

The characteristic-zero de Rham theorem and an equivalent Betti form have been established. Earlier work proved the torus case and numerous special cases and constructed important families of eigensheaves.

Étale, positive-characteristic, quantum, and ramified versions require their own hypotheses and are not all consequences of this theorem. Thus “geometric Langlands” also continues to name a broader program.

Necessary spectral enlargement

Replacing the spectral category by $\operatorname{QCoh}(\operatorname{LocSys}_{\widehat G})$ works in the abelian case but is generally too small. Likewise, the correspondence is not a bijection between individual bundles and individual local systems: both sides are categories on moduli stacks.

Half twisting

The subscript $1/2$ denotes the standard half-density, or square-root of the canonical, twisting on the automorphic side. Precise theorem statements also specify derived and renormalized categories, sheaf theories, coefficient fields, and duality data. Writing an untwisted $D\text{-}\operatorname{mod}(\operatorname{Bun}_G)$ is a useful first approximation, but it suppresses a real normalization rather than a cosmetic choice.

References

Alexander Beilinson and Vladimir Drinfeld, Quantization of Hitchin’s Integrable System and Hecke Eigensheaves, preprint. author manuscript.
Dima Arinkin and Dennis Gaitsgory, “Singular support of coherent sheaves, and the geometric Langlands conjecture,” Selecta Mathematica 21 (2015), 1–199. arXiv.
Dennis Gaitsgory and Sam Raskin, “Proof of the geometric Langlands conjecture I: construction of the functor,” 2024. arXiv.
Dennis Gaitsgory and Sam Raskin, “Proof of the geometric Langlands conjecture V: the multiplicity one theorem,” 2024. arXiv.