Geometric Langlands: Derived / Categorical¶
Job: state the derived / categorical geometric Langlands conjecture in the modern Arinkin–Gaitsgory / Gaitsgory–Rozenblyum form — an equivalence of presentable DG categories between \(\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\) and \(\mathrm{IndCoh}_{\mathrm{N}}(\mathrm{LocSys}_{\widehat{G}})\), the ind-coherent sheaves on the derived stack of \(\widehat{G}\)-local systems with singular support in the nilpotent cone — and explain why the singular support condition is the essential renormalization.
Core¶
The automorphic side¶
The automorphic category is the full (de Rham) D-module category
of \(\mathrm{Bun}_G\), viewed as a presentable DG category (stable, \(\mathbb{C}\)-linear). It is not the bounded derived category of a finite-type abelian category: \(\mathrm{Bun}_G\) is an Artin stack of infinite type, with components indexed by \(\pi_1(G)\) and Harder–Narasimhan strata, and the D-module category is large (presentable, generated by compact objects that are not all perfect). The Hecke action of \(\mathrm{Rep}(\widehat{G})\) (see Hecke Action & Opers) makes \(\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\) a module category over \(\mathrm{Rep}(\widehat{G})\), equivalently a comodule over the spectral category built from \(\mathrm{LocSys}_{\widehat{G}}\).
The spectral side: why \(\mathrm{QCoh}\) is wrong¶
The naïve candidate for the spectral category is \(\mathrm{QCoh}(\mathrm{LocSys}_{\widehat{G}})\), the quasi-coherent sheaves on the moduli stack of \(\widehat{G}\)-local systems. This is wrong for two reasons (Arinkin–Gaitsgory):
- Size / generation: \(\mathrm{QCoh}(\mathrm{LocSys}_{\widehat{G}})\) is too large — it has too many objects, and the natural automorphic-to-spectral functor \(\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\to\mathrm{QCoh}(\mathrm{LocSys}_{\widehat{G}})\) cannot be essentially surjective onto a category this big (the automorphic category has bounded / holonomic / regular singular constraints; \(\mathrm{QCoh}\) does not).
- Hecke compatibility: the Hecke action on \(\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\) is by convolution with Satake sheaves, and the corresponding spectral action is by tensoring with the universal local system \(\mathcal{U}\) on \(\mathrm{LocSys}_{\widehat{G}}\times X\). The tensoring-by-\(\mathcal{U}\) action on \(\mathrm{QCoh}\) does not match the automorphic Hecke action on the nose: one needs the singular-support renormalization to make the actions compatible.
The fix (Arinkin–Gaitsgory) is to use ind-coherent sheaves (not QCoh) on the derived stack \(\mathrm{LocSys}_{\widehat{G}}\) (the derived enhancement of the moduli, retaining the deformation / obstruction theory of flat connections), with a singular support condition cutting out the correct subcategory.
\(\mathrm{IndCoh}_{\mathrm{N}}(\mathrm{LocSys}_{\widehat{G}})\)¶
Let \(\mathrm{LocSys}_{\widehat{G}}(X)\) denote the derived moduli stack of \(\widehat{G}\)-local systems (flat connections) on \(X\) — a derived Artin stack whose truncation is the classical moduli, with derived structure encoding the deformation / obstruction (the cotangent to \(\mathrm{LocSys}\) is the moduli of Higgs bundles / the Hitchin system). The category \(\mathrm{IndCoh}(\mathrm{LocSys}_{\widehat{G}})\) of ind-coherent sheaves (Arinkin–Gaitsgory) is a presentable DG category, the natural recipient of a "D-module-like" pushforward on derived stacks.
Singular support. On a derived Artin stack \(Y\) with a cotangent \(\mathbb{A}^1\)-deformation (the shifted-symplectic / Sp\(_0\) structure), Arinkin–Gaitsgory define the singular support of an object of \(\mathrm{IndCoh}(Y)\) as a closed conical subset of the classical cotangent \(T^\ast Y^{\mathrm{cl}}\). For \(Y=\mathrm{LocSys}_{\widehat{G}}\), the classical cotangent is the moduli of Higgs bundles, and the nilpotent cone \(\mathrm{N}\subset T^\ast\mathrm{LocSys}^{\mathrm{cl}}\) is the closed sub-stack of nilpotent Higgs fields (the nilpotent cone in the sense of Springer). Define
the full subcategory of ind-coherent sheaves whose singular support is contained in the nilpotent cone. This is the renormalized spectral category.
The derived conjecture¶
Conjecture / Theorem (Arinkin–Gaitsgory; Gaitsgory school). There is an equivalence of presentable DG categories $$ \mathfrak{c}^{\mathrm{dR}}G:\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\;\xrightarrow{\sim}\;\mathrm{IndCoh}(X)\bigr) $$ (a) intertwining the Hecke action on the left (by }}\bigl(\mathrm{LocSys}_{\widehat{G}\(\mathrm{Rep}(\widehat{G})\) via convolution) with the tensor action on the right (by the universal local system), (b) compatible with the de Rham / Betti / étale realization functors, (c) sending the Whittaker sheaf on the automorphic side to the structure sheaf on the spectral side, (d) recovering the classical \(\mathrm{GL}_n\) eigensheaf construction on the oper / Miura locus via Beilinson–Drinfeld (see Hecke Action & Opers).
Proven cases:
| Theorem | Statement |
|---|---|
| Gaitsgory (2013–, announcements / series) | the de Rham unramified equivalence for \(G=\mathrm{GL}_n\) over \(\mathbb{C}\) (and the general-\(G\) case, with the full proof structure announced 2020–2024 by Gaitsgory and collaborators — the published status is in the Gaitsgory–Rozenblyum and follow-up manuscripts) |
| Ben-Zvi–Nadler | the Betti version: \(\mathrm{Shv}(\mathrm{Bun}_G)\simeq\mathrm{IndCoh}_{\mathrm{N}}(\mathrm{LocSys}^{\mathrm{Betti}}_{\widehat{G}})\) with nilpotent singular support in the character stack |
| Drinfeld / L. Lafforgue | the \(\ell\)-adic function-field theorem for \(\mathrm{GL}_n\) over \(\mathbb{F}_q(X)\) — a different statement (cuspidal automorphic sheaves \(\leftrightarrow\) irreducible Galois reps), not the categorical equivalence over \(\mathbb{C}\), but the closest arithmetic sibling |
| V. Lafforgue | the general-\(G\) function-field theorem via shtukas — constructs the automorphic-to-spectral functor for arbitrary \(G\) (not a categorical equivalence, but a "realization" / decomposition of the automorphic category by the action of the excursion algebra) |
The singular support conditions table¶
| Support condition | Spectral subcategory of \(\mathrm{IndCoh}(\mathrm{LocSys}_{\widehat{G}})\) | Corresponding automorphic subcategory | Meaning |
|---|---|---|---|
| Nilpotent \(\mathrm{N}\subset T^\ast\mathrm{LocSys}\) | \(\mathrm{IndCoh}_{\mathrm{N}}(\mathrm{LocSys}_{\widehat{G}})\) | the tempered / whole automorphic category \(\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\) | the full derived conjecture; the Springer resolution / Hecke detects nilpotent monodromy |
| Zero support \(\{0\}\subset\mathrm{N}\) | \(\mathrm{IndCoh}_{\{0\}}=\) the lisse / lcc sheaves \(\mathrm{QCoh}(\mathrm{LocSys})^{\mathrm{lcc}}\) | the cuspidal / rigid automorphic category (no parabolic induction; the cuspidal part) | the "no nilpotent monodromy" condition; the spectral objects are the actual local systems (no derived support) |
| Full \(T^\ast\mathrm{LocSys}\) | \(\mathrm{IndCoh}(\mathrm{LocSys}_{\widehat{G}})\) | (no known automorphic interpretation; the category is too large) | the wrong spectral category — the renormalization is required |
| Cone of \(\lambda\in\mathrm{Nil}_{\widehat{\mathfrak{g}}}\) | the sub-stack cut out by the nilpotent orbit closure of \(\lambda\) | the automorphic subcategory matched to the Arthur / unipotent parameters | the partition of the tempered side by nilpotent orbits |
The key point: the nilpotent singular support condition is the renormalization that makes the spectral category the right size and compatible with the Hecke action. It is not a technical add-on — the plain \(\mathrm{IndCoh}\) without support is too large, and the zero support is too small (gives only the cuspidal part). The nilpotent cone is exactly what the Springer resolution / the geometry of the Hecke action detects, and the Arinkin–Gaitsgory conjecture asserts that the tempered automorphic category (the whole \(\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\) in the unramified setting) matches the nilpotent-support spectral category.
See it¶
The abelian base case \(G=\mathrm{GL}_1\): Fourier–Mukai on \(\mathrm{Pic}(X)\). For \(G=\mathrm{GL}_1\) (and \(\widehat{G}=\mathrm{GL}_1\)), the conjecture reduces to geometric class field theory / the Fourier–Mukai transform on the Picard variety. Concretely:
- \(\mathrm{Bun}_1=\mathrm{Pic}(X)=\bigsqcup_{d\in\mathbb{Z}}\mathrm{Pic}^d(X)\), the Picard variety of degree-\(d\) line bundles (each component an abelian variety of dimension \(g\)).
- \(\mathrm{LocSys}_{\mathrm{GL}_1}(X)=\) the moduli of rank-\(1\) local systems \(=\) the dual torus \(\mathrm{Hom}(\pi_1(X),\mathbb{G}_m)\), an abelian variety dual to \(\mathrm{Pic}^0\) (the character variety / dual torus).
- The singular support condition is vacuous: for abelian \(G\) there is no nilpotent cone (the Lie algebra is abelian, all Higgs fields are central, the nilpotent cone is the zero section), so \(\mathrm{IndCoh}_{\mathrm{N}}=\mathrm{QCoh}\) and the conjecture is the Fourier–Mukai transform $$ \mathfrak{c}^{\mathrm{dR}}1:\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Pic}(X))\xrightarrow{\sim}\mathrm{QCoh}(\mathrm{LocSys}(X)), $$ with kernel the }_1Poincaré bundle \(\mathcal{P}\) on \(\mathrm{Pic}(X)\times\mathrm{LocSys}_{\mathrm{GL}_1}(X)\). The Fourier–Mukai transform \(\Phi_{\mathcal{P}}\) sends a D-module / sheaf on \(\mathrm{Pic}\) to its Fourier transform on the dual torus, exactly as in the classical Fourier–Mukai equivalence between an abelian variety and its dual.
This is the base case the derived conjecture generalizes: for \(G=\mathrm{GL}_n\), the categorical equivalence reduces (in the "abelianized" / generic-spectral-curve locus) to the Fourier–Mukai transform on the Prym / Hitchin fiber, with the nilpotent singular support encoding the non-abelian correction. The abelian case being exactly the classical Fourier–Mukai is the sanity check that the nilpotent support is the right renormalization — for \(\mathrm{GL}_1\) it does nothing (vacuous), and the conjecture is the well-known equivalence.
Check¶
Explain why \(\mathrm{QCoh}(\mathrm{LocSys}_{\widehat{G}})\) is the wrong spectral category, with two distinct reasons (size and Hecke compatibility), and identify the geometric object whose existence forces the singular-support renormalization.
(1) Size: \(\mathrm{QCoh}(\mathrm{LocSys}_{\widehat{G}})\) is generated by the structure sheaf and its tensor powers — equivalently by the universal local system and its descendants — but the automorphic side \(\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\) has holonomic / regular-singularity constraints (it is generated by IC sheaves on Harder–Narasimhan strata, which are finite-type). The natural automorphic-to-spectral functor (via the Hecke action) lands in a bounded subcategory, so \(\mathrm{QCoh}\) is too large to be the target of an equivalence. (2) Hecke compatibility: the Hecke action on the automorphic side is by convolution with Satake sheaves, and the Satake category detects nilpotent monodromy (the Springer resolution / weight filtration — see Geometric Satake). On \(\mathrm{QCoh}\), tensoring with the universal local system has no support condition and does not "see" the nilpotent cone; the two actions are incompatible. The geometric object forcing the renormalization is the Springer resolution \(\tilde{\mathrm{N}}=\widehat{G}\times_{\widehat{B}}\mathfrak{n}\to\mathrm{N}\) (resolution of the nilpotent cone) — the Hecke action factors through the Springer sheaf, which lives on \(\tilde{\mathrm{N}}\), and the natural recipient on the spectral side is the IndCoh supported on \(\mathrm{N}\), the nilpotent cone, where the Springer resolution gives the resolution of singularities. Hence \(\mathrm{IndCoh}_{\mathrm{N}}\) (not \(\mathrm{QCoh}\)) is the category on which the Hecke action is reflected correctly.
Why is the nilpotent singular support condition the tempered condition, and why does the zero-support condition give the cuspidal part of the automorphic category?
The tempered automorphic representations are those appearing in the discrete spectrum with no continuous parameter (the discrete automorphic representations of moderate growth in the arithmetic side); categorically, the tempered / cuspidal part is the part of \(\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\) with no Eisenstein / parabolic-induction contribution. The Springer resolution / Hecke action shows that the nilpotent support is what the Hecke functors detect on the spectral side: a local system with nilpotent monodromy at the singular support corresponds, via the Hecke action, to an automorphic object built from the tempered spectrum. The zero-support condition (lisse / lcc sheaves only) forces the spectral object to be a true local system with no nilpotent structure, which corresponds on the automorphic side to the cuspidal part — the local systems with no nilpotent monodromy are the "rigid" ones, the categorical analog of cuspidal representations with zero Arthur parameter. The full tempered category includes the nilpotent-monomy part, hence the nilpotent support condition. The zero support is a strict subcategory (cuspidal), the nilpotent support is the full tempered locus, and the support-free \(\mathrm{QCoh}\) is too large (no tempered constraint).
Pitfalls¶
- \(\mathrm{QCoh}(\mathrm{LocSys})\) is the wrong spectral category: too large, no Hecke compatibility. The renormalization by singular support is not optional — without it the conjectural equivalence is false (and provably so by a counting / generation argument).
- The nilpotent cone is the renormalization, not a technicality: the condition \(\mathrm{supp}\subset\mathrm{N}\) is the substantive input; it is what makes \(\mathrm{IndCoh}_{\mathrm{N}}\) the right size and Hecke-compatible. Treating it as a regularity condition (like holonomicity in the classical story) misses the point — the nilpotent support is the geometric shadow of the Hecke action.
- Derived stack, not classical: \(\mathrm{LocSys}_{\widehat{G}}\) must be the derived moduli stack (retaining the cotangent / deformation theory). Truncating to the classical moduli loses the obstruction theory that \(\mathrm{IndCoh}\) needs; \(\mathrm{IndCoh}\) on the truncation does not match the automorphic side.
- \(\mathrm{IndCoh}\neq\mathrm{QCoh}\) on singular stacks: for smooth / lci stacks \(\mathrm{IndCoh}=\mathrm{QCoh}\), but \(\mathrm{LocSys}_{\widehat{G}}\) is singular (and derived); \(\mathrm{IndCoh}\) is the correct category (Arinkin–Gaitsgory), the recipient of D-module pushforwards.
- Function-field theorem (V. Lafforgue) is a different statement from the de Rham categorical equivalence: the shtuka-based theorem gives a realization / decomposition of the automorphic category by the excursion algebra action, not an equivalence with \(\mathrm{IndCoh}_{\mathrm{N}}\) over \(\mathbb{C}\). The two are siblings, not the same theorem.
- Proven status depends on precise formulation: the de Rham unramified \(\mathrm{GL}_n\) over \(\mathbb{C}\) is the cleanest proven case; the general-\(G\) derived case has announced proof structure (Gaitsgory school, 2020–2024) whose precise published status should be cited carefully — not "proved in full generality" without reference.
Refs¶
- Arinkin–Gaitsgory, Singular support of coherent sheaves and the geometric Langlands conjecture (Duke 2015, Selecta)
- Gaitsgory, On the de Rham version of the geometric Langlands conjecture (series of preprints / announcements, 2013–)
- Gaitsgory–Rozenblyum, A Study in Derived Algebraic Geometry (Mathematical Surveys and Monographs, AMS 2017)
- Ben-Zvi–Nadler, Betti geometric Langlands (Betti version, singular support in character stack)
- V. Lafforgue, Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale (shtukas / function-field, general \(G\))
Next¶
Geometric Satake · Hecke Action & Opers · Ramified & Frontiers · Geometric Langlands index · Foundations · Derived AG · DAG / ind-coherent — the \(\mathrm{IndCoh}\) on the derived \(\mathrm{LocSys}\), the Gaitsgory–Rozenblyum the framework · Higher categories · Langlands foundations · Hitchin mirror symmetry — the geometric Langlands via mirror symmetry of the Hitchin system