Geometric Langlands¶
Job: reformulate Langlands reciprocity as an equivalence between categories of sheaves (D-modules / constructible sheaves) on \(\mathrm{Bun}_G\) and spectral data on the moduli of Langlands parameters — a geometric, categorical upgrade of the function-field Langlands program.
Core¶
For a smooth projective curve \(X\) over \(\mathbb{C}\) (or a finite field, with \(\ell\)-adic coefficients), and a reductive group \(G\), geometric Langlands proposes $$ \mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}G)\ \longleftrightarrow\ \text{``spectral category'' on }\mathrm{LocSys}(X) $$ (or a carefully singular/derived enhancement thereof). Automorphic }\(\leftrightarrow\) Galois becomes sheaves on moduli of \(G\)-bundles \(\leftrightarrow\) sheaves/local systems for the dual group \(\widehat{G}\).
| Layer | Content |
|---|---|
| Automorphic space | \(\mathrm{Bun}_G(X)\): moduli stack of \(G\)-torsors on \(X\) |
| Sheaf theories | D-modules (de Rham); Betti constructible / \(\ell\)-adic sheaves |
| Spectral side | local systems / \(\widehat{G}\)-local systems; Langlands parameters |
| Classical GL | Beilinson–Drinfeld for \(\mathrm{GL}_n\); Hecke eigensheaves |
| Variants | de Rham vs Betti vs étale; quantum / derived / categorical |
Reading map¶
| Page | Focus |
|---|---|
| Foundations | \(\mathrm{Bun}_G\); D-modules; spectral side; \(\mathrm{GL}_n\) sketch; de Rham vs Betti |
| Geometric Satake | affine Grassmannian; Satake category; \(\mathrm{Perv}_{G(\mathcal{O})}(\mathrm{Gr}_G)\simeq\mathrm{Rep}(\widehat{G})\); factorization |
| Hecke Action & Opers | Hecke functors; eigensheaves; opers; Beilinson–Drinfeld construction |
| Derived / Categorical GL | Arinkin–Gaitsgory conjecture; \(\mathrm{IndCoh}_{\mathrm{N}}(\mathrm{LocSys})\); singular support |
| Ramified & Frontiers | ramified / metaplectic GL; Fargues–Scholze \(p\)-adic; quantum GL; recent proof structure |
Dependency order¶
- Algebraic geometry (stacks, moduli of bundles).
- Langlands (classical dictionary; dual group).
- Homological algebra + derived AG (D-modules, ind-coherent sheaves).
- Geometric Langlands package; higher categorical enhancements (higher categories).
Interfaces¶
| Area | Contact |
|---|---|
| Langlands | function-field / categorical shadow of arithmetic LLC |
| Algebraic geometry | \(\mathrm{Bun}_G\), stacks, Hecke correspondences |
| Representation theory | dual group; affine Grassmannian; geometric Satake |
| Derived AG | derived \(\mathrm{Bun}_G\), spectral stacks |
| Mathematical physics | Kapustin–Witten TQFT interpretation |
| Motivic homotopy | parallel “geometry of schemes” but different goals |
Notation (paper-standard)¶
- \(\mathrm{Bun}_G=\mathrm{Bun}_G(X)\); \(\mathrm{LocSys}_H(X)\); \({}^LG\), \(\widehat{G}\)
- \(\mathrm{D}(\mathrm{Bun}_G)\), \(\mathrm{IndCoh}\), \(\mathrm{Shv}_{\mathrm{Betti}}\), \(D^b_c\)
- Hecke functors \(H_V\); eigensheaf \(\mathrm{Aut}_E\) with eigenvalue \(E\in\mathrm{LocSys}_{\widehat{G}}\)
- Geometric Satake: \(\mathrm{Perv}_{G(\mathcal{O})}(\mathrm{Gr}_G)\simeq\mathrm{Rep}(\widehat{G})\)
See it¶
From functions to sheaves. Over \(\mathbb{F}_q\), classical automorphic forms give functions on \(\mathrm{Bun}_G(\mathbb{F}_q)\). Sheaf–function dictionary (Grothendieck): functions arise as traces of Frobenius on stalks of \(\ell\)-adic sheaves. Geometric Langlands lifts the matching of functions to a matching of categories of sheaves, with Hecke operators categorified by correspondences on \(\mathrm{Bun}_G\times\mathrm{Bun}_G\). Arithmetic Langlands is still largely about numbers; geometric Langlands is about categorical equivalences testable over \(\mathbb{C}\).
Check¶
What is a Hecke eigensheaf?
An object \(\mathcal{F}\in\mathrm{D}(\mathrm{Bun}_G)\) together with isomorphisms \(H_V(\mathcal{F})\simeq V_E\otimes\mathcal{F}\) (compatibly in \(V\in\mathrm{Rep}(\widehat{G})\)), where \(V_E\) is the local system associated to a fixed \(E\in\mathrm{LocSys}_{\widehat{G}}\) via the fiber functor — categorified eigenform with “eigenvalue” \(E\).
Why does geometric Satake involve the affine Grassmannian rather than \(\mathrm{Bun}_G\)?
Local Hecke operators at \(x\in X\) factor through the affine Grassmannian \(\mathrm{Gr}_G\) (moduli of \(G\)-torsors on a disc trivialized on the punctured disc). Geometric Satake identifies spherical perverse sheaves on \(\mathrm{Gr}_G\) with \(\mathrm{Rep}(\widehat{G})\), supplying Hecke eigenvalue labels.
Pitfalls¶
- Stacky nature: \(\mathrm{Bun}_G\) is an Artin stack, not a scheme; sheaf theory needs careful coefficients and singularities.
- Derived enhancements: naive \(D^b(\mathrm{LocSys})\) is wrong; spectral side uses IndCoh on a derived stack of local systems (Arinkin–Gaitsgory, …).
- de Rham \(\neq\) Betti: different coefficients and monodromy vs connection data; equivalence packages differ (Ben-Zvi–Nadler, …).
- Proven status: substantial theorems for \(\mathrm{GL}_n\) and many cases; full categorical equivalence in full generality is a research program (Gaitsgory school, recent announcements).
Refs¶
- Frenkel, Lectures on the Langlands Program and Conformal Field Theory
- Gaitsgory–Rozenblyum, A Study in Derived Algebraic Geometry (spectral side tools)
- Beilinson–Drinfeld, Quantization of Hitchin’s integrable system (preprints)
- Arinkin–Gaitsgory, Singular support of coherent sheaves (spectral GL)
Next¶
Foundations · Geometric Satake · Hecke Action & Opers · Derived / Categorical GL · Ramified & Frontiers · Langlands · Langlands foundations · Algebraic geometry · Derived AG · Representation theory · Mathematical physics