Geometric Langlands replaces arithmetic eigenfunctions by sheaf-theoretic eigenobjects and individual representations by categories. Its central architecture relates local systems for the Langlands dual group on a curve to D-modules or sheaf categories on the moduli stack of bundles for the original group.
This chapter continues the BTT routes through Geometric Satake, Automorphic Representations, Microlocal Sheaves, and Affine Lie Algebras.
The two moduli problems
Let X be a smooth projective curve and G a reductive group. The automorphic side uses Bun_G(X), the moduli stack of principal G-bundles. The spectral side uses LocSys_{G^∨}(X), the moduli of local systems for the Langlands dual group G^∨.
The dual group exchanges roots and coroots. Geometric Satake explains its appearance: suitable perverse sheaves on the affine Grassmannian of G form a tensor category equivalent to representations of G^∨.
Hecke modifications
A Hecke modification changes a G-bundle at one point x∈X while leaving it unchanged away from x. The resulting Hecke correspondence defines pull-push functors on sheaves or D-modules on Bun_G.
Representations V of G^∨ label Hecke functors through geometric Satake. If E is a G^∨-local system and E_V its associated local system, a Hecke eigensheaf F satisfies schematically H_V(F)≈F⊠E_V, compatibly for all V.
The GL1 model
For G=GL_1, Bun_G is the Picard stack and G^∨=GL_1. Rank-one local systems determine eigensheaves through an Abel–Jacobi/Fourier-type construction. This abelian case is the geometric analogue of class field theory and provides the cleanest test of the general architecture.
Why stacks and derived geometry matter
Bundles have automorphisms, so coarse moduli spaces discard essential stabilizer data. Bun_G must therefore be treated as a stack. The spectral moduli LocSys_{G^∨} is often singular, so derived geometry records deformation and obstruction spaces that an ordinary variety would lose.
Modern formulations compare D-modules on Bun_G with coherent or ind-coherent sheaves on derived LocSys_{G^∨}, often subject to nilpotent singular-support conditions. This is where the microlocal branch becomes structurally necessary.
Opers, critical level and localization
At critical level, affine Kac–Moody algebras acquire a large center related to opers for G^∨. Localization turns suitable affine Lie-algebra modules into D-modules on Bun_G, providing important constructions of Hecke eigensheaves.
This is one of the strongest bridges between infinite-dimensional representation theory and geometric Langlands: spectral data in the critical-level center becomes automorphic geometry after localization.
Ramification and quantum deformation
Marked points allow parabolic, Iwahori or deeper level structures and corresponding monodromy conditions on local systems. Quantum geometric Langlands introduces a deformation parameter and relates twisted D-module categories for G and G^∨ with dual coupling parameters.
This deformation route leads naturally into Yangians, quantum affine algebras and other quantum representation systems.
What must be stated carefully
- Geometric Langlands is categorical, not a simple bijection of points.
- Hecke eigenvalues are local systems, not scalars.
- G and G^∨ must not be confused.
- Derived enhancements matter on singular spectral moduli.
- Modern formulations can require nilpotent singular support.
- Local, global, ramified and quantum versions have different hypotheses.
- The strongest universal formulations should be versioned carefully rather than presented as one elementary theorem.
Practice and verification
1. Define Bun_G and LocSys_{G^∨}. 2. What does Langlands duality exchange? 3. What is a Hecke modification? 4. State the schematic eigensheaf equation. 5. Why are stacks needed? 6. What role does geometric Satake play? 7. Why does singular support enter modern formulations? 8. How do opers connect affine Lie algebras to eigensheaves?
Answers. Bun_G parametrizes G-bundles; LocSys_{G^∨} parametrizes dual-group local systems. Roots and coroots are exchanged. A Hecke modification changes a bundle at one point. The eigencondition is H_V(F)≈F⊠E_V. Stacks retain automorphisms. Geometric Satake reconstructs G^∨ and labels Hecke operations. Singular support controls allowed derived cotangent directions. Critical-level centers described by opers feed localization constructions.
Representation Mathematics — Batch 13
Continue through Yangians and Quantum Affine Algebras, Macdonald Polynomials and DAHA, and Koszul Duality. Return to the BTT Mathematics Learning Hub.