Geometric Langlands index

Core idea

The geometric Langlands collection develops the categorical correspondence from its curve, bundle, local-system, and Hecke-theoretic dependencies. The correspondence has several formulations and theorem statuses; this index covers the characteristic-zero de Rham formulation and its ramified extensions.

For the ordinary arithmetic, automorphic, endoscopic, and function-field dictionary, see the modern Langlands foundations index.

Curves and moduli stacks

Algebraic curve
Smooth projective curve
Pointed algebraic curve
Algebraic stack
Principal $G$ -bundle on a scheme
$\operatorname{Bun}_G(X)$
Langlands dual group
$G$ -local system
$\operatorname{LocSys}_G(X)$
Global nilpotent cone

Sheaf categories

Quasi-coherent sheaf
Sheaf of differential operators
$D$ -module
Ind-coherent sheaf
Singular support of a coherent sheaf
$\operatorname{IndCoh}_{\mathcal N}$

Hecke structures

Ind-scheme
Loop group
Positive loop group
Dominant coweight
Affine Schubert variety
Hecke modification
Affine Grassmannian
Perverse sheaf
Intersection-cohomology complex
Convolution of sheaves
Geometric Satake equivalence
Hecke correspondence
Geometric Hecke functor
Hecke eigensheaf

Rank one and ramification

Borel subgroup
Flag variety
$\mathbb P^1$ as a rank-one flag variety
Ramification of a $G$ -local system
Riemann–Hilbert correspondence
Regular-singular connection
Irregular-singular connection
Stokes data
Tame ramification
Wild ramification
Level structure on a $G$ -bundle
Parahoric level structure
Iwahori level structure
Ramified geometric Langlands