Geometric Langlands: Foundations¶

Job: define $\mathrm{Bun}_G$ and the automorphic sheaf category, describe the spectral side (local systems / Langlands parameters), sketch classical geometric Langlands for $\mathrm{GL}_n$ over $\mathbb{C}$, and contrast de Rham vs Betti settings.

Core¶

Curve and group¶

Let $X$ be a smooth projective geometrically connected curve over $k=\mathbb{C}$ (or $\overline{\mathbb{F}}_q$ for $\ell$-adic variants). Let $G$ be a reductive algebraic group over $k$, $\widehat{G}$ its Langlands dual.

$\mathrm{Bun}_G=\mathrm{Bun}_G(X)$: the moduli stack of principal $G$-bundles (étale or fppf $G$-torsors) on $X$. For $G=\mathrm{GL}_n$, this is the stack of rank-$n$ vector bundles. It is an Artin stack, locally of finite type, with components labeled by topological / cohomological invariants (e.g.\ degree for $\mathrm{GL}_n$).

Object	Meaning
Point of $\mathrm{Bun}_G$	$G$-bundle $\mathcal{P}$ on $X$
Automorphism group	$H^0(X,\mathrm{Ad}\,\mathcal{P})$ — stabilizers make it a stack
Hecke correspondence at $x$	modify $\mathcal{P}$ on the formal disc at $x$

Automorphic side: sheaves on $\mathrm{Bun}_G$¶

de Rham / complex setting. Work with D-modules on $\mathrm{Bun}_G$ (or suitable Ind-coherent / holonomic categories). Notation varies: $\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)$, $D(\mathrm{Bun}_G)$, etc. Hecke functors act by integral transforms along Hecke correspondences $$ \mathrm{Bun}G\ \leftarrow\ \mathrm{Hecke}_G, $$ labeled by }\ \rightarrow\ \mathrm{Bun$x\in X$ and $V\in\mathrm{Rep}(\widehat{G})$ via geometric Satake.

Geometric Satake (one line). The category of $G(\mathcal{O})$-equivariant perverse sheaves on the affine Grassmannian $\mathrm{Gr}_G$ is tensor-equivalent to $\mathrm{Rep}(\widehat{G})$ (Mirković–Vilonen, …). This supplies the dual group on the automorphic side.

Betti / constructible setting. Replace D-modules by (complexes of) constructible sheaves / Betti sheaves on the complex-analytic or topological moduli — natural for monodromy and character varieties.

Spectral side: Langlands parameters¶

Classically, a Langlands parameter (geometric) is a $\widehat{G}$-local system on $X$ (homomorphism $\pi_1(X)\to\widehat{G}$ up to conjugacy, or flat $\widehat{G}$-connection in de Rham).

Moduli. $\mathrm{LocSys}_{\widehat{G}}(X)$: stack of $\widehat{G}$-local systems. In de Rham: moduli of flat connections / Higgs data related by nonabelian Hodge (Simpson) on the closed points. For categorical geometric Langlands, one uses a derived moduli stack and IndCoh (ind-coherent sheaves) with singular support conditions (Arinkin–Gaitsgory), not merely $\mathrm{QCoh}$ of the coarse moduli space.

Side	Geometric object	Sheaf category (typical)
Automorphic	$\mathrm{Bun}_G$	$\mathrm{D}\text{-}\mathrm{mod}$ / Betti sheaves
Spectral	$\mathrm{LocSys}_{\widehat{G}}$ (derived)	$\mathrm{IndCoh}_{\mathrm{N}}(\mathrm{LocSys})$ / QCoh variants

Conjecture (schematic geometric Langlands). An equivalence of categories (with Hecke / spectral action matching) $$ \mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}G)\;\simeq\;\mathrm{IndCoh}(X)\bigr) $$ (or Betti analogue), intertwining Hecke operators with tensoring by the universal local system.}}\bigl(\mathrm{LocSys}_{\widehat{G}

Classical geometric Langlands for $\mathrm{GL}_n$ (sketch)¶

For $G=\mathrm{GL}_n$, $\widehat{G}=\mathrm{GL}_n$. A rank-$n$ irreducible local system $E$ on $X$ should correspond to a Hecke eigensheaf $\mathrm{Aut}_E$ on $\mathrm{Bun}_n$:

Eigenproperty. For each $x$ and exterior power / standard rep, Hecke modification at $x$ acts on $\mathrm{Aut}_E$ by the fiber $E_x$ (up to shifts/determinants conventions).
Existence/uniqueness. Beilinson–Drinfeld construct $\mathrm{Aut}_E$ for $\mathrm{GL}_n$ using the localization of Kac–Moody / chiral Hecke algebras and the geometry of opers (when $E$ comes from an oper); Frenkel–Gaitsgory–Vilonen and others treat irreducible local systems more generally via Whittaker models and ramified variants.
Abelian case $n=1$. $\mathrm{Bun}_1=\mathrm{Pic}(X)$; geometric class field theory identifies characters of $\pi_1$ with rank-$1$ local systems and with characters of $\mathrm{Pic}$ — categorical abelian geometric Langlands.

Hitchin system. Beilinson–Drinfeld quantization of the Hitchin fibration gives a spectral decomposition of D-modules on $\mathrm{Bun}_G$ over the Hitchin base, matching oper / spectral curve data — the “classical limit” of geometric Langlands.

de Rham vs Betti¶

Feature	de Rham	Betti
Coefficients	D-modules / flat connections	constructible / topological sheaves
Spectral points	flat $\widehat{G}$-connections	monodromy reps $\pi_1\to\widehat{G}$
Nonabelian Hodge	harmonic metrics link Higgs, de Rham, Betti on points	categorical lift more subtle
Natural base field	$\mathbb{C}$ (analytic / algebraic D-modules)	$\mathbb{C}$ topological; also $\ell$-adic étale
Physics pointer	Kapustin–Witten $A$-model flavour	$B$-model / Betti dualities

Betti geometric Langlands (Ben-Zvi–Nadler, …) formulates equivalences with nilpotent singular support in the character stack. Étale / $\ell$-adic geometric Langlands over finite fields connects most tightly to functions-on-$\mathrm{Bun}_G(\mathbb{F}_q)$ and classical automorphic forms.

Relation to arithmetic Langlands¶

Arithmetic	Geometric analogue
Automorphic rep of $G(\mathbb{A})$	Hecke eigensheaf on $\mathrm{Bun}_G$
Galois rep / WD parameter	$\widehat{G}$-local system on curve
$L$-functions matching	categorical equivalence + traces of Frobenius
Unramified Hecke eigenvalues	geometric Satake labels

Function-field Langlands (Drinfeld, L. Lafforgue, V. Lafforgue) is the closest arithmetic sibling; geometric Langlands aims at a stronger categorical statement even over $\mathbb{C}$. See Langlands.

See it¶

Geometric Satake $\Rightarrow$ Hecke eigenvalues take values in $\mathrm{Rep}(\widehat{G})$. Locally at $x$, modifying bundles by $G(\mathcal{K})/G(\mathcal{O})$-data, the spherical Hecke category is $\mathrm{Rep}(\widehat{G})$. Globally, an eigensheaf with eigenvalue $E$ means: each local Hecke action factors through the fiber functor $V\mapsto V_E|_x$. For $G=\mathrm{GL}_2$, modifying a rank-$2$ bundle by upper/lower “elementary” modifications changes degree by $\pm 1$ and the eigensheaf property becomes a precise categorification of Hecke eigenform relations $T_p f=a_p f$.

Opers. A $\widehat{G}$-oper is a flat connection with prescribed reduction to a Borel and transversality — a “special” local system. Beilinson–Drinfeld use the locus of opers inside $\mathrm{LocSys}$ to construct a quantization of Hitchin and many eigensheaves; not every local system is an oper, so full GL needs broader methods.

Check¶

Why can $\mathrm{Bun}_G$ not be replaced by its coarse moduli space of semistable bundles?

Hecke correspondences and unstable strata are essential; the stacky stabilizers and Harder–Narasimhan stratification enter the sheaf theory. Eigensheaves are not sheaves on the GIT moduli space alone. D-modules on stacks remember automorphisms (e.g.\ gerbey structure for $\mathrm{GL}_n$).

What goes wrong if one uses $\mathrm{QCoh}(\mathrm{LocSys})$ without singular support conditions?

The spectral category becomes too large / mismatched with the automorphic side; Arinkin–Gaitsgory singular support (nilpotent cone conditions) cuts out the correct IndCoh subcategory for the conjectural equivalence.

Pitfalls¶

Dual group: $G$ on $\mathrm{Bun}$, $\widehat{G}$ on local systems — swapping is the standard first error.
Ramification: tame/wild ramifications need parabolic / level structures on both sides (parahoric $\mathrm{Bun}$, local systems with monodromy).
Derived vs classical stacks: flat connections moduli are derived; truncating loses obstruction theory needed for IndCoh.
“Proved for $\mathrm{GL}_n$” depends on precise formulation (unramified, irreducible, de Rham vs $\ell$-adic); cite the specific theorem.

Refs¶

Frenkel, Langlands Correspondence for Loop Groups; CFT notes
Beilinson–Drinfeld, Opers; Hitchin quantization manuscripts
Arinkin–Gaitsgory, Singular support of coherent sheaves and the geometric Langlands conjecture
Gaitsgory, outline papers on derived geometric Langlands; Ben-Zvi–Francis–Nadler (Betti)

Next¶

Geometric Langlands index · Langlands · Local Langlands · Algebraic geometry · Derived AG · Representation theory · Mathematical physics · Higher categories · Geometric Satake · Hecke Action & Opers · Derived / Categorical GL · Ramified & Frontiers

Object	Meaning
Point of \(\mathrm{Bun}_G\)	\(G\)-bundle \(\mathcal{P}\) on \(X\)
Automorphism group	\(H^0(X,\mathrm{Ad}\,\mathcal{P})\) — stabilizers make it a stack
Hecke correspondence at \(x\)	modify \(\mathcal{P}\) on the formal disc at \(x\)