Geometric Langlands: Ramified & Frontiers¶

Job: survey ramified geometric Langlands (level structures / parahorics, local geometric Langlands at a point, the Fargues–Scholze $p$-adic categorical local Langlands), metaplectic geometric Langlands, the recent proof structure of the Gaitsgory school (Zastava / Whittaker / abelianized GL / geometric Eisenstein), and the quantum categorical Langlands (Feigin–Frenkel / Gaitsgory) — the active research frontiers beyond the unramified de Rham equivalence.

Core¶

Ramified geometric Langlands¶

Replace $\mathrm{Bun}_G$ by $\mathrm{Bun}_{G,\mathcal{P}}$ for a parahoric / level structure at a divisor $D\subset X$ (a finite set of marked points with a parahoric model $\mathcal{P}_x\subset G(\mathcal{K}_x)$ at each $x\in D$, the parahoric subgroup interpolating between $G(\mathcal{O}_x)$ and its Iwahori). The automorphic side becomes $\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_{G,\mathcal{P}})$, and the spectral side is ramified $\widehat{G}$-local systems: local systems on $X\setminus D$ with prescribed monodromy / regular / irregular singularity at $D$. The local geometric Langlands at a point (Frenkel–Gaitsgory–Vilonen) matches tamely ramified $\widehat{G}$-local systems (with a conjugacy class of monodromy at $x$) with Hecke eigensheaves on the moduli of $G$-bundles with parahoric structure at $x$; the wild case (Frenkel–Gaitsgory) uses the affine Kac–Moody / chiral algebra methods (negative-level modules, irregular Wakimoto modules) to handle irregular singularities of arbitrary slope.

Local geometric Langlands at a point (Frenkel–Gaitsgory–Vilonen, tame). The category of (tamely ramified) $\widehat{G}$-local systems at $x$ with monodromy in a fixed conjugacy class $\mathcal{O}\subset\widehat{G}$ is equivalent (via the Hecke eigensheaf construction on $\mathrm{Bun}_{G,\mathcal{P}}$ for the Iwahoric $\mathcal{P}=I$) to a category of representations of the affine Kac–Moody algebra $\widehat{\mathfrak{g}}$ at the critical level with prescribed principal gradation — the categorical local Langlands at $x$.

The Fargues–Scholze $p$-adic categorical local Langlands¶

A separate, currently-active program (Fargues–Scholze, building on Fargues–Fontaine): the categorical local Langlands correspondence for $p$-adic groups, formulated on the Fargues–Fontaine curve $X_{FF}$ (a one-dimensional $p$-adic geometric object whose $\ell$-adic local systems correspond to Weil–Deligne representations of the local field $F$). The conjecture / theorem:

Fargues–Scholze. For a $p$-adic local field $F$ and a reductive group $G/F$, there is a functor from the category of smooth $\ell$-adic representations of $G(F)$ to the category of sheaves on the stack of $L$-parameters $\mathrm{LocSys}_{\widehat{G}}(X_{FF})$ (on the Fargues–Fontaine curve), realizing the local Langlands correspondence categorically: the image of an irreducible smooth representation is supported on its $L$-parameter, and the fibers recover the local Jacquet–Langlands / Vogan packets.

This is a local (not global) categorical LLC, in the $p$-adic / diamond (Huber / Scholze perfectoid / v-sheaf) setting rather than over $\mathbb{C}$ in the D-module setting. It is related to, but distinct from, the de Rham categorical GL over $\mathbb{C}$: the Fargues–Scholze theorem gives a local correspondence, while the Gaitsgory school equivalence (see Derived / Categorical GL) is a global statement over a smooth complex curve. The bridge between them (the $p$-adic / diamond geometric Langlands of Zhu, Fargues–Scholze, Hansen–Johansson) is an active frontier.

Metaplectic geometric Langlands¶

Twist the automorphic side by a metaplectic / theta cover of $\mathrm{Bun}_G$: a $\mathbb{G}_m$-gerbe (or $K^\ast$-gerbe) $\overline{\mathrm{Bun}}_G\to\mathrm{Bun}_G$ arising from the metaplectic cover of $G$ (the Brylinski–Deligne cover). The metaplectic geometric Langlands (Finkelberg–Lysenko, Lysenko) replaces the dual-group / Satake category by a metaplectic geometric Satake:

Metaplectic geometric Satake (Finkelberg–Lysenko). The category of $G(\mathcal{O})$-equivariant perverse sheaves on $\mathrm{Gr}_G$ twisted by the theta cover (a category of sheaves on the metaplectic gerbe over $\mathrm{Gr}_G$) is tensor-equivalent to the small / Weissauer / neutral-component representation category — a modified representation category in which $\mathrm{Rep}(\widehat{G})$ is replaced by representations of a covering group of $\widehat{G}$ (the metaplectic dual), and the tensor structure is twisted by a half-form correction. The automorphic category $\mathrm{D}\text{-}\mathrm{mod}(\overline{\mathrm{Bun}}_G)$ then matches sheaves on the metaplectic spectral side — the moduli of metaplectic dual local systems.

The half-form / theta twist on the spectral side is the geometric shadow of the metaplectic correction to $L$-functions (the theta factor in the metaplectic $L$-function of Weil / Kubota). This is the geometric Langlands shadow of the arithmetic metaplectic correspondence.

The Gaitsgory school recent proof structure¶

The construction of the automorphic-to-spectral functor $\mathfrak{c}^{\mathrm{dR}}_G$ (see Derived / Categorical GL) in the Gaitsgory school (2020–2024 announcements) proceeds via:

Renormalized Zastava / Zastava-space methods: the Zastava spaces (compactified moduli of maps $X\to\widehat{G}/\widehat{B}$) provide a desingularization / factorization model for $\mathrm{LocSys}_{\widehat{G}}$ and a setting in which the Whittaker sheaf (the "universal" Hecke eigensheaf for the generic Whittaker model) can be constructed as a canonical object.
The Whittaker sheaf as the canonical object: the Whittaker sheaf $\mathrm{Whit}_G$ on $\mathrm{Bun}_G$ (the sheaf with Whittaker equivariance) is the automorphic object whose image under $\mathfrak{c}^{\mathrm{dR}}_G$ is the structure sheaf $\mathcal{O}_{\mathrm{LocSys}}$ on the spectral side. The construction of $\mathrm{Whit}_G$ and proof of its Hecke eigenproperty (with eigenvalue the generic oper / Whittaker local system) is the central technical step.
Reduction to abelianized geometric Langlands (Hitchin / abelianization): the full $\mathfrak{c}^{\mathrm{dR}}_G$ is reduced, via the Hitchin fibration and the geometric Eisenstein series (Drinfeld–Gaitsgory, on the cameral / abelianized side), to the abelianized geometric Langlands — the equivalence on the Prym / generic spectral-curve locus, where the problem reduces to the Fourier–Mukai transform (the $\mathrm{GL}_1$ base case, see Derived / Categorical GL).
The Tate trace / categorical trace (Gaitsgory–Rozenblyum): the passage from the abelianized to the full equivalence uses the categorical trace (the trace of the $\mathbb{G}_m$-action on the spectral category, a Tate-style trace) to recover the full $\mathrm{IndCoh}_{\mathrm{N}}$ from the abelianized locus. The geometric Eisenstein series (Drinfeld–Gaitsgory) construct a large subcategory of the automorphic side (the "Eisenstein part") from the moduli of cameral covers (the spectral curves of the Hitchin fibration); the cuspidal part is then handled by Whittaker methods, and the two glue to the full equivalence.

Geometric Eisenstein series (Drinfeld–Gaitsgory)¶

The geometric Eisenstein sheaf $\mathrm{Eis}_P(\mathcal{F})$ for a parabolic $P\subset G$ and a sheaf $\mathcal{F}$ on $\mathrm{Bun}_M$ ($M$ the Levi) is the integral transform along the Eisenstein correspondence $\mathrm{Bun}_M\xleftarrow{p}\mathrm{Bun}_{P}\xrightarrow{q}\mathrm{Bun}_G$ — the categorification of the Eisenstein series $\sum_{\gamma\in P(F)\backslash G(F)}\phi(\gamma g)$. The constant-term functor $\mathrm{CT}_P:\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\to\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_M)$ is the left adjoint. The geometric Eisenstein / constant-term adjunction realizes the parabolic induction categorically; the spectral counterpart is the pushforward along the Hitchin / cameral cover map. The full automorphic category is generated (under Eisenstein series and extensions) from the cuspidal part on the Levi subgroups, exactly as in the arithmetic Langlands spectral decomposition.

Quantum categorical Langlands (Feigin–Frenkel / Gaitsgory)¶

Introduce the quantum parameter $\hbar\in\mathbb{C}^\ast$ deforming the D-module category to twisted D-modules $\mathrm{D}\text{-}\mathrm{mod}^{\hbar}(\mathrm{Bun}_G)$ (the sheaf of differential operators $\mathcal{D}_\hbar$ twisted by $\hbar$; equivalently, the quantum group $U_\hbar(\mathfrak{g})$ at the global level replacing the critical-level Kac–Moody). The quantum geometric Langlands conjecture (Feigin–Frenkel, Gaitsgory):

Quantum GL. For each $\hbar\in\mathbb{C}^\ast$, there is an equivalence $$ \mathfrak{c}^{\mathrm{dR}}{G,\hbar}:\mathrm{D}\text{-}\mathrm{mod}^{\hbar}(\mathrm{Bun}_G)\;\xrightarrow{\sim}\;\mathrm{IndCoh}(X)\bigr) $$ where the spectral side is the }}\bigl(\mathrm{LocSys}_{\widehat{G},1/\hbartwisted stack of $\widehat{G}$-local systems (twisted by $1/\hbar$). The equivalences form a 1-parameter family self-dual under $\hbar\leftrightarrow 1/\hbar$ — the Langlands duality enhanced by quantum group duality.

The $\hbar=1$ case recovers the unramified de Rham categorical GL (see Derived / Categorical GL); the limit $\hbar\to\infty$ is the classical limit (the Hitchin system / geometric Eisenstein, with D-modules replaced by $\mathcal{O}$-modules / Higgs bundles). The quantum GL is the 1-parameter deformation of the categorical equivalence, and the self-duality $\hbar\leftrightarrow 1/\hbar$ is the categorical shadow of the Feigin–Frenkel duality of the affine Kac–Moody algebra at the critical level.

The variants table¶

Variant	Automorphic category	Spectral category	Coefficient / twist
de Rham (unramified)	$\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)$	$\mathrm{IndCoh}_{\mathrm{N}}(\mathrm{LocSys}_{\widehat{G}})$	$\mathbb{C}$, D-modules
Betti (unramified)	$\mathrm{Shv}_{\mathrm{Betti}}(\mathrm{Bun}_G)$	$\mathrm{IndCoh}_{\mathrm{N}}(\mathrm{LocSys}^{\mathrm{Betti}}_{\widehat{G}})$	constructible sheaves
étale / $\ell$-adic (function-field)	cuspidal automorphic sheaves on $\mathrm{Bun}_G(\mathbb{F}_q)$	irreducible $\ell$-adic Galois reps (Drinfeld / L. Lafforgue / V. Lafforgue via shtukas)	$\overline{\mathbb{Q}}_\ell$
ramified (tame)	$\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_{G,\mathcal{P}})$	ramified $\mathrm{LocSys}$ with monodromy at $D$	D-modules with parahoric level
ramified (wild)	$\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_{G,\mathcal{P}})$ + irregular chiral	irregular $\mathrm{LocSys}$ with arbitrary slope	negative-level Kac–Moody / Wakimoto
$p$-adic / Fargues–Scholze (local)	smooth $\ell$-adic reps of $G(F)$	sheaves on $\mathrm{LocSys}_{\widehat{G}}(X_{FF})$ (Fargues curve)	diamonds / v-sheaves; local statement
metaplectic	$\mathrm{D}\text{-}\mathrm{mod}(\overline{\mathrm{Bun}}_G)$ (twisted gerbe)	metaplectic-dual local systems (half-form twist)	theta cover / small Satake
quantum (parameter $\hbar$)	$\mathrm{D}\text{-}\mathrm{mod}^{\hbar}(\mathrm{Bun}_G)$	$\mathrm{IndCoh}_{\mathrm{N}}(\mathrm{LocSys}_{\widehat{G},1/\hbar})$	$\hbar$-twisted D-modules; self-dual $\hbar\leftrightarrow 1/\hbar$

See it¶

A concrete ramified computation: the Fargues curve and the local LLC for $\mathrm{GL}_n(\mathbb{Q}_p)$. The Fargues–Fontaine curve $X_{FF}$ is the adic / $\ell$-adic geometric object obtained as the quotient of a perfectoid space (the adic space associated to a complete algebraically closed extension $C/\mathbb{Q}_p$, with the Frobenius action) by the Frobenius — a one-dimensional "curve over $\mathbb{Q}_p$" whose $\ell$-adic local systems correspond canonically to Weil–Deligne representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$. The categorical local Langlands functor (Fargues–Scholze)

\[ \mathrm{LL}_{Fargues}:\mathrm{Rep}^{\mathrm{sm}}_{\ell}(\mathrm{GL}_n(\mathbb{Q}_p))\to\mathrm{Shv}_{\ell}(\mathrm{LocSys}_{\mathrm{GL}_n}(X_{FF})) \]

sends an irreducible smooth representation $\pi$ of $\mathrm{GL}_n(\mathbb{Q}_p)$ to a sheaf supported on the point of $\mathrm{LocSys}_{\mathrm{GL}_n}(X_{FF})$ corresponding to its Weil–Deligne parameter $\varphi_\pi:W_{\mathbb{Q}_p}\to\mathrm{GL}_n(\mathbb{C})$ (under the classical Harris–Taylor / Henniart LLC). Concretely for $n=2$ and the Steinberg representation $\mathrm{St}_2$: its parameter is the 2-dimensional indecomposable Weil–Deligne representation $(\rho=|\cdot|^{-1/2}\oplus|\cdot|^{1/2},N\ne 0)$ with non-trivial monodromy $N$; the Fargues–Scholze functor sends $\mathrm{St}_2$ to a sheaf supported on the corresponding nilpotent-support point of $\mathrm{LocSys}_{\mathrm{GL}_2}(X_{FF})$, and the local geometric Jacquet–Langlands recovers the sheaf on the local Shimura variety (the Lubin–Tate / Drinfeld tower) whose trace gives $\mathrm{St}_2$ — the geometric / categorical version of the Carayol / Harris–Taylor local-global compatibility. The Fargues curve is the one-point global object on which local LLC becomes a global geometric statement, the precise sense in which the Fargues–Scholze theorem is "local geometric Langlands."

Check¶

Why is the Fargues–Scholze $p$-adic categorical LLC a local statement rather than a global one, and which geometric object makes it local?

The classical global geometric Langlands (see Derived / Categorical GL) is over a smooth projective curve $X$ over $\mathbb{C}$ (or $\overline{\mathbb{F}}_q$) and is an equivalence $\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G(X))\simeq\mathrm{IndCoh}_{\mathrm{N}}(\mathrm{LocSys}_{\widehat{G}}(X))$. The Fargues–Scholze theorem is over the Fargues–Fontaine curve $X_{FF}$, which is a one-dimensional adic / $\ell$-adic object whose $\ell$-adic local systems are by construction in bijection with Weil–Deligne representations of the local field $F=\mathbb{Q}_p$ (or a finite extension). The functor $\mathrm{LL}_{Fargues}$ has source the category of smooth representations of $G(F)$ (a local object — representations of a $p$-adic Lie group), and the target is sheaves on $\mathrm{LocSys}_{\widehat{G}}(X_{FF})$ (whose points are the $L$-parameters of $F$). Hence the theorem matches local data (local reps) with local parameters (Weil–Deligne data) on a single geometric curve $X_{FF}$ — it is a local categorical LLC, not a global statement over a curve in the sense of the Gaitsgory school. Confusing the two — treating Fargues–Scholze as the "$p$-adic global GL" — is wrong; the global $p$-adic / diamond GL (over a higher-dimensional diamond or a smooth $p$-adic curve) is a separate, partly-conjectural program.

Explain why the quantum geometric Langlands equivalence is a deformation of the unramified de Rham one and not a different equivalence, and identify the parameter and the self-duality.

The quantum parameter $\hbar\in\mathbb{C}^\ast$ deforms the sheaf of differential operators $\mathcal{D}\rightsquigarrow\mathcal{D}_\hbar$ (the algebra of differential operators with a $\hbar$-rescaling of the symbol / principal symbol), and equivalently deforms the critical-level Kac–Moody chiral algebra $V^{\mathrm{crit}}\rightsquigarrow V_\hbar$. At $\hbar=1$ one recovers the unramified de Rham D-module category; at $\hbar\to 0$ the D-module category degenerates to $\mathcal{O}$-modules / Higgs (the classical limit, the Hitchin system); at generic $\hbar$ the category $\mathrm{D}\text{-}\mathrm{mod}^\hbar$ is a non-trivial deformation. The conjectured equivalence $\mathfrak{c}_{G,\hbar}$ for each $\hbar$ is a family of equivalences interpolating these limits, not a separate statement for each $\hbar$ — the unramified de Rham case is the $\hbar=1$ point. The self-duality $\hbar\leftrightarrow 1/\hbar$ comes from the Feigin–Frenkel duality of the affine Kac–Moody algebra at the critical level (the chiral algebra $V^{\mathrm{crit}}_G$ is dual to $V^{\mathrm{crit}}_{\widehat{G}}$ under the quantum group $U_q(\widehat{\mathfrak{g}})$ with $q=\exp(2\pi i\hbar)$), and the Langlands duality $G\leftrightarrow\widehat{G}$ is enhanced by the quantum duality $\hbar\leftrightarrow 1/\hbar$. Hence the quantum GL is the 1-parameter family, with $\hbar=1$ as the de Rham point, and the deformation direction is the quantum group parameter.

Pitfalls¶

Ramified GL requires specifying level / parahoric data: the unramified statement is a restriction of the ramified one, but the ramified setting is not obtained by a simple "restrict the unramified equivalence to the locus of local systems unramified off $D$" — one must replace $\mathrm{Bun}_G$ by $\mathrm{Bun}_{G,\mathcal{P}}$ and the spectral side by ramified local systems with prescribed monodromy / regular / irregular singularity, and the equivalence is a separate theorem (with a different proof using ramified Satake and chiral algebras at negative / critical level).
The Fargues–Scholze theorem is local, not global: it is the categorical LLC for $G(F)$ on the Fargues curve, not a global GL over a higher-dimensional $p$-adic / diamond curve. Treating it as the "$p$-adic global GL" is wrong; the global diamond GL is a separate (partly-conjectural) program.
The recent proof announcements' published status: the de Rham unramified $\mathrm{GL}_n$ over $\mathbb{C}$ has the cleanest published statement (Gaitsgory and collaborators, 2013–, with the full proof structure announced 2020–2024); the general-$G$ derived case is partly in announced / preprint form. Cite the specific theorem (e.g., "Gaitsgory, On the de Rham..., series") rather than "proved in full generality" without reference.
The $\ell$-adic function-field theorem (V. Lafforgue) is a different statement from the de Rham categorical equivalence over $\mathbb{C}$: the shtuka-based theorem gives a realization / decomposition of the automorphic category by the excursion algebra action, not an equivalence with $\mathrm{IndCoh}_{\mathrm{N}}$; the two are siblings, not the same theorem, and one should not conflate "function-field GL" (arithmetic) with "de Rham categorical GL over $\mathbb{C}$" (geometric).
Quantum GL is a deformation, not a different equivalence: each $\hbar$ gives an equivalence, with $\hbar=1$ the de Rham point; the family is the quantum GL, not a separate conjecture. The self-duality $\hbar\leftrightarrow 1/\hbar$ is the enhancement by the quantum group duality (Feigin–Frenkel).
Metaplectic $\neq$ ordinary GL with a twist: the metaplectic Satake replaces $\mathrm{Rep}(\widehat{G})$ by the small / Weissauer category (a covering-group representation category, with the half-form / theta correction); the spectral side is the metaplectic dual local systems, not ordinary $\widehat{G}$-local systems. The theta cover is part of the data, not a coefficient twist on the same category.

Refs¶

Frenkel–Gaitsgory–Vilonen, Whittaker patterns in the geometric Langlands correspondence (ramified / Whittaker)
Frenkel–Gaitsgory, Local geometric Langlands correspondence (ramified, tame / wild)
Fargues–Scholze, Geometrization of the local Langlands correspondence (arXiv 2018 / preprint; the categorical LLC)
Finkelberg–Lysenko, Metaplectic geometric Satake; Lysenko, Metaplectic geometric Langlands
Feigin–Frenkel, Affine Kac–Moody algebras and geometric Langlands duality (quantum GL / Feigin–Frenkel duality)
Drinfeld–Gaitsgory, Compactifications of moduli of $G$-bundles and geometric Eisenstein series (geometric Eisenstein, abelianization)

Next¶

Geometric Satake · Hecke Action & Opers · Derived / Categorical GL · Geometric Langlands index · Foundations · Local Langlands · Langlands foundations · Derived AG

Variant	Automorphic category	Spectral category	Coefficient / twist
de Rham (unramified)	\(\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\)	\(\mathrm{IndCoh}_{\mathrm{N}}(\mathrm{LocSys}_{\widehat{G}})\)	\(\mathbb{C}\), D-modules
Betti (unramified)	\(\mathrm{Shv}_{\mathrm{Betti}}(\mathrm{Bun}_G)\)	\(\mathrm{IndCoh}_{\mathrm{N}}(\mathrm{LocSys}^{\mathrm{Betti}}_{\widehat{G}})\)	constructible sheaves
étale / \(\ell\)-adic (function-field)	cuspidal automorphic sheaves on \(\mathrm{Bun}_G(\mathbb{F}_q)\)	irreducible \(\ell\)-adic Galois reps (Drinfeld / L. Lafforgue / V. Lafforgue via shtukas)	\(\overline{\mathbb{Q}}_\ell\)
ramified (tame)	\(\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_{G,\mathcal{P}})\)	ramified \(\mathrm{LocSys}\) with monodromy at \(D\)	D-modules with parahoric level
ramified (wild)	\(\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_{G,\mathcal{P}})\) + irregular chiral	irregular \(\mathrm{LocSys}\) with arbitrary slope	negative-level Kac–Moody / Wakimoto
\(p\)-adic / Fargues–Scholze (local)	smooth \(\ell\)-adic reps of \(G(F)\)	sheaves on \(\mathrm{LocSys}_{\widehat{G}}(X_{FF})\) (Fargues curve)	diamonds / v-sheaves; local statement
metaplectic	\(\mathrm{D}\text{-}\mathrm{mod}(\overline{\mathrm{Bun}}_G)\) (twisted gerbe)	metaplectic-dual local systems (half-form twist)	theta cover / small Satake
quantum (parameter \(\hbar\))	\(\mathrm{D}\text{-}\mathrm{mod}^{\hbar}(\mathrm{Bun}_G)\)	\(\mathrm{IndCoh}_{\mathrm{N}}(\mathrm{LocSys}_{\widehat{G},1/\hbar})\)	\(\hbar\)-twisted D-modules; self-dual \(\hbar\leftrightarrow 1/\hbar\)