Stack of Langlands Parameters

• The Stack of Langlands Parameters is a moduli stack that parametrizes local Langlands parameters for reductive groups via algebro-geometric and ∞-categorical constructions.
• It features derived and Artin stack structures with explicit quotient presentations and mapping stacks that encode Galois-theoretic invariants and Hecke actions.
• Its rich geometric and categorical properties support local Langlands correspondences, block decompositions, and spectral equivalences across p‑adic, geometric, and quantum settings.

A stack of Langlands parameters is an algebro-geometric or ∞-categorical moduli stack that parametrizes local Langlands (L-)parameters with values in the L-group (or dual group) of a connected reductive group over a local or global field. Such stacks provide the “spectral” or Galois-theoretic side of the categorical and (geometric) Langlands correspondence, and serve as recipients for spectral actions, Hecke operators, and excursion algebras across $p$-adic, geometric, and quantum settings. These stacks admit a wealth of structure: they are typically Artin or derived Artin stacks, often equipped with monoidal actions, torus stratifications, and compatible functoriality with parabolic and inner twist operations. Their geometry encodes deep representation-theoretic invariants such as Bernstein center, depth, or block decomposition.

1. Algebraic and Functorial Constructions

The stack of local Langlands parameters for a reductive group $G$ over a local field $F$ (with Langlands dual $\hat{G}$) is constructed as the quotient stack

$\Par_G = [Z^1(W_F,\hat{G}) / \hat{G}]$

where $Z^1(W_F,\hat{G})$ is the affine scheme (often ind-affine) of (continuous) 1-cocycles from the Weil group $W_F$ to $\hat{G}$, and the group acts by conjugation. For families parametrized by a base ring $A$, points classify homomorphisms $W_F \to \hat{G}(A)$, or cocycles up to $G$0-conjugation (Joshi, 28 Jun 2026, Dat et al., 2024, Fargues et al., 2021, Zhu, 2020).

Presentations vary with context: over $G$1 or $G$2 for $G$3 (Fargues–Scholze, Dat–Helm–Kurinczuk–Moss), or derived enhancements via mapping stacks or formal models (Min's Emerton–Gee stack (Min, 2024)). In the equal-characteristic setting, a closely related construction is the moduli stack of $G$4-shtukas, yielding L-parameters via the cohomology of stacks of restricted shtukas (Genestier et al., 2017).

The functor-of-points for $G$5 is: $G$6 where $G$7 ranges over commutative rings or more general animated algebras.

For tori, $G$8 is a strictly commutative Picard stack, and its points classify $G$9 (Fu, 9 Nov 2025).

2. Geometric, Stacky, and Derived Structures

The stack $F$0 and its variants (e.g., $F$1, the Fargues–Scholze stack) are typically ind-Artin or Artin stacks locally of finite type, smooth outside roots of unity for the center, and decomposed into connected components by semisimple conjugacy data or inertia classes (Shotton, 2023, Fargues et al., 2021, Fu, 29 Mar 2025, Min, 2024).

Derived or animated enhancements arise in two principal frameworks:

• Mapping stacks or derived mapping stacks $F$2, enriching tangent complexes and obstruction theory: cotangent complexes are calculated via continuous group cohomology $F$3 for a parameter $F$4 (Zhu, 2020).
• The Emerton–Gee stack and its derived/prismatic version yield a stack of $F$5-modules or crystalline F-crystals with $F$6-structure, and in the reductive case the derived stack is shown to be classical on truncated animated rings (Min, 2024).

For $F$7-adic Hodge-theoretic settings, v-stacks (Scholze–Weinstein) such as the stack parametrizing mixed-parity L-parameters as the derived mapping stack

$F$8

implement half-integral Hodge–Tate weights and admit a full 6-functor solid sheaf formalism (Tong, 2023).

3. Local and Global Geometric Features

Structural Properties and Stratification

• $F$9 is smooth and of dimension $\hat{G}$0 on the regular semisimple locus.
• Its connected components are indexed by Frobenius-semisimple conjugacy classes or inertial support (block) data; for tori, components correspond to $\hat{G}$1-torsors (Shotton, 2023, Fu, 9 Nov 2025).
• There exist natural maps:
  • $\hat{G}$2 for Levi subgroups $\hat{G}$3, smooth on regular/avoidant loci (Shotton, 2023).
  • $\hat{G}$4 as the inclusion of the "spherical locus" (parameters trivial on inertia), important for the theory of spherical Hecke algebras (Hove, 2024).

Quotient Descriptions

Locally, $\hat{G}$5 admits étale presentations as quotient stacks $\hat{G}$6 where $\hat{G}$7 is affine of finite type (framed moduli or cocycle schemes), and $\hat{G}$8 acts algebraically. This structure is exploited for the computation of coordinate rings, derived categories, and categorical correspondences (Fargues et al., 2021, Fu, 29 Mar 2025).

4. Categorification and Spectral Actions

IndCoh, Perf, and QCoh

The category $\hat{G}$9 (or $\Par_G = [Z^1(W_F,\hat{G}) / \hat{G}]$0, $\Par_G = [Z^1(W_F,\hat{G}) / \hat{G}]$1 depending on context) encodes "spectral" sheaves. These act naturally on categories of sheaves on the stack $\Par_G = [Z^1(W_F,\hat{G}) / \hat{G}]$2 of $\Par_G = [Z^1(W_F,\hat{G}) / \hat{G}]$3-bundles (Fargues–Fontaine, geometric Langlands), organizing the spectral-action formalism:

$\Par_G = [Z^1(W_F,\hat{G}) / \hat{G}]$4

and one has

$\Par_G = [Z^1(W_F,\hat{G}) / \hat{G}]$5

as a monoidal action (via Hecke functors and excursion operators) (Fargues et al., 2021).

Spectral/Bernstein Center

The ring of global functions $\Par_G = [Z^1(W_F,\hat{G}) / \hat{G}]$6 is identified with the "spectral" Bernstein center. The spectral-to-classical Bernstein map $\Par_G = [Z^1(W_F,\hat{G}) / \hat{G}]$7 interpolates the center of the category of representations and is central in the theory of "local Langlands in families" (Dat et al., 2024, Fargues et al., 2021).

Block and Type Decompositions

Irreducible components and block decompositions of $\Par_G = [Z^1(W_F,\hat{G}) / \hat{G}]$8 correspond to depth and supercuspidal supports. For $\Par_G = [Z^1(W_F,\hat{G}) / \hat{G}]$9, explicit spectral block classification and matching with classical depth-zero blocks are verified (Fu, 29 Mar 2025).

Special and Restriction Stacks

The "stack of spherical parameters" $Z^1(W_F,\hat{G})$0 parametrizes those L-parameters that are inertia trivial (spherical representations), and one has closed immersions $Z^1(W_F,\hat{G})$1 aligning with the theory of unramified representations and Hecke congruences (Hove, 2024).

5. Variants and Enhancements

Rigid and Metaplectic Parameters

Rigid enhancements, involving additional gerbe or torsor data (Kaletha's gerbe), lead to stacks of enhanced L-parameters, often as fiber products

$Z^1(W_F,\hat{G})$2

in the Artin stack sense, with orbits classified by twisted extended quotients using generalized Springer theory (Dillery et al., 2023).

For metaplectic or quantum settings, parameter stacks are built from factorization gerbes, often tied to twisted versions of the affine Grassmannian, yielding metaplectic dual groups and supporting the metaplectic geometric Satake equivalence (Gaitsgory et al., 2016, Zhao, 2017).

6. Applications and Interactions

Local Langlands Correspondence (LLC)

Stacks of Langlands parameters furnish the spectral side of the LLC, providing the natural domain for parametrization functors from categories of smooth representations of $Z^1(W_F,\hat{G})$3 (or lisse $Z^1(W_F,\hat{G})$4-adic sheaves) to coherent sheaves over $Z^1(W_F,\hat{G})$5, via "spectral actions" (Fargues–Scholze, Genestier–Lafforgue) (Fargues et al., 2021, Zhu, 2020, Genestier et al., 2017). For tori, the LLC is concretely categorified by the Fourier–Mukai equivalence on Picard stacks (Fu, 9 Nov 2025).

Shtukas, Hecke Operators, and Excursion Algebras

For function fields, the cohomology of stacks of shtukas realizes parameters in stack-theoretic excursion algebras— sheafified incarnations of the Bernstein center over $Z^1(W_F,\hat{G})$6, governing the spectrum of Hecke operators (Genestier et al., 2017).

Spectral–Automorphic Equivalences

The conjectural and in some cases proven equivalences between categories of sheaves on $Z^1(W_F,\hat{G})$7 and coherent (or ind-coherent) sheaves on $Z^1(W_F,\hat{G})$8 form the bedrock of categorical and geometric Langlands programs in both classical and quantum settings (Fargues et al., 2021, Zhu, 2020, Zhao, 2017, Fu, 9 Nov 2025).

Anabelomorphy

Recent results show that the stack $Z^1(W_F,\hat{G})$9 is "amphoric": its isomorphism class depends only on the abstract profinite group $W_F$0, not the particular $W_F$1-adic field $W_F$2. If $W_F$3 are anabelomorphic, there exists an equivalence $W_F$4 as stacks, and the whole categorical Langlands correspondence is conjecturally amphoric in this sense (Joshi, 28 Jun 2026).

7. Special Cases and Further Generalizations

Tori

For $W_F$5, a torus, the stack of L-parameters is a strictly commutative Picard stack, and the LLC is recast as a categorical and geometric Fourier–Mukai equivalence between sheaves on $W_F$6 and coherent sheaves on the stack $W_F$7, including precise Pontryagin duality at the Picard stack level (Fu, 9 Nov 2025).

Real Groups

For real forms $W_F$8, the classical ABV parameter space is shown to arise as the stack of homotopy fixed points $W_F$9 under the Galois involution, furnishing a stack-theoretic version of the classical description (Virk, 2021).

Quantum and Mixed-Parity Parameters

Quantum parameter stacks interpolate between twisted $\hat{G}$0-modules and quasi-coherent sheaves on $\hat{G}$1, parametrized by Artin stacks $\hat{G}$2 of G-invariant Lagrangian subspaces with auxiliary extension data. Similarly, mixed-parity ($\hat{G}$3-Hodge–Tate weights) L-parameter stacks in the $\hat{G}$4-adic world are described as v-stacks of maps from the two-fold cover of the Weil group to $\hat{G}$5 (Zhao, 2017, Tong, 2023).

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Central Constructions and Features (Summary Table)

Stack/Formalism	Base Category	Points Parametrize	Reference Papers
$\hat{G}$6	Artin/Ind-Artin stacks	L-parameters $\hat{G}$7 up to conj.	(Joshi, 28 Jun 2026, Fargues et al., 2021, Zhu, 2020)
$\hat{G}$8	Picard stack	Toral L-parameters $\hat{G}$9	(Fu, 9 Nov 2025)
$A$0	Flat lci ind-schemes	Discrete cocycles, often mod $A$1, integral	(Dat et al., 2024, Shotton, 2023)
Derived/animated Emerton–Gee stack	Formal algebraic/derived stacks	$A$2-modules, crystalline F-crystals	(Min, 2024)
Mapping stack $A$3	Small $A$4-stack, derived	Mixed-parity L-parameters ($A$5-weights)	(Tong, 2023)

• (Gaitsgory et al., 2016): Parameters and duality for the metaplectic geometric Langlands theory
• (Zhao, 2017): Quantum parameters of the geometric Langlands theory
• (Genestier et al., 2017): Chtoucas restreints pour les groupes réductifs et paramétrisation de Langlands locale
• (Zhu, 2020): Coherent sheaves on the stack of Langlands parameters
• (Ben-Zvi et al., 2020): Coherent Springer theory and the categorical Deligne-Langlands correspondence
• (Fargues et al., 2021): Geometrization of the local Langlands correspondence
• (Virk, 2021): Fixed points and Langlands parameter spaces for real groups
• (Shotton, 2023): Irreducible components of the moduli space of Langlands parameters
• (Dillery et al., 2023): A stacky generalized Springer correspondence and rigid enhancements of L-parameters
• (Tong, 2023): $A$6-Categorical Generalized Langlands Program I: Mixed-Parity Modules and Sheaves
• (Dat et al., 2024): Local Langlands in families: The banal case
• (Hove, 2024): The stack of spherical Langlands parameters
• (Min, 2024): Classicality of derived Emerton--Gee stack II: generalised reductive groups
• (Fu, 29 Mar 2025): On the categorical local Langlands conjectures for depth-zero regular supercuspidal representations
• (Fu, 9 Nov 2025): Two Categorifications of the Local Langlands Correspondence for Tori
• (Joshi, 28 Jun 2026): The Categorical Local Langlands Correspondence and Anabelomorphy