Local Geometric Langlands

• Local Geometric Langlands is a web of categorical equivalences that geometrizes the local Langlands correspondence using loop groups, moduli of local systems, and derived categories.
• The framework employs explicit constructions such as affine Grassmannians, Iwahori-level sheaves, and Hecke actions to connect automorphic and spectral sides.
• Recent developments highlight depth-slope relations, functorial theta correspondences, and motivic generalizations that broaden its impact in modern representation theory.

Local Geometric Langlands refers to a web of conjectures, theorems, and categorical constructions that seek to geometrize the local Langlands correspondence for reductive groups over local fields, particularly focusing on local objects such as loop groups, moduli of local systems, and categories of sheaves or representations. Unlike the classical Langlands correspondence, which relates Galois representations and automorphic representations, the geometric version is formulated in terms of equivalences between categories, often enriched with derived, ∞-categorical, or motivic structures, and manifests as a "spectral decomposition" of representation categories according to parameters living over stacks of local systems with possibly ramified or irregular singularities.

1. Foundational Structures: Loop Groups, Local Systems, and Categories

At the heart of the local geometric Langlands program are three classes of objects:

• Loop Groups and Affine Flag Varieties: For a connected reductive group $G$ over a local field $k((t))$, the loop group $LG=G(k((t)))$ and its Iwahori subgroup $I \subset L^+G=G(k[[t]])$ underlie the paper of affine Grassmannians $Gr_G=LG/L^+G$ and affine flag varieties $Fl_G=LG/I$. The derived or dg-categories of sheaves and D-modules on these ind-schemes (e.g., $D\!\operatorname{mod}(LG)$, $Shv_{nilp}(I\backslash LG/I)$) play a principal role (Dhillon et al., 24 Jan 2025, Dhillon, 2022).
• Moduli of Local Systems: The parameter space for the "spectral side" is the moduli stack of de Rham, Betti, or $\ell$-adic $G^\vee$-local systems over the formal punctured disc $D^\times=\operatorname{Spec}k((t))$. This can be realized as the quotient prestack

$$LocSys_{G^\vee}(D^\times)=\{\text{connections }d+A(t)dt\}/G^\vee(k((t)))$$

or the larger prestack $LocSys_G(\!(t)\!)$ for possibly irregular singularities (Raskin, 2015). For tamely ramified situations or fixed monodromy, this stack may be replaced by Springer-type resolutions or Steinberg stacks.

• Categories/Sheaves Parametrized by Local Systems: The central conjectural objects are (derived, dg-, or ∞-) categories over these moduli, such as module categories for $D\operatorname{mod}(LG)$ or $QCoh(LocSys_{G^\vee})$, often restricted to have nilpotent singular support. The relationship between these sides is frequently expressed as an equivalence—or at least a deep correspondence—between their categorical structures (Dhillon, 2022, Dhillon et al., 24 Jan 2025).

2. The Tame Local Betti Geometric Langlands Correspondence

For the case of tame (Iwahori-level) ramification, the correspondence is realized in a particularly explicit and robust form.

• The Main Equivalence: Dhillon–Taylor (Dhillon et al., 24 Jan 2025) prove a monoidal equivalence of $k$-linear $\infty$-categories:

$\Phi:\;\IndCoh_{G}\bigl(\widetilde G\times_{G}\widetilde G\bigr) \xrightarrow{\sim} \Shv_{nilp}(I\backslash LG/I)$

where - $\IndCoh_{G}(\widetilde G\times_G\widetilde G)$ is the dg-category of $G$-equivariant ind-coherent sheaves (the universal spectral affine Hecke category) on the derived Steinberg stack, with monoidal convolution structure. - $\Shv_{nilp}(I\backslash LG/I)$ is the dg-category of analytic Betti sheaves on the Iwahori double coset space $I\backslash LG/I$ with nilpotent singular support, with convolution.

This equivalence settles the Betti version of the Ben-Zvi–Nadler conjecture and extends Bezrukavnikov's affine Hecke category equivalence to the Betti, tamely ramified setting.

• Central and Whittaker Functors: The proof constructs central sheaves (via Gaitsgory's nearby cycles) and Whittaker functors which exhibit commuting spectral and automorphic actions on an intermediate module—the category of Iwahori-Whittaker sheaves—thereby inducing the monoidal functor underlying the equivalence.
• Specialization: Restricting to unipotent monodromy on both sides, the conjecture specializes to a reconstruction of Bezrukavnikov's fundamental theorem for the (coherent) affine Hecke category.
• $GL_n$ Case: For $G=GL_n$ the isomorphism explicitly matches spectral line bundles $O(\lambda)$ to Wakimoto sheaves $W^\lambda$, providing a DG-lift of the classical correspondence for affine Hecke algebras acting on Iwahori-Whittaker models.

3. Spectral Decomposition, Categories, and Ramification

A crucial aspect is the description of spectral decomposition and the role of higher ramification:

• 1-Affineness and Spectral Parameter Stacks: For reductive $G$, the stack $LocSys_G(\!(t)\!)$ (i.e., the stack of $G$-local systems with possibly irregular singularities) is shown to be 1-affine as a prestack (Raskin, 2015). This means that module categories over $QCoh(LocSys_G(\!(t)\!))$ are equivalent to sheaves of categories on $LocSys_G(\!(t)\!)$. Therefore, the "spectral side" of local geometric Langlands is captured by $QCoh$-linear module structure over the stack of local systems, even in the presence of wild ramification.
• Tame vs. Wild: In tamely ramified settings, spectral parameters land in moduli related to the Steinberg stack or Springer resolution. In wild or irregular situations, the stack $LocSys_G(\!(t)\!)$ remains well-behaved (Noetherian, finite-dimensional cohomology), and one can formulate the local geometric Langlands correspondence with arbitrary ramification.
• Spectral Decomposition of Categories: The equivalence

$D\operatorname{mod}(LG)\text{-}\mathbf{mod} \;\simeq\; 2\text{-}\IndCoh_{nilp}(LocSys_{G^\vee}(D^\times))$

expresses that strong categorical representations of the loop group (left-hand side) decompose into fibers—module categories parameterized by the moduli of local systems (right-hand side) (Dhillon, 2022). The precise notion is formalized via $D$-modules on $LG$-spaces, with temperedness, Whittaker, Kac–Moody, and other structures appearing as special cases.

4. Depth, Slope, and Ramification Filtration

A distinguishing feature of the local geometric context is the precise relation between representation-theoretic depth and geometric slope invariants:

• Depth Preservation Conjecture: For a $G$-oper $x$ (i.e., a certain canonical form of a $G$-connection/trivialization upstairs), the conjecture asserts

$$\mathrm{slope}(\nabla) = \mathrm{depth}\bigl(\widehat{\mathfrak{g}_c}{\text{-}}\mathrm{mod}_x\bigr)$$

where $\nabla$ is the underlying connection and $\widehat{\mathfrak{g}_c}{\text{-}}\mathrm{mod}_x$ is the critical-level category at oper $x$ (Chen et al., 2014). Slope captures the order of the leading pole (after possible covering), and depth measures the minimal Moy–Prasad level for which non-trivial fixed vectors exist.

• Half of Conjecture Proven: It is shown that $\mathrm{slope}(\nabla) \le \mathrm{depth}$ for any oper, using explicit action of Segal–Sugawara operators on vacuum modules and the structure of the affine Kac–Moody algebra at critical level. The other inequality is linked to global Hecke eigensheaf nonvanishing (Zhu's conjecture).
• Ramification and Parameters: For regular or epipelagic (e.g., toral) supercuspidals, the local geometric Langlands correspondence attaches to each representation a corresponding formal connection with prescribed slope and refined leading term; this matches with explicit central support calculations and opers via Feigin–Frenkel isomorphism (Yi, 16 Jun 2025).

5. Functoriality, Hecke Actions, and Theta Correspondence

Functoriality, convolution structures, and theta correspondences are geometrically realized in the local setting:

• Hecke Categories and Functoriality: At the Iwahori level, the affine Hecke algebra is geometrized as a convolution algebra of equivariant perverse sheaves or ind-coherent sheaves on affine flag varieties or Steinberg stacks. The categories of sheaves $D_{I_G}(\Fl_G)$ and $$IndCoh_{G^\vee}(\widetilde G \times_G \widetilde G)$$ implement the Hecke correspondence on automorphic and spectral sides, respectively (Dhillon et al., 24 Jan 2025, Farang-Hariri, 2015).
• Geometric Theta Correspondence: For type II dual reductive pairs (e.g., $(GL_n, GL_m)$), the geometric Weil representation and its invariants at the Iwahori level provide a categorification of the theta correspondence. Explicit functoriality is encoded into a bimodule over the affine Hecke algebras of the pair, constructed as the $K$-theory of a convolution-parameter stack built using the dual group morphism (Farang-Hariri, 2015).

6. Categorical and Motivic Generalizations

Local geometric Langlands is cast at several categorical and field-theoretic levels:

• 2-Categories and Sheaves of Categories: The categorical geometric Langlands conjecture is naturally formulated in the setting of $(\infty,2)$-categories, where strong $D$-module actions correspond to sheaves of categories over stacks of local systems (Dhillon, 2022). 1-affineness of these stacks is essential to matching algebraic definitions with "sheaf of categories" perspectives.
• Motivic and $\ell$-adic Realizations: The motivic geometrization of the local Langlands correspondence uses rigid-analytic motives and a six-functor formalism to obtain a universal (integrally defined) construction, with compatibility across all $\ell$-adic realizations and independence of $\ell$ for $L$-parameters (Scholze, 14 Jan 2025). The spectral action of the dual group on categories of (compactly supported) motivic sheaves on Fargues–Fontaine curve v-stacks provides a universal template, subsuming all previously known $\ell$-adic constructions.
• Special Cases and Real Groups: Modified notions of geometric parameters (e.g., Adams–Barbasch–Vogan spaces) and Hodge-theoretical invariants appear for real reductive groups, where geometric techniques lead to purity theorems for extension groups between standard modules in the equivariant derived category and inform Koszul duality conjectures (Virk, 2013).

7. Recent Developments, Open Problems, and Directions

Several open topics and new achievements characterize the current landscape:

• Spectral Structures for Wild Ramification: With the proof of 1-affineness for $LocSys_G(\!(t)\!)$ (Raskin, 2015), the homological algebra on spectral parameter stacks is well-defined for arbitrary ramification. This supports the existence of a full geometric Langlands program, even allowing for irregular singularities.
• Explicit Constructions for Supercuspidals: In the toral case, the correspondence between Adler/Kaletha-type toral supercuspidals and irreducible isoclinic connections has been established, yielding affine parameter spaces for connections with fixed leading term and slope (Yi, 16 Jun 2025).
• Universal and Motivic L-Parameters: The motivic formalism of Scholze provides a universal, integral version of the correspondence with consequences for independence-of-$\ell$ and the possibility of integral and $p$-adic variants (Scholze, 14 Jan 2025).
• Depth/slope equivalence: Only one direction of the depth-slope conjecture is proven in general (Chen et al., 2014); completing the equality is contingent on the non-vanishing of Hecke eigensheaves at all depths, a key open problem.
• Global Compatibility and Further Functorialities: Compatibility with global geometric Langlands, the "excursion" formalism, and further functorial correspondences (e.g., theta and Arthur–Langlands functoriality) are the subjects of ongoing advances.
• Representation-Theoretic Geometric Structure: The "extended quotient" perspective captures the geometric structure of Bernstein blocks in the smooth dual, matching with the corresponding enhanced parameter spaces via generalized Springer theory (Aubert et al., 2012). This organizes blocks as complex tori modulo finite group actions, providing a uniform local geometric template for the LLC.

Local geometric Langlands thus stands as a categorical, derived, and algebro-geometric upgrade of the local Langlands program, integrating representation theory, algebraic geometry, homological algebra, and arithmetic geometry, and providing foundational, spectral, and functorial structures for a range of local categories and parameters.