Universal Monodromic Arkhipov–Bezrukavnikov Equivalence

• The paper establishes a canonical monoidal equivalence between G-equivariant quasi-coherent sheaves and monodromic Iwahori–Whittaker sheaves, bridging algebraic and geometric representation theories.
• It employs explicit rank 1 order-of-vanishing computations and Wakimoto filtrations to verify full faithfulness and compatibility of the monoidal structures.
• The equivalence catalyzes new instances of Koszul duality and advances the tame local Betti geometric Langlands conjecture, offering innovative tools for further research.

The universal monodromic Arkhipov–Bezrukavnikov equivalence is a canonical monoidal equivalence underlying the structure of derived categories of sheaves (both constructible and coherent) associated to a complex reductive group and its Langlands dual. It provides a universal—i.e., monodromic in all torus directions—deformation of the classical Arkhipov–Bezrukavnikov equivalence and fundamentally relates equivariant quasi-coherent sheaves on the Grothendieck alteration of a reductive group to universal monodromic Iwahori–Whittaker sheaves on the enhanced affine flag variety. This construction realizes new instances of Koszul duality, organizes the structure of universal monodromic Hecke categories, and serves as a cornerstone in recent progress on the tame local Betti geometric Langlands conjecture (Taylor, 2023, Dhillon et al., 24 Jan 2025, Dhillon et al., 24 Jan 2025).

1. Categories and Structures Involved

Let $G$ be a pinned split connected reductive group over a field $k$ (characteristic zero unless specified), with Borel $B \subset G$ and Cartan $T=B/N$, and let $LG = G(\mathbb{C}((t)))$ denote the loop group. The principal spaces and categories are:

• Grothendieck alteration $\tilde{G}$:

$$\tilde{G} = \{(g, B') \mid g\in G,\ B'\subset G \text{ a Borel},\ g\in B'\}$$

$\tilde{G} \to G$ is $G$-equivariant under conjugation.

• Equivariant quasi-coherent sheaves
  • $\mathrm{QCoh}_G(\tilde{G})$: The DG-category of $k$0-equivariant quasi-coherent sheaves on $k$1, monoidal under $k$2-linear tensor product.
  • $k$3: Adjoint-equivariant sheaves on $k$4 itself.
• Universal monodromic sheaf category:
  • $k$5: $k$6-equivariant constructible sheaves on the enhanced affine flag variety $k$7.
  • $k$8: The universal monodromic Iwahori–Whittaker category, constructed as a base change along the monoidal Whittaker generator for the finite flag variety, endowing the category with universal monodromy.
• Hecke and Soergel categories:
  • $k$9: Soergel bimodules over $B \subset G$0.
  • $B \subset G$1: Derived category of “weakly $B \subset G$2–constructible” complexes on $B \subset G$3, with $B \subset G$4-monodromic data.
• Bi-Iwahori–Whittaker sheaves and their convolution algebra:
  • $B \subset G$5, with convolution monoidal structure.

2. Statement of the Universal Monodromic Equivalence

There are two central monoidal equivalences described:

• (A) Grothendieck alteration to monodromic Whittaker sheaves:

$B \subset G$6

This functor is characterized by: 1. $B \subset G$7, the universal Whittaker generator, 2. Intertwining of monoidal structures: $B \subset G$8, 3. Admittance of continuous left and right adjoints via Whittaker averaging functors.

• (B) Adjoint-equivariant sheaves to bi-Whittaker category:

$B \subset G$9

with $T=B/N$0-linearity and compatibility so that $T=B/N$1.

This equivalence identifies universal monodromic Hecke categories and places the “big tilting object” (Whittaker averaged) as the kernel of the functor, intertwining the module and monoidal structures.

3. Construction and Fully Faithfulness

The proof proceeds by relating Hom spaces and monoidal structures:

• Hom-space comparison: Fully faithfulness of $T=B/N$2 reduces to checking that

$T=B/N$3

is an isomorphism of $T=B/N$4-modules for all dominant coweights $T=B/N$5.

• Flatness, rank, and Hartogs’ lemma: Both sides are shown to be finite-rank free $T=B/N$6-modules in degree zero. Hartogs’ theorem ensures it suffices to verify isomorphism after inverting all but one root hyperplane, which reduces the problem to checks on semisimple rank 1 Levi (i.e., $T=B/N$7).
• Rank 1 calculation: Explicit order-of-vanishing computations in $T=B/N$8 establish agreement of localizations, fitting the Wakimoto filtration structure and confirming the monoidal equivalence globally (Dhillon et al., 24 Jan 2025).

4. Key Objects: Tiltings, Wakimoto Sheaves, and Monodromy

• Universal monodromic big tilting sheaf ($T=B/N$9): Constructed via Whittaker averaging, $LG = G(\mathbb{C}((t)))$0 in $LG = G(\mathbb{C}((t)))$1 is a tilting object whose endomorphism algebra is

$LG = G(\mathbb{C}((t)))$2

establishing a link to the fiber product $LG = G(\mathbb{C}((t)))$3 (Taylor, 2023).

• Soergel functor: The functor $LG = G(\mathbb{C}((t)))$4, $LG = G(\mathbb{C}((t)))$5, is strictly monoidal, carrying tilting objects (Bott–Samelson type) to Soergel bimodules and inducing the derived equivalence $LG = G(\mathbb{C}((t)))$6.
• Wakimoto filtration: Crucial for understanding the gradings and associated graded pieces, both in geometric and categorical constructions, and essential to connect to representation-theoretic data under the equivalence.
• Bi-Whittaker modules and identities: These serve as the categorical incarnation of the endomorphism algebras, and bi-Whittaker averaging connects to the coherent group algebra side.

5. Connections to Langlands Duality and Hecke Categories

• Universal affine Hecke category: Both automorphic (sheaf-theoretic) and spectral (coherent-theoretic) sides are realized as monoidal DG-categories,

$LG = G(\mathbb{C}((t)))$7

$LG = G(\mathbb{C}((t)))$8

and are shown to be canonically equivalent as monoidal categories (Dhillon et al., 24 Jan 2025).

• Betti geometric Langlands conjecture: The universal equivalence implies Ben-Zvi–Nadler's tame local Betti geometric Langlands, identifying Iwahori-monodromic Betti sheaves on the affine flag variety with ind-coherent sheaves on a Steinberg-type stack, interpolating between all monodromic situations.

6. Applications, Consequences, and Universality

Several foundational consequences and applications arise:

• Koszul duality and “uncompleted” setting: The universal equivalence realizes a strong form of “uncompleted” Koszul duality beyond the classical (completed, unipotent) setting, resolving conjectures of Eberhardt regarding the extension of BGS dualities to universal monodromic and K-motive frameworks (Taylor, 2023).
• Prounipotent endomorphismensätze: Formal completions recover the previously known results of Soergel and Bezrukavnikov–Riche, but now with fully explicit universal deformation and valid over arbitrary fields.
• Monodromy and functoriality: The functors constructed are equivariant with respect to all torus monodromies, not just unipotent classes, establishing a universal symmetry that encompasses all possible geometric conjugacy classes.
• Methodological innovation: Unlike previous approaches, the proof of full faithfulness uses localization in semisimple (as opposed to unipotent) directions, reducing computations to order-of-vanishing checks in rank 1, and providing new techniques for the equivalence of derived categories (Dhillon et al., 24 Jan 2025).

7. Mixed Characteristic Variants and Further Developments

The framework admits mixed characteristic analogs:

• Nearby cycles and Wakimoto filtrations: The $LG = G(\mathbb{C}((t)))$9-adic variant employs nearby cycles on the $\tilde{G}$0-adic Hecke stack, perverse t-exactness, and Wakimoto filtrations at Iwahori level to construct analogous functors in the mixed characteristic setting (Anschütz et al., 2023).
• Central functor and E2-monoidality: Gaitsgory's central functor extends to the $\tilde{G}$1-adic context, producing $\tilde{G}$2–central structures and monodromic (Wakimoto-filtered) sheaves, with monoidal equivalences holding for all classical and certain exceptional groups.
• Universality: Inertia actions and monodromy are identified on both sides of the equivalence, demonstrating the robustness and universality of the monodromic Arkhipov–Bezrukavnikov framework in both function field and $\tilde{G}$3–adic settings (Anschütz et al., 2023).

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References:

• "Universal monodromic tilting sheaves" (Taylor, 2023)
• "The universal monodromic Arkhipov–Bezrukavnikov equivalence" (Dhillon et al., 24 Jan 2025)
• "Tame local Betti geometric Langlands" (Dhillon et al., 24 Jan 2025)
• "Gaitsgory's central functor and the Arkhipov–Bezrukavnikov equivalence in mixed characteristic" (Anschütz et al., 2023)