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Iwahori Fundamental Local Equivalence

This paper constructs three tamely ramified local equivalences of factorization module categories, providing factorization versions of the Arkhipov-Bezrukavnikov and Bezrukavnikov equivalences at a point, as well as an Iwahori-ramified version of the factorizable Fundamental Local Equivalence.

Original authors: Taeuk Nam

Published 2026-08-06

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CC BY 4.0

Original authors: Taeuk Nam

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a massive, cosmic puzzle where two completely different languages describe the exact same universe. On one side, you have "geometry," which deals with shapes, curves, and how things move through space. On the other side, you have "spectral theory," which deals with patterns, frequencies, and the hidden algebraic rules that govern those shapes. For decades, mathematicians have been trying to prove that these two languages are actually just different dialects of the same truth. This is the heart of the "Geometric Langlands" program. Think of it like realizing that a map of a city drawn with streets and buildings is mathematically identical to a map of the same city drawn with sound waves and musical notes. If you can translate perfectly between the two, you can solve problems in one world by using the tools of the other.

Recently, mathematicians proved that this translation works perfectly for a "smooth" version of the city, where everything is uniform and unbroken. But real life (and real math) is rarely that smooth. Often, things have "kinks," "twists," or specific points where the rules change abruptly. In the language of this puzzle, these are called "ramified" points. The big question was: Does the perfect translation still work when we zoom in on these messy, twisted spots? This paper tackles that exact question for a specific type of twist called "Iwahori ramification," which is like a very specific, intricate knot in the fabric of the mathematical city.

The author, Taeuk Nam, sets out to build a new kind of bridge between these two worlds, specifically designed to handle these messy knots. The paper's main finding is that yes, the translation works even here, but you have to build the bridge differently. Instead of just connecting two static points, the author constructs "factorization module categories." To use an analogy, imagine that in the smooth world, you just needed a single bridge to cross a river. But in this twisted world, the river is made of many smaller streams that merge and split as you walk along it. You can't just build one bridge; you need a system of bridges that can snap together or fall apart depending on how the streams merge. The author proves that for every "geometric" category (the shape side) with these knots, there is a matching "spectral" category (the pattern side) that behaves exactly the same way when these streams merge.

The paper explicitly rules out the idea that you can simply take the old, smooth bridges and stretch them to fit the knots. The author shows that the old methods fail because the "knots" (the Iwahori subgroups) don't play nicely with the way points move around in families. Instead, the paper argues that you must treat the knots as fixed anchors while letting other points move around them, colliding and interacting in a specific way. The author doesn't just suggest this works; they provide a rigorous, step-by-step proof that these new "factorization" bridges are mathematically equivalent. They construct three specific equivalences (which are like perfect translation dictionaries) and prove that they hold up under the most complex conditions, including when points crash into each other.

The journey to this proof involves three main steps, each building a different part of the bridge. First, the author constructs a bridge for a category called "Whit!(FlG)" to "QCoh(čn/čB)." Think of this as translating a complex dance of geometric flags into a language of algebraic sheaves. They prove this works by showing that if the translation is correct at a single point, it is correct everywhere, thanks to a property called "fusability," which ensures the pieces snap together perfectly.

Second, they build a bridge for the "Spherical Hecke" categories, which are like the master keys that unlock the structure of the whole city. They show that the way these keys work on the geometric side is identical to how they work on the spectral side. This is crucial because these keys are what allow the different parts of the puzzle to talk to each other.

Finally, the author tackles the "Fundamental Local Equivalence" (FLE), which is the grand unification of the whole system. They prove that the "Iwahori" version of this equivalence works by showing that the categories involved are "tempered." In our analogy, "tempered" means the categories are stable enough that they don't explode or collapse when you apply the translation. By proving that both sides of the equation are tempered and that they match up perfectly at the point of the knot, the author confirms that the entire system is consistent.

In short, this paper is a masterclass in building a new kind of mathematical infrastructure. It takes a known, beautiful theory and extends it into a messy, complex territory where it was previously unknown if the theory would hold. The author doesn't just guess; they construct the machinery, prove it works at the smallest scale, and then show that the machinery scales up to handle the entire system. The result is a confirmed, robust translation between the geometric and spectral worlds, even in the presence of the most intricate mathematical knots. This doesn't just solve a local problem; it provides the tools needed to understand the global behavior of these systems, paving the way for future discoveries in how the universe's hidden patterns are woven together.

Technical Summary: Iwahori Fundamental Local Equivalence

Problem Statement
The paper addresses the local formulation of the geometric Langlands correspondence in the presence of tame (Iwahori) ramification. While the global geometric Langlands conjecture for the unramified case has been recently established via a series of works by Arinkin, Beraldo, Campbell, Chen, Faergeman, Gaitsgory, Lin, Raskin, and Rozenblyum, the local behavior of the Langlands functor at a point with Iwahori level structure requires a spectral description. Specifically, the author seeks to construct a spectral description for the DG category of D-modules on the stack $\text{Bun}_G^I$ (bundles with Iwahori level structure at a fixed point $x_0$ ).

The unramified local theory is governed by three fundamental equivalences at a point $x$ : the geometric Casselman-Shalika equivalence ( $CS_{G,x}$ ), the geometric Satake equivalence ( $Sat_{G,x}$ ), and the Fundamental Local Equivalence ( $FLE_{G,x}$ ). The paper aims to construct Iwahori-ramified counterparts to these equivalences and, crucially, to upgrade them from pointwise equivalences to factorization module equivalences. This allows the equivalences to encode the behavior of points "moving around" and "colliding" with the fixed ramification point $x_0$ , a necessity for controlling the global Langlands functor.

Methodology
The paper employs the framework of Factorization Module Categories. Instead of treating the local categories at $x_0$ in isolation, the author treats them as modules over the corresponding unramified factorization categories defined on the Ran space.

Factorization Module Structures: The author defines six key factorization module categories (pairs $(C, M)$ where $C$ is a factorization category and $M$ is a factorization $C$ -module at $x_0$ ). These include:
- Geometric side: $(Whit!(G), Whit!(Fl_G))$ , $(Sph_G, Aff_G)$ , and $(KL(G)_{crit}, IKL(G)_{crit})$ .
- Spectral side: $(Rep(\check{G}), QCoh(\check{n}/\check{B}))$ , $(Sph^{spec}_{\check{G}}, Aff^{spec}_{\check{G}})$ , and $(IndCoh(Op^{mon-free}_{\check{G}}), IndCoh(Op^{mer}_{\check{G}} \times_{LS^{mer}_{\check{G}}} \check{n}/\check{B}))$ .
Approximation via Fusability: A central technical tool is the notion of Fusability (introduced in the context of Bogdanova, Chen, and Gaitsgory's upcoming work [BCG26]). A factorization module category is "fusable" if it can be reconstructed from its fiber at $x_0$ via a specific combinatorial condition. The author proves that the relevant categories (e.g., $Whit!(Fl_G)$ and $Aff_G$ ) are fusable. This property allows the reduction of proving an equivalence of factorization module categories to proving an equivalence of their fibers at the point $x_0$ .
Construction of Equivalences:
- Arkhipov-Bezrukavnikov ($AB$): Constructed by relating $Whit!(Fl_G)$ to a semi-infinite version via the Jacquet functor and utilizing the equivalence of the semi-infinite version established in [CCF+24].
- Bezrukavnikov ( $B$ ): Constructed by leveraging the action of the spherical Hecke category on the affine Hecke category and the previously constructed $AB$ equivalence. The pointwise equivalence is verified by adapting arguments from Dhillon and Taylor [DT25] from the Betti setting to the de Rham setting.
- Iwahori Fundamental Local Equivalence ($IFLE$): Constructed using the coaction of the spectral category on the geometric category, mediated by the equivalence $B$ .
Temperedness: To prove the pointwise equivalence of $IFLE$, the author introduces Iwahori-Hecke Temperedness. This condition ensures that tensoring with the Whittaker category is conservative. The proof involves embedding both sides of the equivalence into a larger ambient category ( $IndCoh(Op^{mer}_{\check{G}, x_0})$ ) and showing they share the same essential image.

Key Contributions and Results
The paper establishes three main theorems, upgrading pointwise local equivalences to equivalences of factorization module pairs in $\text{FactModCat}_{x_0}$ :

The Arkhipov-Bezrukavnikov Equivalence ($AB$):
The morphism of pairs $AB: (Whit!(G), Whit!(Fl_G)) \to (Rep(\check{G}), QCoh(\check{n}/\check{B}))$ is an equivalence. This extends the classical Arkhipov-Bezrukavnikov equivalence to the factorization setting, compatible with the geometric Casselman-Shalika equivalence.
Bezrukavnikov's Equivalence ( $B$ ):
The morphism of monoidal pairs $B: (Sph_G, Aff_G) \to (Sph^{spec}_{\check{G}}, Aff^{spec}_{\check{G}})$ is an equivalence. This provides a spectral description of the affine Hecke category compatible with the geometric Satake equivalence.
The Iwahori Fundamental Local Equivalence ($IFLE$):
The morphism of pairs $IFLE: (KL(G)_{crit}, IKL(G)_{crit}) \to (IndCoh(Op^{mon-free}_{\check{G}}), IndCoh(Op^{mer}_{\check{G}} \times_{LS^{mer}_{\check{G}}} \check{n}/\check{B}))$ is an equivalence. This is the Iwahori-ramified version of the Fundamental Local Equivalence, linking the Iwahori-ramified Kazhdan-Lusztig category to ind-coherent sheaves on a specific stack of opers with monodromy conditions.

Significance
The paper claims significance in providing the necessary local ingredients to study the global Langlands functor with Iwahori ramification, denoted $L^I_G$ . Just as the unramified local equivalences ($CS$, $Sat$, $FLE$) were essential for the proof of the global unramified conjecture, these Iwahori-ramified factorization equivalences are the requisite tools to understand the behavior of $L^I_G$ when points vary in families and collide with the ramification point.

The work demonstrates that the "factorization module" framework is the correct technical device to encode the interaction between mobile points and a fixed Iwahori level structure. By proving these equivalences, the paper completes the local picture for tame ramification, mirroring the structure of the unramified case and enabling the potential extension of global Langlands results to the tamely ramified setting. The results rely heavily on the compact generation of $D\text{-mod}(\text{Bun}_G^I)$ (established in [Nam24]) and the tamely ramified spectral action (from [Fae24]).

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