The DG Category of D-modules on the Moduli Stack of Principal G-bundles with Iwahori Level Structure is Compactly Generated

מושגי ליבה

The DG category of D-modules on the moduli stack of principal G-bundles with Iwahori level structure is proven to be compactly generated, extending the result of Drinfeld and Gaitsgory for the unramified case.

תקציר

This research paper delves into the intricate realm of algebraic geometry, specifically focusing on the compact generation of DG categories of D-modules. The authors extend a pivotal result by Drinfeld and Gaitsgory, who demonstrated the compact generation of the DG category of D-modules on the moduli stack of principal G-bundles (BunG).

The paper meticulously examines the moduli stack of principal G-bundles with Iwahori level structure (BunI
G), a more complex structure than BunG. The authors establish a stratification of BunI
G analogous to the Harder-Narasimhan stratification of BunG, dissecting it into manageable quasicompact open substacks.

Through a series of technical arguments, the paper demonstrates that the closed complements of these substacks are contractive, implying the co-truncativeness of the open substacks. This property, coupled with the local QCA nature of BunI
G, leads to the central theorem: D-mod(BunI
G) is compactly generated.

Furthermore, the authors assert that their proof seamlessly generalizes to the moduli stack of principal G-bundles with Iwahori level structure at multiple points, Bun(I;x1,...,xk)
G, establishing its compact generation as well.

This result holds significant implications for the broader field of geometric Langlands correspondence. The compact generation of D-mod(BunI
G) serves as a crucial step towards formulating and proving an Iwahori-ramified version of the geometric Langlands conjecture.

סטטיסטיקה

deg(L′−1 ⊗L−1) = −2n < 0.
deg(L⊗2) = 2n > 2g −2.
deg(L⊗2(−x)) ≥2g −1.
⟨θ, ˇαi⟩≥2g −2 for all simple roots ˇαi ∈ΓG.

ציטוטים

"Being compactly generated is a highly desirable finiteness condition on a DG category for the following two reasons, one internal and one external."
"The compact generation of D-mod(BunG) is a necessary condition for the geometric Langlands conjecture to hold."
"As a step towards the Iwahori ramified geometric Langlands conjecture, in the body of this paper we will show that D-mod(BunI
G) is compactly generated."

תובנות מפתח מזוקקות מ:

$\mathrm{D}-\mathrm{mod}(\mathrm{Bun}_G^\mathrm{I})$ is Compactly Generated

by Taeuk Nam (1... ב- arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.03057.pdf

שאלות מעמיקות

How does the understanding of D-mod(BunI

G) being compactly generated contribute to the broader research landscape of the geometric Langlands program beyond the Iwahori case?
Answer:
The compact generation of D-mod(BunI
G) is a significant stepping stone in the geometric Langlands program, particularly in the pursuit of understanding the ramified version of the conjecture. Here's how it contributes beyond the Iwahori case:

Foundation for Ramified Langlands: The unramified geometric Langlands conjecture, now a theorem, establishes an equivalence between categories related to representations of a reductive group G and sheaves on moduli spaces of G-bundles. The natural next step is to explore the ramified setting, where we introduce level structures on these G-bundles. The Iwahori case is a crucial first step towards this ramified picture. Proving D-mod(BunI
G) is compactly generated provides a solid foundation for extending the Langlands correspondence to more general ramification types.

Tame Ramification as a Bridge: Iwahori level structure represents a "tame" form of ramification. By understanding this tamer case, we gain insights and develop techniques that could potentially be adapted to tackle wilder ramification scenarios. The strategies employed in proving compact generation in the Iwahori case, such as the analysis of strata and contractiveness, might offer a blueprint for approaching more intricate level structures.

Implications for Moduli Spaces: The study of D-mod(BunI
G) and its compact generation deepens our understanding of the geometry and structure of the moduli stack BunI
G itself. This stack, parametrizing G-bundles with Iwahori level structure, is a fundamental object in the Langlands program. Knowing its category of D-modules is compactly generated provides valuable information about its properties and behavior.

Connections to Representation Theory: The geometric Langlands program has profound connections to the representation theory of reductive groups. Compact generation results often have implications for the structure and properties of categories of representations. Understanding the compact objects in D-mod(BunI
G) could shed light on the representation theory of affine Lie algebras and their connections to the Langlands dual group.
In essence, the compact generation of D-mod(BunI
G) is not merely a technical result but a conceptual advance. It opens doors to exploring the ramified geometric Langlands conjecture, provides a testing ground for new ideas, and deepens our understanding of the intricate relationship between representation theory, geometry, and number theory.

Could there be alternative approaches to proving the compact generation of D-mod(BunI

G) that circumvent the need for explicit stratification and analysis of contractiveness?
Answer:
While the approach using explicit stratification and contractiveness is the standard method for proving compact generation in the context of algebraic stacks like BunI
G, it's certainly conceivable that alternative approaches could exist. Here are some speculative possibilities:

Abstract Characterizations of Compact Objects: One could try to develop more abstract criteria for identifying compact objects in DG categories arising from geometric contexts. If one could find a characterization that doesn't rely on explicit presentations of the stack, it might be possible to prove compact generation of D-mod(BunI
G) more directly. This would likely require a deeper understanding of the categorical structures involved.

Deformation Theory and Tannakian Formalism: The stack BunI
G can be viewed as a deformation of BunG. It might be fruitful to explore whether deformation-theoretic techniques could be used to relate the compact generation of D-mod(BunG) to that of D-mod(BunI
G). Additionally, the Tannakian formalism, which connects categories of representations to their underlying algebraic structures, could potentially offer a different perspective on compact generation in this setting.

Methods from Derived Algebraic Geometry: Derived algebraic geometry provides a powerful framework for studying moduli spaces and stacks. It's possible that techniques from this area, such as the theory of derived stacks and their functoriality, could lead to a more conceptual proof of compact generation that avoids some of the explicit computations involved in the stratification approach.

Connections to Other Geometric Structures: BunI
G is related to other geometric objects, such as affine Grassmannians and loop groups. Exploring these connections and leveraging known results about the compact generation of categories associated with these related structures might offer an alternative route to proving the desired result for D-mod(BunI
G).
It's important to note that these are just potential avenues for exploration, and it's unclear whether they would ultimately lead to a simpler or more elegant proof. The explicit stratification and contractiveness approach, while technically involved, has the advantage of being concrete and providing a clear geometric picture of the situation. Nevertheless, exploring alternative approaches could potentially yield new insights and connections within the geometric Langlands program.

If the concept of "compactness" reflects a form of finiteness in the context of infinite-dimensional categories, what mathematical objects or structures could be considered "compact" in other areas of mathematics or theoretical physics?

Answer:
The notion of "compactness" as a reflection of finiteness in infinite-dimensional settings extends beyond DG categories and appears in various guises across mathematics and theoretical physics. Here are some examples:
Functional Analysis:

Compact Operators: In the theory of Banach spaces, compact operators are linear operators that map bounded sets to relatively compact sets (sets whose closure is compact). They generalize the notion of finite-rank operators and exhibit "finiteness" by "shrinking" the dimension in some sense.

Finite-Rank Operators: As mentioned above, finite-rank operators on vector spaces have finite-dimensional images. They are the prototypical examples of "compact" objects in functional analysis.
Topology and Geometry:

Compact Spaces: In topology, compact spaces are spaces where every open cover has a finite subcover. This generalizes the notion of closed and bounded subsets of Euclidean space. Compact spaces exhibit "finiteness" by ensuring that we can always "control" their behavior with a finite amount of information.

Finite CW-Complexes: In algebraic topology, finite CW-complexes are spaces built by attaching finitely many cells of increasing dimension. They are "compact" in the sense that they can be constructed from a finite number of building blocks.
Representation Theory:

Finite-Dimensional Representations: In the representation theory of groups and algebras, finite-dimensional representations are those where the vector space on which the group or algebra acts is finite-dimensional. These representations are "compact" in the sense that they are completely determined by a finite amount of data.
Theoretical Physics:

Bound States in Quantum Mechanics: In quantum mechanics, bound states of a system, like the electron in a hydrogen atom, are states with quantized energy levels and spatially localized wavefunctions. They can be viewed as "compact" objects within the infinite-dimensional Hilbert space of all possible states.

Effective Field Theories: In quantum field theory, effective field theories describe physics at a particular energy scale by focusing on a finite set of degrees of freedom relevant at that scale. They provide a "compact" description of the system by ignoring irrelevant high-energy details.
In each of these cases, the notion of "compactness" captures a sense of finiteness, manageability, or controllability within an otherwise infinite-dimensional or complex setting. It highlights the importance of identifying and understanding objects that exhibit such "finiteness" properties, as they often play a crucial role in simplifying analysis and revealing underlying structures.