A Langlands duality of elliptic Hecke algebras

Technical Summary: A Langlands Duality of Elliptic Hecke Algebras

Problem Statement
The paper addresses the representation theory of the elliptic affine Hecke algebra (specifically the GKV algebra introduced by Ginzburg, Kapranov, and Vasserot), which is defined as a coherent sheaf on an abelian variety associated with a root datum. A central question arises from the observation that representations of this algebra possess an underlying coherent sheaf structure: Can the Fourier-Mukai transform of the underlying abelian variety be extended to establish a duality between the representation categories of the GKV algebra and the GKV algebra associated with the Langlands dual root datum? While the Fourier-Mukai transform is well-understood for coherent sheaves on abelian varieties, extending this equivalence to the specific module categories of these elliptic Hecke algebras requires a suitable intermediary structure.

Methodology
The authors construct a bridge between the representation categories of the original GKV algebra and its Langlands dual by introducing an intermediate algebra: the elliptic Hecke algebra with dynamical parameters ($H^d$). The methodology proceeds in three main stages:

1. Construction of the Dynamical Algebra: The authors define $H^d$ as a sheaf on the product of the abelian variety $A$ and its dual $A^\vee$. This algebra is constructed using a "residue definition" analogous to the GKV construction but adapted for dynamical parameters. The definition involves rational sections of a specific line bundle $L$ (a twist of the Poincaré line bundle) and imposes specific residue conditions (analogous to Demazure-Lusztig operators) to ensure the algebra structure is well-defined.
2. Functorial Connections:
   • Step 1 (GKV to Dynamical): Using the Poincaré line bundle, the authors construct a functor from the category of modules over the original GKV algebra ($Mod\text{-}H$) to the category of modules over the dynamical algebra ($Mod\text{-}H^d$). This is achieved by establishing a coproduct-like algebra homomorphism $\Psi: H^d \to (H^d + S) \otimes p^*H$, where $S$ is the twisted group algebra. This allows the dynamical algebra to act on the pullback of a GKV module tensored with the line bundle $L$.
   • Step 2 (Dynamical to Langlands Dual): The authors demonstrate that the dynamical Demazure-Lusztig operators satisfy the relations of the Weyl group of the Langlands dual root datum. By utilizing these operators, they construct an algebra homomorphism from the Langlands dual GKV algebra ($H_L$) to the pushforward of the dynamical algebra. This induces a functor from $Mod\text{-}H^d$ to $Mod\text{-}H_L$.
3. Synthesis: By composing these functors, the authors realize the Fourier-Mukai transform on the level of representation categories.

Key Contributions and Results

• Definition of the Dynamical Elliptic Hecke Algebra: The paper provides a rigorous integral structure for the elliptic Hecke algebra with dynamical parameters, defining it via residue conditions on rational sections. This algebra serves as the crucial intermediary for the duality.
• The Fourier-Mukai Functor: The primary result (Corollary 5.1) is the construction of a functor $FM: Mod\text{-}H \to Mod\text{-}H_L$. This functor makes the diagram commutative with the standard Fourier-Mukai transform on the underlying abelian varieties. Specifically, it maps the category of GKV modules to the category of modules over the Langlands dual GKV algebra.
• Inverse Functor and Equivalence: The paper constructs an inverse functor (Corollary 5.2) using the inverse Fourier-Mukai transform and the dual dynamical algebra.
• Equivalence of Subcategories: The authors prove that when restricted to specific subcategories—homogeneous vector bundles ($Mod\text{-}fl\text{-}H$) and finitely supported coherent sheaves ($Mod\text{-}fin\text{-}H_L$)—the functors induce an equivalence of abelian categories (Corollary 5.6).
• Classification of Irreducible Representations: As a consequence of the established equivalence and prior results, the paper states that if the underlying group $G$ is of adjoint type, the irreducible objects in $Mod\text{-}fl\text{-}H$ are in one-to-one correspondence with the set of triples $(P, x, \chi)$, where $(P, x)$ is a nilpotent $G$-Higgs bundle and $\chi$ is an irreducible representation of the component group of its automorphism group (Corollary 5.7).

Significance and Claims
The paper claims to provide an affirmative answer to the question of whether the Fourier-Mukai transform extends to the representation categories of elliptic affine Hecke algebras. The significance lies in:

1. Unifying Framework: It connects the representation theory of elliptic Hecke algebras with the geometric Langlands program via the Fourier-Mukai transform, utilizing the dynamical parameters as the necessary "glue."
2. Explanation of Weyl Relations: The work offers a conceptual explanation for why the dynamical Demazure-Lusztig operators (previously studied by Rimányi and Weber in the context of elliptic Schubert calculus) satisfy the Weyl group relations. The paper attributes this to their embedding within the Hecke algebra of the Langlands dual root datum.
3. Structural Insight: It establishes that the Fourier-Mukai transform is not merely a geometric operation on sheaves but a structural duality between the representation theories of dual elliptic Hecke algebras.

The authors explicitly note that while the results are expected to be compatible with the Fourier transform of the double affine Hecke algebra (related to S-duality), the specific relation to the double affine case is left for future investigation. The paper focuses strictly on the elliptic setting and the construction of the duality functor.