Elliptic classes via the periodic Hecke module and its Langlands dual

Technical Summary of "Elliptic Classes via the Periodic Hecke Module and Its Langlands Dual"

Problem Statement
The paper addresses the construction and structural properties of elliptic classes associated with the Springer resolution $T^*G/B$. These classes, which arise in the study of symplectic resolutions and elliptic cohomology, are rational sections of specific vector bundles over abelian varieties. While previous approaches by Aganagic-Okounkov (via stable envelopes) and Rimányi-Weber (via Bott-Samelson resolutions and elliptic genera) have established iteration formulas involving elliptic Demazure-Lusztig (DL) operators with dynamical parameters, this work seeks a purely algebraic framework. The central problem is to define these classes using the structures of root systems and the Poincaré line bundle, and to elucidate the relationship between the elliptic classes of a root system and those of its Langlands dual.

Methodology
The authors employ a purely algebraic approach grounded in the theory of root data, elliptic curves, and coherent sheaves. The methodology proceeds through the following steps:

1. Geometric Setup: Let $E$ be an elliptic curve over $\mathbb{C}$. Given a root datum with dual lattices $X^*$ and $X_*$, the authors define abelian varieties $A = X^* \otimes E$ and $A^\vee = X_* \otimes E$. The Weyl group $W$ acts on both, denoted as the spectral action ($W$) and the dynamical action ($W^d$).
2. The Periodic Hecke Module: Utilizing the Poincaré line bundle $\mathcal{P}$ on $A \times A^\vee$ and a specific $\rho$-shift, the authors define a line bundle $L$. They construct the periodic module $M = \bigoplus_{w \in W} w^*L$, which serves as a model for the equivariant elliptic cohomology of the Springer resolution.
3. Elliptic Twisted Group Algebra: An algebra structure is defined on the direct sum of line bundles $S = \bigoplus_{v,w \in W} S_{v,w}$, where $S_{v,w} = L \otimes (w^{-1})^*(v^{-1})^{d*}L^{-1}$. This algebra, referred to as the elliptic twisted group algebra, acts on the periodic module. The algebra structure corresponds to the Kostant-Kumar twisted product, augmented by dynamical Weyl group actions.
4. Elliptic DL Operators: The authors define elliptic Demazure-Lusztig (DL) operators $T_\alpha^{z,\lambda}$ with dynamical parameters as rational sections of the twisted group algebra. These operators satisfy braid relations and are constructed using only the elliptic curve, the Poincaré line bundle, and the root datum.
5. Langlands Duality: A parallel construction is performed for the Langlands dual root datum, yielding a dual periodic module $M^d$ and dual operators. The authors utilize duality functors and Poincaré pairings to relate the original system to its Langlands dual.

Key Contributions and Results

• Algebraic Definition of Elliptic Classes: The paper provides a rigorous algebraic definition of elliptic classes as rational sections of the periodic module. Explicit formulas for the restriction of these classes to fixed points are derived.
• The Main Theorem (Langlands Duality Isomorphism): The primary result establishes a natural assembly of the DL operators, denoted as $T$, which defines a rational isomorphism between the periodic module $M$ and the dual periodic module $M^d$. Specifically:
  1. The map $T: M \dashrightarrow M^d$ is a rational isomorphism of vector bundles.
  2. The map $T$ sends the fixed-point basis $\tilde{f}_v$ of $M$ to the elliptic classes $E^d_v$ of the Langlands dual system.
  3. Conversely, $T$ sends the elliptic classes $E_v$ of the original system to the fixed-point basis $\tilde{f}^d_v$ of the dual system.
  4. The inverse map is given by the dual operator assembly $T^d$.
• Poincaré Pairings and Adjunction: The authors introduce Poincaré pairings on the periodic modules and prove adjunction properties for the elliptic DL operators. These properties are crucial for proving the main theorem and for establishing the duality between elliptic classes and their "opposite" classes (defined via dual line bundles).
• Restriction Formulas: The paper derives explicit formulas for the restriction coefficients of elliptic classes to fixed points. These coefficients are expressed as sums over subsets of the reduced word of a Weyl group element, involving products of theta functions.
• Logarithmic Degree: A new invariant, the "logarithmic degree" $\Theta_w$, is defined for elements of the Weyl group. This element in $X^* \otimes X_*$ encodes the zero and pole structure of the restriction coefficients.

Significance and Claims
The paper claims that its main theorem provides a canonical isomorphism between the equivariant elliptic cohomology of the Springer resolution $T^*G/B$ and that of its Langlands dual $T^*G^\vee/B^\vee$. This isomorphism maps the fixed-point basis of one space to the elliptic classes of the other, effectively intertwining the geometric structures of dual root data.

The authors position this work as a distinct perspective on 3d-mirror symmetry (symplectic duality) within the context of elliptic cohomology. While previous works (e.g., Rimányi-Weber) have compared restriction formulas and linked them to mirror symmetry, this paper offers a structural explanation via the periodic Hecke module and the interplay of DL operators.

The paper also notes that the restriction coefficients can be viewed as "elliptic double Schubert polynomials" in type A, drawing an analogy to Allen Knutson's work on cohomological Schubert classes. The authors state that a "root polynomial formula" for the restriction coefficients (focusing on $b$-coefficients rather than the $a$-coefficients discussed here) will be presented in a forthcoming paper, suggesting that the current work lays the foundational algebraic machinery for these future developments.

The construction is noted to be well-behaved with respect to Langlands duality, providing a unified framework for studying elliptic classes without relying on specific geometric resolutions like Bott-Samelson, though it recovers known iteration formulas. The paper concludes by acknowledging that the integral structure of the elliptic Hecke algebra (without localization) remains an interesting object for representation theory, referencing separate work for its definition and applications in Fourier-Mukai transforms.