Elliptic classes via the periodic Hecke module and its Langlands dual

TL;DR

This paper constructs elliptic classes of the Springer resolution using the periodic Hecke module and establishes a rational isomorphism with the Langlands dual system, revealing deep dualities in elliptic cohomology.

Contribution

It introduces the elliptic twisted group algebra and demonstrates a rational isomorphism linking elliptic classes with the Langlands dual system.

Findings

Elliptic classes are explicitly constructed as rational sections.

A rational isomorphism intertwines elliptic classes with the dual system.

The Demazure-Lusztig operators act as rational sections in this framework.

Abstract

This paper explores a construction of the elliptic classes of the Springer resolution using the periodic Hecke module. The module is established by employing the Poincar\'e line bundle over the product of the abelian variety of elliptic cohomology and its dual. Additionally, we introduce the elliptic twisted group algebra, which acts on the periodic module. The construction of the elliptic twisted group algebra is such that the Demazure-Lusztig (DL) operators with dynamical parameters are rational sections. We define elliptic classes as rational sections of the periodic module, and give explicit formulas of the restriction to fixed points. Our main result shows that a natural assembly of the DL operators defines a rational isomorphism between the periodic module and the one associated to the Langlands dual root system. This isomorphism intertwines the (opposite) elliptic classes with the fixed point basis in the dual system.