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

Elliptic classes via the periodic Hecke module and its Langlands dual