Geometric representations of the formal affine Hecke algebra

TL;DR

This paper establishes a geometric interpretation of the formal affine Hecke algebra associated with any formal group law, connecting it to oriented cohomology theories and Springer fiber cohomology, generalizing classical results.

Contribution

It identifies the formal affine Hecke algebra with a convolution algebra from oriented cohomology, extending Lusztig's K-theory action to a broader formal group law context.

Findings

Algebra acts on Springer fiber cohomology.

Provides a residue interpretation matching known constructions.

Generalizes classical affine Hecke algebra actions.

Abstract

For any formal group law, there is a formal affine Hecke algebra defined by Hoffnung, Malag\'on-L\'opez, Savage, and Zainoulline. Coming from this formal group law, there is also an oriented cohomology theory. We identify the formal affine Hecke algebra with a convolution algebra coming from the oriented cohomology theory applied to the Steinberg variety. As a consequence, this algebra acts on the corresponding cohomology of the Springer fibers. This generalizes the action of classical affine Hecke algebra on the $K$-theory of the Springer fibers constructed by Lusztig. We also give a residue interpretation of the formal affine Hecke algebra, which coincides with the residue construction of Ginzburg, Kapranov, and Vasserot when the formal group law comes from a 1-dimensional algebraic group.