Equivariant oriented homology of the affine Grassmannian

TL;DR

This paper extends the understanding of equivariant homology of the affine Grassmannian to general oriented cohomology theories, using formal affine Demazure algebras to establish new structural results.

Contribution

It generalizes the small-torus equivariant K-homology property to all oriented (co)homology theories and proves the GKM condition for these theories on the affine Grassmannian.

Findings

Equivariant oriented cohomology satisfies the GKM condition.

Homology is isomorphic to the centralizer in the formal affine Demazure algebra.

Results unify various oriented cohomology theories in the context of the affine Grassmannian.

Abstract

We generalize the property of small-torus equivariant K-homology of the affine Grassmannian to general oriented (co)homology theory in the sense of Levine and Morel. The main tool we use is the formal affine Demazure algebra associated to the affine root system. More precisely, we prove that the small-torus equivariant oriented cohomology of the affine Grassmannian satisfies the GKM condition. We also show that its dual, the small-torus equivariant homology, is isomorphic to the centralizer of the equivariant oriented cohomology of a point in the the formal affine Demazure algebra.