Homology of the affine Grassmannian and quantum cohomologies

Let G be a complex reductive group, and X be any smooth projective G-variety. In this talk, we will construct an algebra homomorphism from the G-equivariant homology of the affine Grassmannian Gr_G to the G-equivariant quantum cohomology of X. The construction uses shift operators in quantum cohomologies. We will also discuss the possible extension to the loop rotation equivariant setting and the relation with the Peterson isomorphism when X is the flag variety associated with G. This is based on joint work with Alexander Braverman.