Let G be a complex semisimple linear algebraic group, and X be a smooth projective G-variety. In this talk, we will construct an action of the equivariant homology of the affine Grassmannian Gr_G on the G-equivariant quantum cohomology of X. The construction uses shift operators in quantum cohomologies. When G = SL(2) and X is the projective line, this action is compatible with the Peterson isomorphism. Based on joint work with Alexander Braverman.

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