Affine symmetries of the equivariant quantum cohomology ring of rational homogeneous spaces

Let X be a rational homogeneous space and let QH(X)× loc be the group of invertible elements in the small quantum cohomology ring of X localised in the quantum parameters. We generalise results of [ChMaPe06] and realise explicitly the map π1(Aut(X)) → QH (X)× loc described in [Se97]. We even prove that this map is an embedding and realise it in the equivariant quantum cohomology ring QH T (X)× loc . We give explicit formulas for the product by these elements. The proof relies on a generalisation, to a quotient of the equivariant homology ring of the affine Grassmannian, of a formula proved by Peter Magyar [Ma07]. It also uses Peterson’s unpublished result [Pe97] — recently proved by Lam and Shimozono in [LaSh07] — on the comparison between the equivariant homology ring of the affine Grassmannian and the equivariant quantum cohomology ring.