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

Abstract

Let $X$ be a rational homogeneous space and let $QH^*(X)_{loc}^\times$ be the group of invertible elements in the small quantum cohomology ring of $X$ localised in the quantum parameters. We generalise results of \cite{cmp2} and realise explicitly the map $\pi_1({\rm Aut}(X))\to QH^*(X)_{loc}^\times$ described in \cite{seidel}. We even prove that this map is an embedding and realise it in the equivariant quantum cohomology ring $QH^*_T(X)_{loc}^\times$. 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 \cite{Magyar}. It also uses Peterson's unpublished result \cite{Peterson} --- recently proved by Lam and Shimozono in \cite{Lam-Shi} --- on the comparison between the equivariant homology ring of the affine Grassmannian and the equivariant quantum cohomology ring.