Peterson-Lam-Shimozono’s theorem is an affine analogue of quantum Chevalley formula

We give a new proof of an unpublished result of Dale Peterson, proved by Lam and Shimozono, which identifies explicitly the structure constants, with respect to the quantum Schubert basis, for the T -equivariant quantum cohomology QH•T(G/P) of any flag variety $G/P$ with the structure constants, with respect to the affine Schubert basis, for the T -equivariant Pontryagin homology HT•(Gr) of the affine Grassmannian Gr of G , where G is any simple simply-connected complex algebraic group. Our approach is to construct an HT•(pt) -algebra homomorphism by Gromov-Witten theory and show that it is equal to Peterson’s map. More precisely, the map is defined via Savelyev’s generalized Seidel representations, which can be interpreted as certain Gromov-Witten invariants with input HT•(Gr)⊗ QHT•(G/P) . We determine these invariants completely, in a way similar to how Fulton and Woodward did in their proof of the quantum Chevalley formula.