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

Abstract We give a new proof of an unpublished result of Dale Peterson, proved by Lam and Shimozono, which identifies explicitly the structure constants, with respect to the quantum Schubert basis, for the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.