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

TL;DR

This paper provides a new proof linking the structure constants of quantum cohomology of flag varieties with those of the affine Grassmannian, using Gromov-Witten theory and generalized Seidel representations.

Contribution

It introduces a novel approach to identify structure constants in quantum cohomology and affine homology via Gromov-Witten invariants, confirming Peterson's unpublished result.

Findings

Explicit construction of an algebra homomorphism matching Peterson's map

Complete determination of relevant Gromov-Witten invariants

Establishment of the affine analogue of the 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We give a new proof of an unpublished result of Dale Peterson, proved by Lam and Shimozono, which identifies explicitly the structure constants, with respect to the quantum Schubert basis, for the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 $QH^{\bullet}_T(G/P)$ of any flag variety $G/P$ with the structure constants, with respect to the affine Schubert basis, for the $T$-equivariant Pontryagin homology $H^T_{\bullet}(\mathcal{G}r)$ of the affine Grassmannian $\mathcal{G}r$ of $G$, where $G$ is any simple simply-connected complex algebraic group. Our approach is to construct an $H_T^{\bullet}(pt)$-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 $H^T_{\bullet}(\mathcal{G}r)\otimes QH_T^{\bullet}(G/P)$. We determine these invariants completely, in a way similar to how Fulton and Woodward did in their proof of quantum Chevalley formula.