Peterson-Lam-Shimozono's theorem is an affine analogue of quantum Chevalley formula
Chi Hong Chow
TL;DR
The paper provides a geometric proof of the Peterson–Lam–Shimozono correspondence, linking the $T$-equivariant quantum cohomology of flag varieties to the $T$-equivariant Pontryagin homology of the affine Grassmannian via a Savelyev–Seidel construction. It introduces a graded $H_T^{\bullet}(pt)$-algebra homomorphism $\Phi_{SS}$ built from Gromov–Witten invariants of $G/P$-bundles induced by maps to the affine Grassmannian, and proves that this map reproduces Peterson’s map after localization. The analysis hinges on regularity and transversality of moduli spaces of sections, precise degree computations, and a detailed combinatorial description of zero-dimensional components, culminating in an explicit formula for $\Phi_{SS}(\xi_{wt_{\lambda}})$ that matches the affine Schubert data with quantum Schubert data. The approach offers an alternate, geometric proof of the corresponding parabolic version and provides a framework to interpret structure constants in both calculi via counts of section moduli.
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We give a new proof of an unpublished result of Dale Peterson, proved by Lam and Shimozono, which identifies explicitly the structure constants, with respect to the quantum Schubert basis, for the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.