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

Chow, Chi Hong

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e62

Résumé

We give a new proof of an unpublished result of Dale Peterson, proved by Lam and Shimozono, which identifies explicitly the structure constants, with respect to the quantum Schubert basis, for the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

Détail

DOI : 10.1017/fms.2025.24

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