On D. Peterson's presentation of quantum cohomology of $G/P$

TL;DR

This paper proves a general isomorphism between the T-equivariant quantum cohomology of flag varieties and coordinate rings of certain strata in the Peterson scheme, confirming a conjecture originally discovered by Dale Peterson.

Contribution

It establishes the full generality of Peterson's unpublished result linking quantum cohomology and the Peterson scheme for any flag variety G/P.

Findings

Isomorphism between quantum cohomology and Peterson scheme coordinate ring

Use of Yun-Zhu's isomorphism and Peterson-Lam-Shimozono's homomorphism

General proof applicable to all flag varieties G/P

Abstract

We prove in full generality that the $T$-equivariant quantum cohomology of any flag variety $G/P$ is isomorphic to the coordinate ring of a stratum of the Peterson scheme associated to the Langlands dual group scheme $G^{\vee}$. This result was discovered by Dale Peterson but remains unpublished. Our isomorphism is constructed using Yun-Zhu's isomorphism and Peterson-Lam-Shimozono's homomorphism.