Coordinate rings of regular nilpotent Hessenberg varieties in the open opposite Schubert cell

Dale Peterson has discovered a surprising result that the quantum cohomology ring of the flag variety $\mbox{GL}_n(\mathbb{C})/B$ is isomorphic to the coordinate ring of the intersection of the Peterson variety $\mbox{Pet}_n$ and the opposite Schubert cell associated with the identity element $\Omega_e^\circ$ in $\mbox{GL}_n(\mathbb{C})/B$. This is an unpublished result, so papers of Kostant and Rietsch are referred for this result. An explicit presentation of the quantum cohomology ring of $\mbox{GL}_n(\mathbb{C})/B$ is given by Ciocan-Fontanine and Givental-Kim. In this paper we introduce further quantizations of their presentation so that they reflect the coordinate rings of the intersections of regular nilpotent Hessenberg varieties $\mbox{Hess}(N,h)$ and $\Omega_e^\circ$ in $\mbox{GL}_n(\mathbb{C})/B$. In other words, we generalize the Peterson's statement to regular nilpotent Hessenberg varieties via the presentation given by Ciocan-Fontanine and Givental-Kim. As an application of our theorem, we show that the singular locus of the intersection of some regular nilpotent Hessenberg variety $\mbox{Hess}(N,h_m)$ and $\Omega_e^\circ$ is the intersection of certain Schubert variety and $\Omega_e^\circ$ where $h_m=(m,n,\ldots,n)$ for $1<m<n$. We also see that $\mbox{Hess}(N,h_2) \cap \Omega_e^\circ$ is related with the cyclic quotient singularity.