Closure relations among generalized Peterson strata

Prove that for Schubert varieties \(X_{w,P}\) and \(X_{w',P'}\), the conditions \(P'\subseteq P\) and \(w\equiv w'\pmod{W_P}\) imply the closure relation \(Y^{\mathrm{Div}}_{w,P}\subseteq\overline{Y^{\mathrm{Div}}_{w',P'}}\) between the corresponding generalized Peterson varieties.

Background

The classical Peterson variety has closure relations among its strata indexed by parabolic subgroups. The paper derives necessary conditions for a closure relation between generalized Peterson varieties, including a Bruhat-order condition and a root-theoretic nonintersection condition.

It then proposes the displayed sufficient condition as a conjectural extension of the classical closure relations. Such relations are expected to underpin functorial relationships among quantum cohomology rings of Schubert varieties.