Toric orbifolds associated with partitioned weight polytopes in classical types

Given a root system $\Phi$ of type $A_n$, $B_n$, $C_n$, or $D_n$ in Euclidean space $E$, let $W$ be the associated Weyl group. For a point $p \in E$ not orthogonal to any of the roots in $\Phi$, we consider the $W$-permutohedron $P_W$, which is the convex hull of the $W$-orbit of $p$. The representation of $W$ on the rational cohomology ring $H^\ast(X_\Phi)$ of the toric variety $X_\Phi$ associated to (the normal fan to) $P_W$ has been studied by various authors. Let $\{s_1,\ldots,s_n\}$ be a complete set of simple reflections in $W$. For $K \subseteq [n]$, let $W_K$ be the standard parabolic subgroup of $W$ generated by $\{s_k:k \in K\}$. We show that the fixed subring $H^\ast(X_\Phi)^{W_K}$ is isomorphic to the cohomology ring of the toric variety $X_\Phi(K)$ associated to a polytope obtained by intersecting $P_W$ with half-spaces bounded by reflecting hyperplanes for the given generators of $W_K$. By a result of Balibanu--Crooks, the cohomology rings $H^\ast(X_\Phi(K))$ are isomorphic with cohomology rings of certain regular Hessenberg varieties.