Toric orbifolds associated with partitioned weight polytopes in classical types

Given a root system Φ 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∈E not orthogonal to any of the roots in Φ, 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^∗(X_Φ) of the toric variety X_Φ associated to (the normal fan to) P_W has been studied by various authors. Let {s₁,…,s_n} be a complete set of simple reflections in W. For K⊆[n], let W_K be the standard parabolic subgroup of W generated by {s_k:k∈K}. We show that the fixed subring H^∗(XΦ)W^K is isomorphic to the cohomology ring of the toric variety X_Φ(K) associated to a polytope obtained by intersecting P_W with half-spaces bounded by reflecting hyperplanes for the given generators of W_K. We also obtain explicit formulas for h-vectors of these polytopes. By a result of Balibanu–Crooks, the cohomology rings H^∗(X_Φ(K)) are isomorphic with cohomology rings of certain regular Hessenberg varieties.