On invariants of representations of Weyl groups associated with the cohomology of toric varieties

For a Weyl group $W$ and a $W$-permutohedron $P$, there are associated toric varieties $X_P$ and $X_{P\mathrm{/}{W}_K}$ for any parabolic subgroup $W_K$ of $W$, since the quotient $P\mathrm{/}{W}_K$ can be identified with a polytope inside $P$. We construct an explicit algebra isomorphism between $H^{\ast }(X_{P\mathrm{/}{W}_K};\mathbb{Q})$ and $H^{\ast }(X_P;\mathbb{Q}{)}^{W_K}$. We further generalize this isomorphism to intermediate lattices, to finite Coxeter groups, and to non-degenerate $W$-symmetric polytopes. Our results give affirmative answers to two open questions of Horiguchi–Masuda–Shareshain–Song.