Complete simplicial fans, Stanley--Reisner rings, and equivariant h-polynomials

We derive a graded character formula for the action of any finite group on the Artinian reduction of the Stanley–Reisner ring of any complete simplicial fan, which is given by an equivariant version of the classical h-polynomial. This gives the graded character formula for the representation of the group on the cohomology of the associated toric variety when the fan is rational. As an application, we use a navel tool, which we called hybrid fan, to compute the Poincaré polynomial of the invariants of the Artinian reduction of the Stanley–Reisner ring of any complete simplicial fan under a finite reflection group action. This implies that the Poincaré polynomial of the quotient of a compact toric orbifold by any finite reflection group is equal to the Poincaré polynomial of the compact toric orbifold associated to the hybrid fan.