On Generalized $h$--Vectors of Rational Polytopes with a Symmetry of Prime Order

We prove tight lower bounds for the coefficients of the generalized $h$-vector of a rational polytope with a symmetry of prime order that is fixed–point–free on the boundary. These bounds generalize results of R.~Stanley and R.~Adin for the $h$–vector of a simplicial rational polytope with a central symmetry or a symmetry of prime order respectively.