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We study the equivariant Hilbert series of the complex Stanley-Reisner ringof a simplicial complex $\Sigma$ under the action of a finite group viasimplicial automorphisms. We prove that, when $\Sigma$ is Cohen-Macaulay andthe group action is translative, i.e., color-preserving for some propercoloring of $\Sigma$, the equivariant Hilbert series of $\Sigma$ admits arational expression whose numerator is a positive integer combination ofirreducible characters.  We relate such a result to equivariant Ehrhart theory via an equivariantanalogue of the Betke-McMullen theorem developed by Jochemko, Katth\"an andReiner: namely, if a lattice polytope admits a lattice unimodular triangulationthat is invariant under the action of a finite group, then the equivariantEhrhart series of the polytope equals the equivariant Hilbert series of such atriangulation.  As an application, we study the equivariant Ehrhart series of alcovedpolytopes in the sense of Lam and Postnikov and derive explicit results in thecase of order polytopes and of Lipschitz poset polytopes.

Equivariant Hilbert and Ehrhart series under translative group actions