Generalizing a result of Stanley [13] on centrally symmetric polytopes, Adin [2], [3] has derived tight lower bounds for the face numbers of a rational simplical polytope equipped with a fixed-point-free linear action of a cyclic group G of prime power order. The main goal of this paper is to extend these results further by replacing Adin's fixedpoint-free condition with the assumption that the action of G is proper. As corollaries, we obtain a generalization of Adin's equivariant lower bound theorem and of a condition by Stanley [13] implying combinatorial isomorphism with a minimal polytope. Finally, we prove sufficiency of an equivariant version of the McMullen and Walkup generalized lower bound conjecture.

The object of this paper is to obtain a set of inequalities relating the face numbers of different orbit types of a simplicial polytope P with a finite solvable group G of linear symmetries. It is assumed that (1) for each subgroup H of G, the fixed point set PH is a subpolytope of P, and (2) the toric variety X(P) associated to P is nonsingular. The action of G on P induces an action on X(P), and we describe a set of Smith-type inequalities between the Betti numbers of X(P) H , where H ranges through the set of subgroups of G. By relating each X(P) H with X(P H ), we then express these inequalities in terms of the face numbers of the different orbit types of P and the rank of fixed point sets of certain compact tori. This rank is determined explicitly when G is abelian. Moreover, assumption (2) is removed for a polytope of dimension 2.