Index of elliptic operators associated with discrete groups and fixed points

Given an action of a discrete group on a closed smooth manifold, we consider the class of nonlocal elliptic operators generated by pseudodifferential operators on the manifold and shift operators. The operators in this class are Fredholm under suitable ellipticity conditions. In the present paper, we give an index formula for these operators for the case of the group $\mathbb{Z} \times F$, where $F$ is an arbitrary finite group. To solve this index problem, we explicitly write the contribution to the index by the fixed points of finite-order elements of the group. The index formula is stated in terms of characteristic classes in periodic cyclic cohomology.