A Formula of Atiyah–Bott–Lefschetz Type and Its Application to Operators with a Finite Group of Shifts

Abstract

The Lefschetz number of an endomorphism of an elliptic complex is expressed in terms of regularized traces of the operators defining the endomorphism. This result is obtained under certain conditions on the wavefront sets of the operators in question. In the particular case of geometric endomorphisms of the complex, we obtain the classical Atiyah–Bott formula. As an application, we compute the Lefschetz numbers of nonlocal elliptic operators associated with an action of a finite group on a closed smooth manifold. For the de Rham complex, this gives a formula for the Lefschetz number in the cohomology of the orbit space in terms of fixed points.