A Lefschetz fixed point formula for elliptic differential operators

Introduction. The classical Lefschetz fixed point formula expresses, under suitable circumstances, the number of fixed points of a continuous map ƒ : X-+X in terms of the transformation induced by ƒ on the cohomology of X. If X is not just a topological space but has some further structure, and if this structure is preserved by ƒ, one would expect to be able to refine the Lefschetz formula and to say more about the nature of the fixed points. The purpose of this note is to present such a refinement (Theorem 1) when X is a compact differentiable manifold endowed with an elliptic differential operator (or more generally an elliptic complex). Taking essentially the classical operators of complex and Riemannian geometry we obtain a number of important special cases (Theorems 2,3) . The first of these was conjectured to us by Shimura and was proved by Eichler for dimension one.