Introduction The purpose of this book is to use the hypoelliptic Laplacian to evaluate semisimple orbital integrals, in a formalism that unifies index theory and the trace formula. 0.1 The trace formula as a Lefschetz formula Let us explain how to think formally of such a unified treatment, while allowing ourselves a temporarily unbridled use of mathematical analogies. Let X be a compact Riemannian manifold, and let X be the corresponding Laplace-Beltrami operator. For t > 0, consider the trace of the heat kernel Tr exp t X /2 . If L X is the Hilbert space of square-integrable functions 2 on X, Tr exp t X /2 is the trace of the `group element' exp t X /2 acting on L X . 2 Suppose that L X is the cohomology of an acyclic complex R on which X 2 acts. Then Tr exp t X /2 can be viewed as the evaluation of a Lefschetz