*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 Δ^{X} be the corresponding Laplace-Beltrami operator. For *t* > 0, consider the trace of the heat kernel Tr [exp (*t*Δ^{X}/2)]. If is the Hilbert space of square-integrable functions on *X*, Tr [exp (*t*Δ^{X}/2)] is the trace of the ‘group element’ ...