Introduction

This introductory chapter reviews the hypoelliptic Laplacian. It first explains how to mathematically conceive of a union between index theory and the trace formula through the Lefschetz fixed point formulas. The chapter then embarks on a brief history of the hypoelliptic Laplacian, hereafter turning to the construction of the hypoelliptic Laplacian that is carried out in this volume. Moreover, it discusses the analysis of the hypoelliptic orbital integrals, and its overlap with the analysis of the hypoelliptic Laplacian in previous literature, in which the Riemannian manifold X was assumed to be compact, and genuine traces or supertraces were considered. Here in this chapter X is noncompact, and the orbital integrals that appear are defined using explicit properties of the corresponding heat kernels. After this review, the chapter gives a short overview on the following chapters.