Hypoelliptic laplacian and orbital integrals -- Contents -- Acknowledgments -- Introduction -- Chapter One: Clifford and Heisenberg algebras -- Chapter Two: The hypoelliptic Laplacian on X = G/K -- Chapter Three: The displacement function and the return map -- Chapter Four: Elliptic and hypoelliptic orbital integrals -- Chapter Five: Evaluation of supertraces for a model operator -- Chapter Six: A formula for semisimple orbital integrals -- Chapter Seven: An application to local index theory -- Chapter Eight: The case where [ƭ (γ), po]=0 -- Chapter Nine: A proof of the main identity -- Chapter Ten: The action functional and the harmonic oscillator -- Chapter Eleven: The analysis of the hypoelliptic Laplacian -- Chapter Twelve: Rough estimates on the scalar heat kernel -- Chapter Thirteen: Refined estimates on the scalar heat kernel for bounded b -- Chapter Fourteen: The heat kernel Xb,t for bounded b -- Chapter Fifteen: The heat kernel Xb,t for b large -- Bibliography -- Subject Index -- Index of Notation

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed.