Hypoelliptic Laplacian and orbital integrals / Jean-Michel Bismut.

Format:

Book

Author/Creator:

Bismut, Jean-Michel.

Series:

Annals of mathematics studies ; no. 177.

Annals of mathematics studies ; no. 177

Language:

English

Subjects (All):

Differential equations, Hypoelliptic.

Laplacian operator.

Definite integrals.

Orbit method.

Physical Description:

1 online resource (320 p.)

Edition:

Course Book

Other Title:

Odilo Préstam-e (Servicio en línea)

Place of Publication:

Princeton : Princeton University Press, 2011.

Language Note:

English

Summary:

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. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

Contents:

Frontmatter

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 [k (γ) ; p0] = 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 qXb;t for bounded b

Chapter Fifteen. The heat kernel qXb;t for b large

Bibliography

Subject Index

Index of Notation

Notes:

Description based upon print version of record.

Includes bibliographical references and indexes.

ISBN:

9786613163875

9781283163873

128316387X

9781400840571

1400840570

OCLC:

747411206