Hypoelliptic Laplacian and Orbital Integrals

AM-177

Jean-Michel Bismut

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English

Princeton University Press
28 August 2011

Analytic geometry; Algebraic geometry

Series: Annals of Mathematics Studies

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.

By: Jean-Michel Bismut
Imprint: Princeton University Press
Country of Publication: United States
Volume: 200
Dimensions: Height: 235mm, Width: 152mm, Spine: 20mm
Weight: 482g
ISBN: 9780691151304
ISBN 10: 069115130X
Series: Annals of Mathematics Studies
Pages: 344
Publication Date: 28 August 2011
Audience: College/higher education , Professional and scholarly , Primary , Undergraduate
Format: Paperback
Publisher's Status: Active

Jean-Michel Bismut is professor of mathematics at the Universit Paris-Sud, Orsay.