Download e-book for iPad: Hypoelliptic Laplacian and Orbital Integrals (AM-177) by Jean-Michel Bismut

This publication makes use of the hypoelliptic Laplacian to guage semisimple orbital integrals in a formalism that unifies index idea and the hint formulation. The hypoelliptic Laplacian is a kin of operators that's alleged to interpolate among the normal Laplacian and the geodesic circulation. it's primarily the weighted sum of a harmonic oscillator alongside the fiber of the tangent package, and of the generator of the geodesic circulate. during this booklet, semisimple orbital integrals linked to the warmth kernel of the Casimir operator are proven to be invariant lower than an appropriate hypoelliptic deformation, that is built utilizing the Dirac operator of Kostant. Their particular evaluate is acquired by way of localization on geodesics within the symmetric house, in a formulation heavily relating to the Atiyah-Bott fastened element formulation. Orbital integrals linked to the wave kernel also are computed.

Estimates at the hypoelliptic warmth kernel play a key position within the proofs, and are received by means of combining analytic, geometric, and probabilistic recommendations. Analytic options emphasize the wavelike facets of the hypoelliptic warmth kernel, whereas geometrical issues are had to receive right regulate of the hypoelliptic warmth kernel, particularly within the localization approach close to the geodesics. Probabilistic recommendations are specifically proper, simply because underlying the hypoelliptic deformation is a deformation of dynamical structures at the symmetric house, which interpolates among Brownian movement and the geodesic stream. The Malliavin calculus is used at serious phases of the proof.