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.
