Tractor Bundles in Cr Geometry

This is an outline of my lecture at the Hayama Symposium on several complex variables on December 21, 2002. The subject of the talk are new applications of the canonical Cartan connection for CR manifolds. Most of the results I will report on are based on joint research (partly in progress) with A.R. Gover and with J. Slovák and V. Souček. Partly, these results are specific for the class of CR structures but many of them have analogs in the much more general setting of Cartan geometries corresponding to arbitrary parabolic subgroups in semisimple Lie groups, the so–called parabolic geometries. A central ingredient in these new developments is a special class of natural vector bundles called tractor bundles. These are equivalent to the Cartan bundle and Cartan connection and directly lead to an invariant calculus.