Invariant operators for parabolic geometries I

The parabolic geometries are curved deformations of (real) homogeneous

spaces G/P, such that the complexification

is a parabolic subgroup in a semisimple Lie group. The general ideas go back

to Cartan's 'generalized spaces'. Examples involve the conformal,

projective, almost quaternionic, and CR geometries.

The first talk will present an introduction, survey of recent existence

results, and a more detailed explanation of new techniques leading to an

effective calculus for invariant operators. The explicit construction of

curved analogues of the famous Bernstein-Gel'fand-Gel'fand resolutions for

all these geometries (without any use of its representational theoretical

version) will be presented as the first major application of our new

approach.

The second talk will focus on the parabolic geometries with irreducible

tangent bundles. In this case, all the standard geometrical structures known

from conformal Riemannian geometries admit nice general counterparts and,

in particular, there is a more straightforward explicit construction of the

operators from the above mentioned resolutions based on finite dimensional

representation theory. It is remarkable that the closed formulae for all

these operators of a given order do not depend on the choice of the

structure groups.