Abstract: Let G be a semisimple Lie group and P a closed subgroup of G. Any representation of P gives rise to a homogeneous vector bundle over the homogenous space G/P and a representation of G on the space of smooth sections of this bundle. If the subgroup P is parabolic, then this is the process of parabolic induction and starting from irreducible representations of P, one obtains principal series representations of G. On smooth sections, also the Lie algebra of G acts naturally, and the corresponding Casimir operator is a canonical G-equivariant map on the representation.

We show that this Casimir operator naturally extends to an invariant operator acting on sections of the natural bundle on parabolic geometries of type (G,P) defined by the representation. We prove that the resulting operator is always of first order. For an irreducible inducing representation, the operator acts by a scalar which can be computed using (finite dimensional) representation theory.

We show how these rather simple observations can be used to construct a large number of invariant differential operators, among them the splitting operators used in the construction of BGG sequences and conformally invariant powers of the Laplician.

Reference: A. Cap, V. Soucek, arXiv:0708.3180