Andreas Cap - "BGG sequences and geometric overdetermined systems".

Parabolic geometries form a large class of geometric
structures which admit an equivalent description as Cartan geometries
with homogeneous model a generalized flag manifold, the quotient of a
semisimple Lie group $G$ by a parabolic subgroup $P$. This family
contains many well known examples like conformal structures,
hypersurface type CR structures, almost quaternionic structures, path
geometries, quaternionic contact structures, and several types of
generic distributions.

The technique of BGG sequences associates to each irreducible
representation of $G$ a sequence of differential operators which are
intrinsic to the given structure. The first operator(s) in this
sequence always form an overdetermined system. Many well known
geometric systems arise in this way, for example the conformal Killing
equations on all types of tensors. The construction of the BGG
sequences automatically relates the overdetermined system in the
beginning of the sequence to a linear connection on a bigger bundle.

After outlining this construction, I will indicate how these methods
can be extended to obtain an invariant prolongation of these
overdetermined system (i.e. the construction of an equivalent
invariant first order system in closed form) as well as prolongations
of arbitrary semi--linear systems with the same principal symbol.