Prolongation on regular infinitesimal flag manifolds

Many interesting geometric structures can be described as regular infinitesimal flag structures, which occur as the underlying structures of parabolic geometries. Among these structures we have for instance conformal structures, contact structures, certain types of generic distributions and partially integrable almost CR-structures of hypersurface type. The aim of this article is to develop for a large class of (semi-)linear overdetermined systems of partial differential equations on regular infinitesimal flag manifolds $M$ a conceptual method to rewrite these systems as systems of the form $\tilde{\mathrm{\nabla }}(\mathrm{\Sigma })+C(\mathrm{\Sigma })=0$, where $\tilde{\mathrm{\nabla }}$ is a linear connection on some vector bundle $V$ over $M$ and $C:V\to T^{\ast }M\otimes V$ is a (vector) bundle map. In particular, if the overdetermined system is linear, $\tilde{\mathrm{\nabla }}+C$ will be a linear connection on $V$ and hence the dimension of its solution space is bounded by the rank of $V$. We will see that the rank of $V$ can be easily computed using representation theory.