On a new normalization for tractor covariant derivatives

A regular normal parabolic geometry of type $G/P$ on a manifold $M$ givesrise to sequences $D_i$ of invariant differential operators, known as thecurved version of the BGG resolution. These sequences are constructed from thenormal covariant derivative $\na^\om$ on the corresponding tractor bundle $V,$where $\om$ is the normal Cartan connection. The first operator $D_0$ in thesequence is overdetermined and it is well known that $\na^\om$ yields theprolongation of this operator in the homogeneous case $M = G/P$. Our first mainresult is the curved version of such a prolongation. This requires a newnormalization $\tilde{\na}$ of the tractor covariant derivative on $V$.Moreover, we obtain an analogue for higher operators $D_i$. In that case oneneeds to modify the exterior covariant derivative $d^{\na^\om}$ by differentialterms. Finally we demonstrate these results on simple examples in projectiveand Grassmannian geometry. Our approach is based on standard techniques of theBGG machinery.