A regular normal parabolic geometry of type on a manifold gives rise to sequences of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative on the corresponding tractor bundle where is the normal Cartan connection. The first operator in the sequence is overdetermined and it is well known that yields the prolongation of this operator in the homogeneous case . Our first main result is the curved version of such a prolongation. This requires a new normalization of the tractor covariant derivative on . Moreover, we obtain an analogue for higher operators . In that case one needs to modify the exterior covariant derivative by differential terms. Finally we demonstrate these results on simple examples in projective, conformal and Grassmannian geometry. Our approach is based on standard techniques of the BGG machinery.
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Matthias Hammerl, Petr Somberg, Vladimír Souček, Josef Šilhan, On a new normalization for tractor covariant derivatives. J. Eur. Math. Soc. 14 (2012), no. 6, pp. 1859–1883DOI 10.4171/JEMS/349