# On a new normalization for tractor covariant derivatives

### Matthias Hammerl

Universität Wien, Austria### Petr Somberg

Charles University, Prague, Czech Republic### Vladimír Souček

Charles University, Prague, Czech Republic### Josef Šilhan

Masaryk University, Brno, Czech Republic

## Abstract

A regular normal parabolic geometry of type $G/P$ on a manifold $M$ gives rise to sequences $D_i$ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative $\nabla^\omega$ on the corresponding tractor bundle $V,$ where $\omega$ is the normal Cartan connection. The first operator $D_0$ in the sequence is overdetermined and it is well known that $\nabla^\omega$ yields the prolongation of this operator in the homogeneous case $M = G/P$. Our first main result is the curved version of such a prolongation. This requires a new normalization of the tractor covariant derivative on $V$. Moreover, we obtain an analogue for higher operators $D_i$. In that case one needs to modify the exterior covariant derivative $d^{\nabla^\omega}$ 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.

## Cite this article

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. 1859–1883

DOI 10.4171/JEMS/349