We introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type {C_{n}} and admits a contact grading. We show that a structure of each of these types on a smooth manifold M determines a canonical compatible linear connection on the tangent bundle {\mathrm{TM}}. This connection is characterized by a normalization condition on its torsion. The algebraic background for this result is proved using Kostant’s theorem on Lie algebra cohomology. For each type, we give an explicit description of both the geometric structure and the normalization condition. In particular, the torsion of the canonical connection naturally splits into two components, one of which is exactly the obstruction to the underlying structure being conformally symplectic. This article is the first in a series aiming at a construction of differential complexes naturally associated to these geometric structures.

Parabolic almost conformally symplectic structures were introduced in the first part of this series of articles as a class of geometric structures which have an underlying almost conformally symplectic structure. If this underlying structure is conformally symplectic, then one obtains a PCS-structure. In the current article, we relate PCS-structures to parabolic contact structures. Starting from a parabolic contact structure with a transversal infinitesimal automorphism, we first construct a natural PCS-structure on any local leaf space of the corresponding foliation. Then we develop a parabolic version of contactification to show that any PCS-structure can be locally realized (uniquely up to isomorphism) in this way. In the second part of the paper, these results are extended to an analogous correspondence between contact projective structures and so-called conformally Fedosov structures. The developments in this article provide the technical background for a construction of sequences and complexes of differential operators which are naturally associated to PCS-structures by pushing down BGG sequences on parabolic contact structures. This is the topic of the third part of this series of articles.

Given a contact manifold $M_#$ together with a transversal infinitesimal automorphism $\xi$, we show that any local leaf space $M$ for the foliation determined by $\xi$ naturally carries a conformally symplectic (cs-) structure. Then we show that the Rumin complex on $M_#$ descends to a complex of differential operators on $M$, whose cohomology can be computed. Applying this construction locally, one obtains a complex intrinsically associated to any manifold endowed with a cs-structure, which recovers the generalization of the so-called Rumin-Seshadri complex to the conformally symplectic setting. The cohomology of this more general complex can be computed using the push-down construction.Comment: 13 pages v2: added reference [7] to earlier construction of the Rumin-Seshadri complex in a conformally symplectic setting; changed terminonolgy from "locally conformally symplectic" to "conformally symplectic"; some other minor changes; finaly version to appear in Differential Geom. App

The principal group of a Klein geometry has canonical left action on the homogeneous space of the geometry and this action induces action on the spaces of sections of vector bundles over the homogeneous space. This paper is about construction of differential operators invariant with respect to the induced action of the principal group of a particular type of parabolic geometry. These operators form sequences which are related to the minimal resolutions of the k-Dirac operators studied in Clifford analysis.

Abstract. This is the first paper in a series of two papers. In this paper we construct complexes of invariant differential operators which live on homogeneous spaces of |2|-graded parabolic geometries of some particular type. We call them k-Dirac complexes. More explicitly, we will show that each k-Dirac complex arises as the direct image of a relative BGG sequence and so this fits into the scheme of the Penrose transform. We will also prove that each k-Dirac complex is formally exact, i.e., it induces a long exact sequence of infinite (weighted) jets at any fixed point. In the second part of the series we use this information to show that each k-Dirac complex is exact at the level of formal power series at any point and that it descends to a resolution of the k-Dirac operator studied in Clifford analysis.