On Relative Tractor Bundles

TL;DR

This paper advances the relative BGG machinery for parabolic geometries by constructing explicit relative tractor bundles and sheaves, especially for Legendrean contact structures and generalized path geometries, under weak torsion assumptions.

Contribution

It introduces a broad family of relative tractor bundles with explicit sheaf descriptions, extending the applicability of the BGG machinery to more geometries with minimal torsion restrictions.

Findings

Explicit description of resolved sheaves for relative tractor bundles

Application to Legendrean contact structures and generalized path geometries

Most relative tractor bundles are obtained by the new construction

Abstract

This article contributes to the relative BGG-machinery for parabolic geometries. Starting from a relative tractor bundle, this machinery constructs a sequence of differential operators that are naturally associated to the geometry in question. In many situations of interest, it is known that this sequence provides a resolution of a sheaf that can locally be realized as a pullback from a local leaf space of a foliation that is naturally available in this situation. An explicit description of the latter sheaf was only available under much more restrictive assumptions. For any geometry which admits relative tractor bundles, we construct a large family of such bundles for which we obtain a simple, explicit description of the resolved sheaves under weak assumptions on the torsion of the geometry. In particular, we discuss the cases of Legendrean contact structures and of generalized path geometries, which are among the most important examples for which the relative BGG machinery is available. In both cases, we show that essentially all relative tractor bundles are obtained by our construction and our description of the resolved sheaves applies whenever the BGG sequence is a resolution.