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Multicontact vector fields on Hessenberg manifolds. (English) Zbl 1077.22019

Recently, there has been much interest in differential geometry modelled on homogeneous spaces of the form \(G/P\) where \(G\) is a semisimple Lie group and \(P\) is a parabolic subgroup. This ‘parabolic differential geometry’ generalises several classical cases including conformal and CR geometry. There are many features of the classical cases that hold in general for parabolic geometries. The existence of a preferred Cartan connection is closely related to a ‘Liouville theorem’ on the flat model: the local symmetries of \(G/P\), i.e.those preserving its parabolic structure, all extend to global symmetries induced by the action of \(G\).
As a potentially weaker notion, F. De Mari, M. Cowling, A. Korányi and H. M. Reimann [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 13, 219–232 (2002; Zbl 1225.22012)] have suggested that, for a ‘Liouville theorem’, it is enough to insist that a natural collection of special sub-bundles of the tangent bundle of \(G/P\) be preserved. Since these sub-bundles have the property that repeated Lie brackets of their sections generate the full tangent bundle, they call them ‘multicontact’ structures. These authors have proved such a Liouville-type theorem in the case that \(P\) is minimal parabolic and \(G\) has rank greater than one.
In this article, the author shows that an infinitesimal version of this result extends to Hessenberg submanifolds of \(G/P\) with \(P\) minimal. Hessenberg submanifolds are defined by combinatorial data, mainly a proper subset \({\mathcal{R}}\) of the positive roots defining \(P\) (subject to certain further conditions). These submanifolds come equipped with preferred vector fields whose Lie brackets generate the whole tangent bundle. This allows one to define the local notion of multicontact mapping. Provided \({\mathcal{R}}\) satisfies further technical conditions, it is shown that the vector fields corresponding to infinitesimal multicontact mappings all arise from the expected subalgebra of \({\mathfrak{g}}\).

MSC:

22E46 Semisimple Lie groups and their representations
22F30 Homogeneous spaces
53D10 Contact manifolds (general theory)
57S20 Noncompact Lie groups of transformations

Citations:

Zbl 1225.22012
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