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arXiv · 1303.1307

The gap phenomenon in parabolic geometries

Boris Kruglikov·Dennis The
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Abstract

The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P), we use Tanaka theory to derive a universal upper bound on the submaximal symmetry dimension. We use Kostant's version of the Bott-Borel-Weil theorem to show that this bound is in fact sharp in almost all complex and split-real cases by exhibiting (abstract) models. We explicitly compute all submaximal symmetry dimensions when G is any complex or split-real simple Lie group.

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Boris Kruglikov, Dennis The. 2013-03-06. The gap phenomenon in parabolic geometries. https://doi.org/10.1515/crelle-2014-0072

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