  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.
