Abstract: For Cartan geometries of a given type, the maximal amount of symmetry is realized by the flat model. However, if the geometry is not (locally) flat, how much symmetry can it have? Understanding this "gap" between maximal and submaximal symmetry in the case of parabolic geometries is the subject of this talk. We show how representation-theoretic considerations involving a combination of Kostant's version of the Bott-Borel-Weil theorem and Tanaka prolongation lead to (often sharp) results on the submaximal dimension. (Joint work with Boris Kruglikov.)