On uniqueness of submaximally symmetric parabolic geometries

Abstract

Among the (regular, normal) parabolic geometries of type [Formula: see text], there is a locally unique maximally symmetric structure and it has the symmetry dimension [Formula: see text]. The symmetry gap problem concerns the determination of the next realizable (submaximal) symmetry dimension. When [Formula: see text] is a complex or split-real simple Lie group of rank at least three or when [Formula: see text], we establish a local uniqueness result for submaximally symmetric structures of type [Formula: see text].