Essential Killing Fields of Parabolic Geometries

We study vector fields generating a local flow by automorphisms of a parabolic geometry with higher-order fixed points. We develop general tools extending the techniques of [22], [13], and [14], and we apply them to almost-Grassmannian, almost-quaternionic, and contact parabolic geometries, including CR structures. We obtain descriptions of the possible dynamics of such flows near the fixed point and strong restrictions on the curvature; in some cases, we can show vanishing of the curvature on a non-empty open set. Deriving consequences for a specific geometry entails evaluating purely algebraic and representation-theoretic criteria in the model homogeneous space.