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 [1], [2], and [3], 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 nonempty open set. Deriving consequences for a specific geometry entails evaluating purely algebraic and representation-theoretic criteria in the model homogeneous space. Dedicated to Michael Eastwood on the occasion of his 60th birthday.