Essential Killing fields of parabolic geometries

TL;DR

This paper investigates the behavior of vector fields generating automorphisms in parabolic geometries with higher order fixed points, providing new tools and results that restrict curvature and describe local dynamics.

Contribution

It develops general methods to analyze automorphism flows with fixed points in various parabolic geometries, extending previous techniques and applying them to specific structures.

Findings

Restrictions on curvature near fixed points

Conditions for curvature vanishing on open sets

Descriptions of local flow dynamics

Abstract

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]. We apply these tools to almost Grassmannian, almost quaternionic, and contact parabolic geometries, including CR structures, to 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. Published in Indiana University Mathematics Journal.