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Journal of Lie Theory 25 (2015), No. 3, 677--715
Copyright Heldermann Verlag 2015

On Automorphisms with Natural Tangent Actions on Homogeneous Parabolic Geometries

Jan Gregorovic
Department of Mathematics, Aarhus University, Ny Munkegade 118, Aarhus C 8000, Denmark

Lenka Zalabová
Institute of Mathematics and Biomathematics, Faculty of Science, University of South Bohemia, Branisovská 31, Ceské Budejovice 370 05, Czech Republic

We consider automorphisms of homogeneous parabolic geometries with a fixed point. Parabolic geometries carry the distinguished distributions and we study those automorphisms which enjoy natural actions on the distributions at the fixed points. We describe the sets of such automorphisms on homogeneous parabolic geometries in detail and classify whether there are non-flat homogeneous parabolic geometries carrying such automorphisms. We present two general constructions of such geometries and we provide complete classifications for the types (G,P) of the parabolic geometries that have G simple and the automorphisms are of order 2.

Keywords: Parabolic geometries, homogeneous spaces, automorphisms with fixed points, harmonic curvature restrictions.

MSC: 53C10; 53C30, 58J70

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