Geometric properties of homogeneous parabolic geometries with generalized symmetries

TL;DR

This paper studies homogeneous parabolic geometries with generalized symmetries, showing they can be simplified to known structures, providing explicit interpretations, and classifying non-trivial cases.

Contribution

It introduces a reduction method for these geometries, linking them to generalized symmetric spaces and offering explicit classifications.

Findings

Reductions correspond to known generalized symmetric spaces

Explicit example illustrating the reduction process

Complete classification of non-trivial cases

Abstract

We investigate geometric properties of homogeneous parabolic geometries with generalized symmetries. We show that they can be reduced to a simpler geometric structures and interpret them explicitly. For specific types of parabolic geometries, we prove that the reductions correspond to known generalizations of symmetric spaces. In addition, we illustrate our results on an explicit example and provide a complete classification of possible non--trivial cases.