A Cartan decomposition for non-symmetric reductive spherical pairs of rank-one type and its application to visible actions

TL;DR

This paper introduces a new Cartan decomposition for non-symmetric reductive spherical pairs of rank-one type and demonstrates the strong visibility of certain group actions on these spaces, expanding understanding of their geometric and harmonic properties.

Contribution

The paper provides the first examples of Cartan decompositions for non-symmetric reductive spherical pairs of rank-one type, extending the framework beyond symmetric cases.

Findings

New Cartan decomposition for non-symmetric rank-one spherical pairs

Strong visibility of compact group actions on these spaces

Enhanced understanding of orbit geometry and harmonic analysis

Abstract

A Cartan decomposition for symmetric pairs plays an important role to study not only orbit geometry of the symmetric spaces but also harmonic analysis on them. For non-symmetric reductive pairs, there are examples of generalizations of Cartan decompositions for some spherical complex homogeneous spaces such as complex line bundles over the complexified Hermitian symmetric spaces and triple spaces. This paper provides new examples of a Cartan decomposition for non-symmetric reductive pairs, namely, reductive non-symmetric spherical pairs of rank-one type. We also show that the action of some compact group on a non-symmetric reductive spherical homogeneous space of rank-one type is strongly visible.