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A Cartan Decomposition for Non-symmetric Reductive Spherical Pairs of Rank-One Type and Its Application to Visible Actions

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 290)

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.

Keywords

Spherical pair Cartan decomposition Cayley algebra Exceptional Lie group 

2010 Mathematics Subject Classification

Primary: 22E46 Secondary: 32M05 22E60 14M17 

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceTokai UniversityHiratsukaJapan

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