Cartan decomposition and K-orbits in spherical varieties

Cartan decomposition is a generalization of the well-known Singular Values Decomposition from linear algebra to semisimple Lie groups. After covering some background material I will discuss ongoing work to parametrize K-orbits in a so-called spherical homogeneous space. When the homogeneous space is a semisimple group one recovers Cartan decomposition. This is a joint work with Megumi Harada, Johannes Hofscheier and Victor Batyrev. I will try to make (main parts of) the talk accessible to graduate students in algebra and geometry.