This paper presents new characterization for a non-compact Hermitian symmetric space $G/K$ to be of tube type (or non-tube type) by multiplicities in some branching laws and visible actions. The study in this paper gives an example of a kind of the Cartan decomposition for non-symmetric homogeneous spaces.
"Admissible representations, multiplicity-free representations and visible actions on non-tube type Hermitian symmetric spaces." Proc. Japan Acad. Ser. A Math. Sci. 91 (5) 70 - 75, May 2015. https://doi.org/10.3792/pjaa.91.70