Geometry of the Shilov boundary
Let D be a bounded symmetric domain, G its group of holomorphic diffeomorphisms, S its Shilov boundary. After recalling the description of the G-orbits in SXS (two versions), we look at the G-orbits in SxSxS. Using the Kähler form of D and a limit process to the boundary S, we first produce an invariant on SxSxS. We then explain the difference between tube-type domains and non tube-type domains. In the case of tube-type domain, there is a finite number of G-orbits in SxSxS, the invariant co?ncides with the generalized Maslov index and a complete classification of G-orbits in SxSxS is obtained.