A remark on the orbit structure of the complexification of a semisimple symmetric space

We consider the action of a real semisimple Lie group G   on the complexification GC/HCGC/HC of a semisimple symmetric space G/HG/H and we present a refinement of Matsukiʼs results (Matsuki, 1997 [1]) in this case. We exhibit a finite set of points in GC/HCGC/HC, sitting on closed G-orbits of locally minimal dimension, whose slice representation determines the G  -orbit structure of GC/HCGC/HC. Every such point p¯ lies on a compact torus and occurs at specific values of the restricted roots of the symmetric pair (g,h)(g,h). The slice representation at p¯ is equivalent to the isotropy representation of a real reductive symmetric space, namely ZG(p4)/Gp¯. In principle, this gives the possibility to explicitly parametrize all G  -orbits in GC/HCGC/HC.