Real orbits of Complex Flag Varieties

Even though there is already a good answer to the question I would like to add that Aomoto's paper just marks the beginning of an extensive body of research on open $G_0$-orbits on $G/B$ or, more generally, on $G/P$ for any parabolic $P\subseteq G$. These open orbits are often called flag domains.

After Aomoto's paper one should mention a paper by Wolf (BAMS, 1969) in which general parabolics where considered. In particular, he derives a parameterization of flag domains in terms of double cosets inside the Weyl group of $G$. There is also a monograph by Fels-Huckleberry-Wolf on this topic and a survey by Akhiezer (Real group orbits on flag manifolds).

Finally, there is the Matsuki correspondence, setting up a bijection between the set of $G_0$- and the set of $K$-orbits in $G/P$. Here $K$ is the complexification of a maximal compact subgroup $K_0\subseteq G_0$. Hereby, open $G_0$-orbits correspond to compact (i.e. closed) $K$-orbits. In general, $K$-orbits on $G/B$ have been studied by Springer (some of them with Richardson) in a series of papers.

Aomoto (1966) gives a formula for the number of open orbits (page 15, between (46) and (47)), and specializes it to various special cases (just search the paper for "open").