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Let $G$ be a semi-simple algebraic group over $\mathbb{C}$ with Borel subgroup $B$ and consider the flag variety $G/B$. If $G_0 \subset G$ is some real form, then $G_0$ acts on $G/B$ and decomposes the space into a disjoint union of orbits. In the case of $SL_2$ one gets 2 open orbits while in $SL_3$ my calculations seem to show a single open orbit. Can one characterize the number of open orbits in general? For what groups is this number larger than 1?

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2 Answers 2

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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").

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  • $\begingroup$ Many thanks for the answer. I apologize if this question is too basic for the site. $\endgroup$
    – Alexander
    Commented Aug 16, 2016 at 1:45
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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.

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  • $\begingroup$ Dear Friedrich, many thanks for your response! $\endgroup$
    – Alexander
    Commented Aug 20, 2016 at 2:07

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