5
$\begingroup$

If $G$ is a complex semisimple Lie group,and $B$ is a Borel group, we can form the flag variety $G/B$. If $G_R$ is a real form of $G$, we can then let $G_R$ act of $G/B$ on the left and consider the orbit space $G_R\setminus G/B$. I have seen discussions of the open orbits, but is there a reference that classifies all the orbits? In particular I am interested in the case $SL_n(H)\setminus SL_{2n}(C)/B$

$\endgroup$
4
$\begingroup$

Classic paper: Joseph A. Wolf (1969), The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components.

Recent survey: Dmitri Akhiezer (2013), Real group orbits on flag manifolds.

$\endgroup$
  • $\begingroup$ This is good but what I really would like is an exact characterization of the orbits, and if I'm not mistaken these sources only give an upper bound. In particular the first source (page 1131) gives some orbits indexed by $P_{\phi}$ and then says they may not be distinct. Is there some way to tell which of these are equal to each other? $\endgroup$ – A.D. Sep 8 '16 at 19:30
1
$\begingroup$

For the real forms of G=SL(n,C) you can also find concrete descriptions, i.e. in terms of flags/ combinatorial parametrization,of the open orbits on G/P and for SU(p,q) of the unique closed orbit, here http://arxiv.org/abs/1411.0202. If I'm not mistaken there might be some people working now on writing down concretely all of the G_0-orbits for the classical semisimple Lie groups. Of course most of these can already be found in the papers of Joseph A. Wolf.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.