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Let $G$ be a reductive group, $K$ a symmetric subgroup of $G$ (e.g., fixed point of an involution), and $B$ a Borel subgroup of $G$. Then it is well known that $G/B$ has finitely many $K$-orbits. There is a well-known combinatorial parametrisation of these orbits by Matsuki and Oshima, in a paper Embeddings of discrete series into principal series using the notion of "clans". However, this paper does not include any proofs. Is there a reference for the proofs of this result?

The main difficulty seems to be proving that any two flags with identical clans are $K$-conjugate.

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  • $\begingroup$ My apologies for my edit with the wrong article. I searched the authors, saw that the title seemed to match, and didn't even notice that I had the wrong year. $\endgroup$
    – LSpice
    Commented Mar 9 at 1:29

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A good reference for this is "On Rationality Properties of Involutions of Reductive Groups" by Helminck and Wang (Advances in Math. 1993). The decomposition of $K\backslash G/B$ is studied in all generality (over any base field and for any parabolic subgroup).

For an algebraically closed base field, the description of the double cosets is due to T.A. Springer ("Some Results on Algebraic Groups with Involutions", adv. studies in pure math. 1985).

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  • $\begingroup$ Thanks for the references. However, I am not sure how those papers are related to the Matsuki-Oshima parameterisation by "clans". For example, my understanding is that the work of Springer (and Richardson-Springer) provides a correspondence between orbits and twisted involutions but this correspondence might not always be a bijection. If you could explain where one can find the connection between these works, or where the more concrete method of Matsuki-Oshima is done in more detail, that would be great. $\endgroup$
    – Hadi
    Commented Mar 9 at 1:12
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For what its worth the Atlas of Lie Groups and Representation software https://www.liegroups.org computes the K orbits on G/B for any (connected complex reductive) G and any (algebraic) involution. The algorithm we use is closely related to the one in Richardson-Springer.

In fact this is a fundamental part of our algorithm for computing the unitary dual.

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  • $\begingroup$ Thanks Jeff. If I understand correctly, in general the Richardson-Springer map from $K\backslash G/B$ to the set of twisted involutions in the Weyl group is neither injective nor surjective. Can you please explain how one can characterize set for the orbits? For $(\mathrm{GL}_n,\mathrm{O}_n)$ one can prove bijectivity by an explicit construction, but I don't know what happens in geneal. It seems to me that existence of $K$-orbits corresponding to the "clans" of Oshima-Matsuki can be proved by a similar technique, but it is not clear to me why this method is supposed to produce ALL orbits. $\endgroup$
    – Hadi
    Commented Mar 15 at 12:34
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    $\begingroup$ See Section 11 of Algorithms for Representation Theory of Real Groups, by myself and Fokko du Cloux. In particular Remark 11.5 says that the fiber of the map from K\G/B to twisted involutions is naturally in bijection with characters of the component group of the dual group. For $GL(n,\mathbb R)$ the dual group is $U(p,q)$ whose Cartans are connected. $\endgroup$
    – Jeffrey Adams
    Commented Apr 21 at 14:23
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Coming in late here, but it's possible that what you are looking for is in Yamamoto's paper "ORBITS IN THE FLAG VARIETY AND IMAGES OF THE MOMENT MAP FOR CLASSICAL GROUPS I" which I'm remembering provides some elementary and linear algebraic arguments to prove that the parameterization claimed by Matsuki--Oshima works in several important special cases.

Despite the article's title, unfortunately it seems there has been no sequel to date.

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