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Let $G$ be a compact connected Lie group and T be it's maximal torus. Let $\theta: G \rightarrow G$ be an involution on $G$ and let $G^\theta = \lbrace g \in G , \theta(g)=g \rbrace $.

I'm looking for an a simple the proof of the claim which says that $G/T$ has finitely many $G^\theta (\mathbb{C})$-orbits or just a sketch of the proof because I'm not very familiar with algebraic geometry and I would like to just know what are the ideas used to prove it.

Any help would be greatly appreciated! Thanks

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  • $\begingroup$ where did you see this claim? $\endgroup$ Apr 15, 2021 at 16:35
  • $\begingroup$ @Venkataramana, thank you for your question because it makes me realize that I was wrong. Actually the claim I'm looking for is that the set $\lbrace x \in G/T, \theta(G_x)=G_x \rbrace $, where $G_x$ is the stabilizer of $x$ has finitely many $G^\theta$-orbits and these orbits are in bijection with the $G^\theta(\mathbb{C})$-orbits of $G/T$. $\endgroup$
    – Mira
    Apr 15, 2021 at 16:56
  • $\begingroup$ Could you please give me an outline of the proof (hopefully a simple one because I'm familiar with algebraic geometry) of the claim that there is finite number of orbits of $G/T$ under the action of $G^\theta(\mathbb{C})$? $\endgroup$
    – Mira
    Apr 15, 2021 at 17:00
  • $\begingroup$ perhaps you can correct your question( Since you say what you asked for is wrong). $\endgroup$ Apr 15, 2021 at 17:14
  • $\begingroup$ Yes , sure I'll edit my question. $\endgroup$
    – Mira
    Apr 15, 2021 at 17:20

1 Answer 1

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First of all, one has to worry about how $H:=G^\theta(\mathbb C)$ is supposed to act on $G/T$. The only way which comes to my mind is to use the well-known fact that $G/T\cong G(\mathbb C)/B=:X$ where $B$ is a Borel subgroup. This holds because $G$ acts transitively on $G(\mathbb C)/B$ with isotropy group $T$.

Assuming this then your statement is clear: $H$ has finitely many orbits in $X$ iff $G(\mathbb C)$ consists of finitely many $H\times B$-double cosets iff (by symmetry) $B$ has finitely many orbits in $G/H$. Complex homogeneous spaces with this latter property are called spherical and it is well known that $G(\mathbb C)/H$ where $H$ is the centralizer of an involution is spherical.

As a starter you could check the papers of Richardson-Springer who worked on a classification of the orbits you are interested in. But all of this probably much older and was known even before the term "spherical" was coined.

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  • $\begingroup$ Thank you so much for your answer and for the reference. $\endgroup$
    – Mira
    Apr 16, 2021 at 11:13

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