1
$\begingroup$

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

$\endgroup$
5
  • $\begingroup$ where did you see this claim? $\endgroup$ Apr 15 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 $G/T$. $\endgroup$
    – asma
    Apr 15 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 $G^\theta(\mathbb{C})$? $\endgroup$
    – asma
    Apr 15 at 17:00
  • $\begingroup$ perhaps you can correct your question( Since you say what you asked for is wrong). $\endgroup$ Apr 15 at 17:14
  • $\begingroup$ Yes , sure I'll edit my question. $\endgroup$
    – asma
    Apr 15 at 17:20
8
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Thank you so much for your answer and for the reference. $\endgroup$
    – asma
    Apr 16 at 11:13

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.