# $G/T$ has finitely many $G^\theta$ orbits

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

• where did you see this claim? Apr 15 at 16:35
• @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 $G/T$.
– asma
Apr 15 at 16:56
• 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 $G^\theta(\mathbb{C})$?
– asma
Apr 15 at 17:00
• perhaps you can correct your question( Since you say what you asked for is wrong). Apr 15 at 17:14
• Yes , sure I'll edit my question.
– asma
Apr 15 at 17:20

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.