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



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


