I've posted the following question some days ago in math.stackexchange https://math.stackexchange.com/questions/4155747/co-adjoint-orbit but I didn't get any answer!

While I was trying to teach my self about the slice theorem in symplectic geometry, I've come across this paragraph:

Let $$G$$ be a compact Lie group with Lie algebra $$\mathfrak{g}$$ and let $$\mathcal{O}$$ be a co-adjoint orbit of $$\mathfrak{g}^*$$. Let T be a maximal torus of $$G$$ with Lie algebra $$\mathfrak{t}$$, and let $$W$$ be the Weyl group associated. We make a choice of a Weyl chambers $$\mathfrak{t}_+^*$$ in $$\mathfrak{t}^*$$. Let $$\xi$$ be the unique point in $$\mathfrak{t}_+^*$$ such that $$\mathcal{O} = G. \xi$$, and let $$\sigma$$ be the unique open face of $$\mathfrak{t}_+^*$$ which contains $$\xi$$. The stabilizer subgroup $$G^{\xi'} \subset G$$ doesn't depend on the choice of $$\xi' \in \sigma$$, and is denoted $$G_\sigma$$.

My questions are the following:

Let $$\xi$$ be the unique point in $$\mathfrak{t}_+^*$$ such that $$\mathcal{O} = G. \xi$$

Why it's guaranteed that there is a unique element $$\xi$$ in $$\mathfrak{t}_+^*$$ which satisfies $$\mathcal{O} = G. \xi$$ ?

The stabilizer subgroup $$G^{\xi'} \subset G$$ doesn't depend on the choice of $$\xi' \in \sigma.$$

Why this is true ?

Thank you!

• The stabiliser subgroup definitely does depend on $\xi'$, but the various choices are conjugate; this is elementary group theory. Jun 7 at 12:48
• As for conjugacy into the chosen chamber (assuming it's closed), this is a basic structure theorem about compact Lie groups. I don't know a completely elementary proof, but it will be in any textbook on them, so I am voting to close. There are so many books it's hard to pick one, but I think tom Dieck's book is well received. Jun 7 at 12:50
• @LSpice, thank you for your comments! The reference I'm using to learn about representation theory of Compact lie groups is Sepanski's book but it doesn't contain the answer to my questions, I just took a look to the reference that you suggested of Tom dieck, it seems that I can't find these propositions, but I'll read it carefully later! I'll be very grateful to you or to anyone who give just the sketch of the proof to those propositions.
– asma
Jun 7 at 13:24
• I was typing earlier on my phone, so a bit terse. I want to be clear that these are good questions, just probably not MO-level; I have answered on MSE, and can give more details there if needed. The relevant references in Sepanski are Theorem 5.9 and Theorem 6.43(c). Jun 7 at 13:52
• I got the answer to my question by LSpice here in math.stackexchange.com/questions/4155747/co-adjoint-orbit/…. so I want to close this post but I don't know how, if someone else could please close it !
– asma
Jun 7 at 16:53