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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!

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    $\begingroup$ The stabiliser subgroup definitely does depend on $\xi'$, but the various choices are conjugate; this is elementary group theory. $\endgroup$
    – LSpice
    Jun 7 at 12:48
  • $\begingroup$ 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. $\endgroup$
    – LSpice
    Jun 7 at 12:50
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    $\begingroup$ @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. $\endgroup$
    – asma
    Jun 7 at 13:24
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    $\begingroup$ 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). $\endgroup$
    – LSpice
    Jun 7 at 13:52
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    $\begingroup$ 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 ! $\endgroup$
    – asma
    Jun 7 at 16:53

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