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YCor
YCor
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Let$\newcommand{\g}{\mathfrak{g}}$Let $G$ be a compact Lie group and $g$$\g$ its Lie algebra. I came across the the very important result that $G/T$ ($T$ a maximal torus of $G$) iscan be identified to a coadjoint orbit. However it is not at all clear to me how one can show this result. I guess we must somehow prove that there always exists an $F\in g^*$$F\in \g^*$ such that $\operatorname{Stab}(F)\cong T$. But this is not clear at all from the definition of $\operatorname{Stab}(F)$ so I guess the proof for that fact must be using more elegant ideas than that. So can anyone explain what ideas are used to show this or reference somewhere?

Let $G$ be a compact Lie group and $g$ its Lie algebra. I came across the the very important result that $G/T$ ($T$ a maximal torus of $G$) is a coadjoint orbit. However it is not at all clear to me how one can show this result. I guess we must somehow prove that there always exists an $F\in g^*$ such that $\operatorname{Stab}(F)\cong T$. But this is not clear at all from the definition of $\operatorname{Stab}(F)$ so I guess the proof for that fact must be using more elegant ideas than that. So can anyone explain what ideas are used to show this or reference somewhere?

$\newcommand{\g}{\mathfrak{g}}$Let $G$ be a compact Lie group and $\g$ its Lie algebra. I came across the the very important result that $G/T$ ($T$ a maximal torus of $G$) can be identified to a coadjoint orbit. However it is not at all clear to me how one can show this result. I guess we must somehow prove that there always exists an $F\in \g^*$ such that $\operatorname{Stab}(F)\cong T$. But this is not clear at all from the definition of $\operatorname{Stab}(F)$ so I guess the proof for that fact must be using more elegant ideas than that. So can anyone explain what ideas are used to show this or reference somewhere?

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RobPratt
RobPratt
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How can one show G$G/TT$ is a coadjoint orbit for G a compact lieLie group $G$ and T it's$T$ its maximal torus?

Let $G$ be a compact lieLie group and $g$ it's lieits Lie algebra. I came across the the very important result that $G/T$ ($T$ a maximal torus of $G$) is a coadjoint orbit. However it is not at all clear to me how one can show this result. I guess we must somehow prove that there always exists an $F\in g^*$ such that $Stab(F)\cong T$$\operatorname{Stab}(F)\cong T$. But this is not clear at all from the definition of $Stab(F)$$\operatorname{Stab}(F)$ so I guess the proof for that fact must be using more elegant ideas than that. So can anyone explain what ideas are used to show this or reference somewhere.

Thanks in advance?

How can one show G/T is a coadjoint orbit for G a compact lie group and T it's maximal torus

Let $G$ be a compact lie group and $g$ it's lie algebra. I came across the the very important result that $G/T$ ($T$ a maximal torus of $G$) is a coadjoint orbit. However it is not at all clear to me how one can show this result. I guess we must somehow prove that there always exists an $F\in g^*$ such that $Stab(F)\cong T$. But this is not clear at all from the definition of $Stab(F)$ so I guess the proof for that fact must be using more elegant ideas than that. So can anyone explain what ideas are used to show this or reference somewhere.

Thanks in advance

How can one show $G/T$ is a coadjoint orbit for a compact Lie group $G$ and $T$ its maximal torus?

Let $G$ be a compact Lie group and $g$ its Lie algebra. I came across the the very important result that $G/T$ ($T$ a maximal torus of $G$) is a coadjoint orbit. However it is not at all clear to me how one can show this result. I guess we must somehow prove that there always exists an $F\in g^*$ such that $\operatorname{Stab}(F)\cong T$. But this is not clear at all from the definition of $\operatorname{Stab}(F)$ so I guess the proof for that fact must be using more elegant ideas than that. So can anyone explain what ideas are used to show this or reference somewhere?

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spyros
spyros
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How can one show G/T is a coadjoint orbit for G a compact lie group and T it's maximal torus

Let $G$ be a compact lie group and $g$ it's lie algebra. I came across the the very important result that $G/T$ ($T$ a maximal torus of $G$) is a coadjoint orbit. However it is not at all clear to me how one can show this result. I guess we must somehow prove that there always exists an $F\in g^*$ such that $Stab(F)\cong T$. But this is not clear at all from the definition of $Stab(F)$ so I guess the proof for that fact must be using more elegant ideas than that. So can anyone explain what ideas are used to show this or reference somewhere.

Thanks in advance