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Given a Lie group $G$, it acts smoothly on the dual $\mathfrak g^*$ of its Lie algebra $\mathfrak g$ by the coadjoint action. The orbits of that action are called coadjoint orbits.
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Interesting properties of "coadjoint" orbits inside $V\in \operatorname{Rep}G$
Let $G$ be a reductive group over $\mathbf{C}$. It acts on the dual of its Lie algebra $\mathfrak{g}^*$ by conjugation.
One can describe the orbits of $\mathfrak{g}^*$ explicitly (e.g. using Jordan …