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Given a Lie group $G$ there is a natural representation of $G$ on the dual of its Lie algebra $\mathfrak{g}^*$ given by the coadjoint representation. This representation is obtained by differentiating the diffeomorphisms given by conjugation $\text{con}_g(h) = g h g^{-1}$ and then taking the adjoint. From this one gets two important sets: for $\mu \in \mathfrak{g}^*$ the coadjoint orbits $G \cdot \mu$ and the coadjoint isotropy subgroups $G_\mu = \{ g \in G \ | \ g \cdot \mu = \mu \}$.

From a purely symbolic perspective, this is a very natural construction: the representation is an intrinsic property of the Lie group itself, since no external objects are needed to define it. However I don't quite see why this representation is interesting from an intuitive point of view - it actually seems quite arcane. How can one imagine elements in $\mathfrak{g}^*$ and the action of $G$ on them? What is the meaning of the coadjoint orbits and the coadjoint isotropy subgroups, how can they help to understand the Lie group $G$ better?

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    $\begingroup$ One can study irreducible unitary representations of certain Lie groups by looking at their coadjoint orbits, this was first discovered by Kirillov: en.m.wikipedia.org/wiki/Orbit_method $\endgroup$ Aug 8 '21 at 18:03

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