# Meaning of the coadjoint representation and its orbits

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?

• 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 Aug 8 '21 at 18:03