Nilpotent Group Case
Let G be a connected, simply connected nilpotent Lie group. Kirillov proved that the equivalence classes of irreducible unitary representations of G are parametrized by the coadjoint orbits of G, that is the orbits of the action G on the dual space of its Lie algebra. The Kirillov character formula expresses the Harish-Chandra character of the representation as a certain integral over the corresponding orbit.
Read more about this topic: Orbit Method
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