Compact Lie Group Case
Complex irreducible representations of compact Lie groups have been completely classified. They are always finite-dimensional, unitarizable (i.e. admit an invariant positive definite Hermitian form) and are parametrized by their highest weights, which are precisely the dominant integral weights for the group. If G is a compact semisimple Lie group with a Cartan subalgebra h then its coadjoint orbits are closed and each of them intersects the positive Weyl chamber h*+ in a single point. An orbit is integral if this point belongs to the weight lattice of G. The highest weight theory can be restated in the form of a bijection between the set of integral coadjoint orbits and the set of equivalence classes of irreducible unitary representations of G: the highest weight representation L(λ) with highest weight λ∈h*+ corresponds to the integral coadjoint orbit G·λ. The Kirillov character formula amounts to the character formula earlier proved by Harish-Chandra.
Read more about this topic: Orbit Method
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