Coherent state representations. A survey

The coherent state (CS) representation of a Lie group G is an irreducible unitary representation such that G has a complex orbit on the projective space corresponding to the representation space. We show how the fact that any representation of a compact Lie group is a CS representation leads to the orbit method for such groups. Next, we describe CS representations of Heisenberg groups and semisimple groups of Hermitian type. Finally, we sketch a proof of the theorem that a unimodular Lie group which admits CS representations is locally isomorphic to a semidirect product of a Heisenberg group and a group of Hermitian type.