The overlap of Generalized Coherent States of any rank one simple Lie algebra and its classical limit

We provide a formula for computing the overlap between two Generalized Coherent States of any rank one simple Lie algebra. Then, we apply our formula to spin coherent states (i.e. su(2) algebra), pseudo-spin coherent states (i.e. su(1,1) algebra), and the sl(2,R) subalgebras of Virasoro. In all these examples, we show the emergence of a semi-classical behaviour from the set of coherent states and verify that it always happens when some parameter, depending on the algebra and its representation, becomes large.