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 behavior from the set of coherent states and verify that it always happens when some parameter, depending on the algebra and its representation, becomes large.