  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. $\mathfrak{su}(2)$ algebra), pseudo-spin coherent
states (i.e. $\mathfrak{su}(1,1)$ algebra), and the
$\mathfrak{sl}(2,\mathbb{R})$ 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.
