  Do co-adjoint orbits of Lie groups support a K\"{a}hler structure? We study
this question from a point of view derived from coherent states. We examine
three examples of Lie groups: the Weyl-Heisenberg group, $\mathrm{SU(2)}$ and
$\mathrm{SU(1,1)}$. In cases, where the orbits admit a K\"{a}hler structure, we
show that coherent states give us a K\"{a}hler embedding of the orbit into
projective Hilbert space. In contrast, squeezed states, (which like coherent
states, also saturate the uncertainty bound) only give us a symplectic
embedding. We also study geometric quantisation of the co-adjoint orbits of the
group $\mathrm{SUT(2,\mathbb{R})}$ of real, special, upper triangular matrices
in two dimensions. We glean some general insights from these examples. Our
presentation is semi-expository and accessible to physicists.
