Coherent States as Kahler embeddings in ${mathbb C}P^{infty}$

In this semi-expository paper we study two examples of coherent states based on the Weyl- Heisenberg group and the group of $2 times 2$ upper triangular matrices. It is known that sometimes the coherent states provide us with a Kahler embedding of a coadjoint orbit into the projective Hilbert space ${mathbb C}P^n$ or ${mathbb C}P^{infty}$. We show an explicit computation of this in the above two examples. We also note the presence of other coadjoint orbits which only embed symplectically into the projective Hilbert space. These correspond to squeezed states, which have several applications in physics. Our exposition includes a detailed study of the geometric quantisation of the coadjoint orbits of the Lie Algebra of upper triangular matrices. This reveals the presence of distinguished orbits which correspond to coherent states, as well as others corresponding to squeezed states. The coadjoint orbit of $SUT^{+}$ we consider is intimately connected to the $2$-dimensional Toda system.