Let $(L,\nabla )$ be a prequantum line bundle over a symplectic manifold $X$, and $S$ its symplectization. Kostant showed that the classical Poisson bracket on $S$ is simply prequantization on $X$. C. Duval and I have taken this a step farther to obtain a quantization of $X$ using a generalized star-product on $S$.
References:

1. Kostant, B. [2003], Minimal coadjoint orbits and symplectic induction, arXiv: SG/0312252.