From action-angle coordinates to geometric quantization

The philosophy of geometric quantization is to ¯nd and understand a \(one-way) dictionary" that \translates" classical systems into quantum systems . In this way, a quantum system is associated to a classical system in which observables (smooth functions) become operators of a Hilbert space and the classical Poisson bracket becomes the commutator of operators. In this process, the choice of additional geometric structures (polarizations) plays an important r^ole. A desired property is that the quantization obtained does not depend on the polarization. Another rule in the game is that of keeping track of the symmetries on both sides. This is the deep link of geometric quantization with representation theory. The quanti- zation commutes with reduction \principle" becomes realistic in some geometric quantization set-ups. Our point of view in this big endeavour is very modest. We plan to construct a \representation space" in the case the polarization is given by a real polarization. For this, we follow the de¯nition of Kostant of the representation spaces via higher cohomology groups with coe±cients in the sheaf of °at sections. In this short note, we will not discuss either the (pre)Hilbert structure of this space nor the quantization rules.Preprin