Quantization in singular real polarizations: K\"ahler regularization, Maslov correction and pairings

We study the Maslov correction to semiclassical states by using a K\"ahler regularized BKS pairing map from the energy representation to the Schr\"odinger representation. For general semiclassical states, the existence of this regularization is based on recently found families of K\"ahler polarizations degenerating to singular real polarizations and corresponding to special geodesic rays in the space of K\"ahler metrics. In the case of the one-dimensional harmonic oscillator, we show that the correct phases associated with caustic points of the projection of the Lagrangian curves to the configuration space are correctly reproduced.