Geometric quantization of Kähler manifolds successfully leads to the Segal-Bargmann, or holomorphic, representation. The polarized sections naturally lead to a distribution weight given by the Husimi function, which is everywhere positive. The Husimi function, however, has several disadvantages in analyzing semiclassical behaviour. By reproducing the ordinary marginals, the Wigner pseudo-distribution is much more desirable, no matter its negativity.
The Segal-Bargmann representation arrises from a rather natural polarization choice: the sections are nothing but the holomorphic functions on a (complexified) manifold. Since inversion formulas for the Weierstrass transform are not usually sufficient to recover the Wigner's pseudo-distribution, I am curious about the possibility of devising a geometric quantization procedure, together with a specific polarization, where the weight function is the Wigner pseudo-distribution itself, not Husimi's. This might require weakening pre-quantization and polarization axioms.
Does anyone know anything about it? Any reference? Is this possible?