Note on duality on polarized symplectic manifolds (Q1081918)

We use some of our earlier results of [Trans. Am. Math. Soc. 252, 163-195 (1979; Zbl 0423.14008)] to derive a general duality theorem for the cohomologies of foliated structures on a manifold. The result is applied to the special case of symplectic manifold M on which the foliation is given by a complex polarization F in the sense of geometric quantization. We obtain, for example, a rigorous proof of the fact that for a smooth function f on M whose Hamiltonian vector field leaves F invariant, the spectrum of the corresponding prequantization operator v(f) coincides with the spectrum of its transpose, under the above duality. This latter result was obtained by D. J. Simms.