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We study the dependence of geometric quantization of the standard symplectictorus on the choice of invariant polarization. Real and mixed polarizations areinterpreted as degenerate complex structures. Using a weak version of theequations of covariant constancy, and the Weil-Brezin expansion to describedistributional sections, we give a unified analytical description of thequantization spaces for all nonnegative polarizations.  The Blattner-Kostant-Sternberg (BKS) pairing maps between half-form correctedquantization spaces for different polarizations are shown to be transitive andrelated to an action of $Sp(2g,\R)$. Moreover, these maps are shown to beunitary.

Quantization of Abelian Varieties: distributional sections and the transition from K\"ahler to real polarizations