Quantization of compact Riemannian symmetric spaces

The phase space of a compact, irreducible, simply connected, Riemannian symmetric space admits a natural family of Kähler polarizations parametrized by the upper half plane S. Using this family, geometric quantization, including the half-form correction, produces the field Hcorr→S of quantum Hilbert spaces. We show that projective flatness of Hcorr implies, that the symmetric space must be isometric to a compact Lie group equipped with a biinvariant metric. In the latter case the flatness of Hcorr was previously established.