Quantization of compact Riemannian symmetric spaces

The phase space of a compact, irreducible, simply connected, Riemannian symmetric space admits a natural family of K\"ahler polarizations parametrized by the upper half plane $S$. Using this family, geometric quantization, including the half-form correction, produces the field $H^{corr}\rightarrow S$ of quantum Hilbert spaces. We show that projective flatness of $H^{corr}$ 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 $H^{corr}$ was previously established.