  We study the interplay between the Fourier-Mukai transform and the
decomposition theorem for an integrable system $\pi: M \rightarrow B$. Our main
conjecture is that the Fourier-Mukai transform of sheaves of K\"ahler
differentials, after restriction to the formal neighborhood of the zero
section, are quantized by the Hodge modules arising in the decomposition
theorem for $\pi$. For an integrable system, our formulation unifies the
Fourier-Mukai calculation of the structure sheaf by Arinkin-Fedorov, the
theorem of the higher direct images by Matsushita, and the "perverse = Hodge"
identity by the second and the third authors.
  As evidence, we show that these Fourier-Mukai images are Cohen-Macaulay
sheaves with middle-dimensional support on the relative Picard space, with
support governed by the higher discriminants of the integrable system. We also
prove the conjecture for smooth integrable systems and certain 2-dimensional
families with nodal singular fibers. Finally, we sketch the proof when cuspidal
fibers appear.
