I think Fourier-Mukai transform is related to the Fourier transforms you described through the space A \times \check A which is symplectic and somehow relevant, though I don't know the details.

The reasoning is that when x, y, ... live in A, the vector fields live in \check A, so that's analogous to teh space C[x, y, ..., d/dx, d/dy, ...] you mentioned above.

I think Fourier-Mukai transform is related to the Fourier transforms you described through the space A \times \check A which is symplectic and somehow relevant, though I don't know the details.

The reasoning is that when x, y, ... live in A, the vector fields kind of live in \check A, so that's analogous to the space C[x, y, ..., d/dx, d/dy, ...] you mentioned above.