Differential operators on blow-ups and Fourier transform.

For a general algebraically closed field k of characteristic 0, we compute
the ring of differential operators on the blow-up of k^n at the origin
twisted by any power of the canonical bundle associated to the exceptional
divisor of the blow-up.
We show the existence of a surjective map from the universal enveloping
algebra of sl(n+1,k) onto the above ring, and the Fourier transform gives an
explicit isomorphism between the above twisted ring and some twisted ring of
differential operators on the n-dimensional projective space P^n.
We further elaborate on the Fourier transform in the context of Goncharov's
construction and toric varieties, if time permits.