I think saying "constant sheaf" isn't quite right.  You want to take the functions on each orbit, and do the intermediate extension of that D-module.

Now, that's a bit unexplicit, so let me try a different description.  Recall that there is a Grothendieck simultaneous resolution, a map from `G x_B b` (the adjoint bundle for b on G/B) to g extending the inclusion of b in the obvious way.  If one takes functions on `G x_B b`, and pushes them forward by this map, then one gets a D-module on g.  The isotypic summands of this D-module are in bijection with the representations of `S_n`. Over regular semi-simple elements, the map `G x_B b -> g` is a Galois S_n-cover, and so we're just decomposing the local system.  By some geometric magic, this decomposition extends over the rest of the Lie algebra.

Now, we have to do Fourier transform, which for D-modules, just means change your mind about which variables in the Weyl algebra are coordinate and which ones and vectors.  Then the summand corresponding to the S_n representation for a Young diagram becomes exactly the intermediate extension for the orbit with corresponding Jordan type (or is it the transpose?  I always forget).