I am reading some papers which involve D-modules on a Lie algebra g, which are supported on the nilpotent cone n.  They are equivariant for the action of g.  (In particular, I consider g=sl_n).

It was explained to me (the statement, not the proof) that the category of such D-modules is semi-simple, and that the simple objects are given by the constant sheaf on each g-orbit on n, so they are in bijection with Jordan decompositions with all zeroes, so just partitions of n.

I'm not strong with D-modules (I'm learning!).  My question is this:  Since g is affine, D(g) is an associative algebra, namely the Weyl algebra on the vector space g.  How can I describe the D(g)-module M corresponding to partition \lambda explicitly as a module over D(g)?