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 semisimple, and that the simple objects are given by (the intermediate extension) of the constant sheaf sheaf of functions 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)?



D-modules supported on the nilpotent cone
Watch out that there are more simple objects than it looks like at first glance, even for sl\_n. Although the orbits are parameterized by partitions, they can carry nontrivial local systems whose intermediate extensions to the nilpotent cone will be new simple D-modules. 


I have heard that the general problem of writing down intermediate extensions explicitly, say by generators and relations, is weirdly difficult. Kari Vilonen did this for isolated singularities in his thesis. For the nilpotent cone I wonder how well Ben's suggestion works: is it a simple matter to pick out the isotypic components of this pushforward D-module, using the bare fact (geometric magic) that it's an intermediate extension of a D-module you know how to decompose?


