TITLE: Making D-modules on affine varieties more explicit
QUESTION [10 upvotes]: This is a follow up to my question about D-modules supported on the nilpotent cone.  I got some good answers there but now I have a more basic question.
Consider an affine algebraic variety X, a closed subvariety i:Y-->X, and the intermediate extension of the structure sheaf on Y to all of X (do I denote this i!*OY ?  For that matter, explaining either the * or ! extension instead would be a helpful start if its easier).
My question is this:  Since X is affine, D(X) is just an associative algebra, generated by O(X) and Vect(X) by the usual construction. My question is how can I understand i!*OY as a module the associative algebra D(X), supposing I understand D(X)?
In other words, what is the vector space underlying i!*OY, how do functions in O(X) and vector fields act?
I probably need to invest some serious time with a textbook to answer this question myself, but any help getting started would be most appreciated!

REPLY [2 votes]: Since Y is closed, the !, *, and !* extensions coincide, and there should be a straightforward (but confusing for me) way of doing what you want.  If Y is smooth, then I wonder if you can identify the vector space you're talking about with something like sections of the conormal bundle to Y in X, but this is not quite right (e.g. when Y = X).  When Y is singular, there is a more complicated story, having to do with the fact that the obvious definition of D-module on a singular variety is no good.
When Y is only locally closed, describing the !* extension is a difficult problem.  Vilonen's thesis, the only reference I know about this, is available here: 
http://gdz.sub.uni-goettingen.de/en/dms/load/img/?PPN=PPN356556735_0081&DMDID=dmdlog12
There is even something nontrivial to say about the ! and * extensions--the characteristic cycles of such D-modules were computed by Schmid and Vilonen.  Maybe there is a more elementary answer to your question about these extensions, though.