Making D-modules on affine varieties more explicit 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!
 A: Have you looked at Bernstein's lectures on D-modules? He proves a result relevant to your question in Lec. 3, Sec. 14: for an affine embedding Y --> X with Y irreducible, if E is an OY-coherent DY module (i.e. a vector bundle with a flat connection) then the !* direct image from E can be characterized as the unique irreducible subquotient of either the * or ! direct image which has nonzero restriction to Y. Since it can be easy in such situations to compute the * direct image, this may be a good way to get a handle on the other functors too. For example, I think one can see from this that if we take the embedding of the origin into the affine line, then both the * and !* direct images of OY=k (the ground field) are A(1)/A(1)t, where t is the coordinate on the affine line and A(1)=DX is the 1-dimensional Weyl algebra k[t,d/dt]. This quotient is rightly considered the "delta-function" A(1)-module, since its generator δ satisfies tδ=0. I'm not sure how similar to this case your general situation will be. But certainly by Kashiwara's theorem one knows that the * direct image (and hence the !* direct image) will be supported on the subvariety Y. 
A: 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.
A: I think I can answer my subquestion about * extension after thinking about it, but it's really the ! and !* extensions I'd like to understand.  I think I remember the * extension is just the pushforward.
So i*O(Y) = O(X) ⊗i O(Y), and the D-module action is always happening on the O(X) factor.
Sorry to answer (a tiny part of) my own question, but I couldn't do it in a comment because of the formatting, and maybe this will save some kind respondent some time.
