This is a follow up to my question [about D-modules supported on the nilpotent cone](https://mathoverflow.net/questions/2971/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!



Making D-modules on affine varieties more explicit
Have you looked at Bernstein's [lectures](http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein-dmod.ps) 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. 


