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<sub>!*</sub>O<sub>Y</sub> ?  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<sub>!*</sub>O<sub>Y</sub> as a module the associative algebra D(X), supposing I understand D(X)?

In other words, what is the vector space underlying i<sub>!*</sub>O<sub>Y</sub>, 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!