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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!

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# 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 ?).

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!