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I am looking for a reference for basic facts about actions of linear algebraic groups and their Lie-algebras on $\mathcal O_X$-modules.

For example I could not find a reference the following:

Let $G$ be a connected, complex linear algebraic group. Then an $\mathcal O_X$-linear map between two $G$-equivariant $\mathcal O_X$-modules commutes with the group actions iff it commutes with the Lie-algebra actions.

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Chapter 5, section 1 of Chriss and Ginzburg contains many nice facts about $G$-equivariant $\mathcal{O}$-modules. However, they do not discuss Lie algebra actions at all. –  Mike Skirvin Nov 18 '10 at 15:52
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