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.