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This question is basically about comparison of IC sheaves (or their sheaf cohomologies) for the settings: 1. variety is over $\mathbb{C}$ and sheaf is $\mathbb{C}$-linear, 2. variety is over a finite ...

The exposition in the famous J. Bernstein's lecture notes on D-modules is rather sketchy. Fortunately, there are many other sources for the information from this text (like Hotta's excellent "D-...