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Let $G$ be an algebraic group (over $\mathbb{C}$) acting algebraically on a variety $X$. Bernstein and Lunts then define in [BL94] the equivariant derived category $D^b_G(X,\mathbb{C})$ (of $\mathbb{C}$-valued sheaves).

They also mention that it should be possible to define $\displaystyle D^b_{G(\overline{\mathbb{F}_p})}(X(\overline{\mathbb{F}_p}), \overline{\mathbb{Q}_l})$

So my question: Do the comparison theorems of [BBD] Chapter 6 also hold in the equivariant case?
Is there any "nice" reference?

[BL94] Equivariant Sheaves and Functors, Bernstein, J.; Lunts. V. Springer Lecture Notes. [BBD] Beilinson, A. A.; J. Bernstein, P. Deligne (1982). "Faisceaux pervers". Astérisque (Société Mathématique de France, Paris) 100.

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Oliver this a very late comment. But regardless, certainly if you are using approximations (in the $\mathbb{F}_q$) setting that can be spread out to approximation over $\mathbb{C}$, then the usual comparison theorem will give you comparison for the equivariant setup. For instance, certainly for $G_m$ using projective spaces for your approximations. No? –  Reladenine Vakalwe Jun 10 at 19:20

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