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this question is trivial. We know from this paper link text, Springer constructed rep of the Weyl group $W$ on the cohomology of the Springer fibre. Also, Deligne-Lusztig constructed the linear rep of finite group of Lie type.

They all consider the ℓ-adic field,

hence, I want to know why they must consider the field $Q_{l}$, and consider the ℓ-adic cohomology?? and If I want to study this theory, whether I must study the theory of ℓ-adic field? and whether the ℓ-adic cohomology is the basic tool in dealing these theories?

Thank you.

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In the introduction of Milne's lecture notes on étale cohomology, he explained the motivations for étale (or $l$-adic) cohomology, including why the field $Q_l$ is necessary, and mentioned Deligne-Lusztig's theory as an application in the end. You may find it helpful. –  shenghao Jun 18 '11 at 14:25
    
I don't think you need to know much about $\ell$-adic fields, but it is probably a good idea to make yourself comfortable with locally constant sheaves. –  S. Carnahan Jun 18 '11 at 16:40
    
@shenghao, Carnahan, thank you. I have got Milne's lecture from his homepage –  wison Jun 19 '11 at 4:40

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