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Maybe this question is trivial.

We know from this paper at Inv. Math 1976 (DOI link), T. A. Springer constructed representation of the Weyl group $W$ on the cohomology of the Springer fibre. Also, Deligne-Lusztig constructed the linear representation of finite group of Lie type.

They all consider the $\ell$-adic field, hence, I want to know why they must consider the field $\mathbb{Q}_{\ell}$, and consider the $\ell$-adic cohomology? and if I want to study this theory, should I study the theory of $\ell$-adic fields? and is the $\ell$-adic cohomology the basic tool in dealing these theories?

Improve this question
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    $\begingroup$ 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. $\endgroup$
    – shenghao
    Commented Jun 18, 2011 at 14:25
  • $\begingroup$ 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. $\endgroup$
    – S. Carnahan ♦
    Commented Jun 18, 2011 at 16:40
  • $\begingroup$ @shenghao, Carnahan, thank you. I have got Milne's lecture from his homepage $\endgroup$
    – wison
    Commented Jun 19, 2011 at 4:40

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