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.