Deligne–Lusztig variety
In mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ-adic cohomology with compact support, introduced by Deligne & Lusztig (1976).
Lusztig (1984) used these representations to find all representations of all finite simple groups of Lie type.
Suppose that G is a reductive group defined over a finite field, with Frobenius map F.
Macdonald conjectured that there should be a map from general position characters of F-stable maximal tori to irreducible representations of GF (the fixed points of F). For general linear groups this was already known by the work of Green (1955). This was the main result proved by Deligne and Lusztig; they found a virtual representation for all characters of an F-stable maximal torus, which is irreducible (up to sign) when the character is in general position.
When the maximal torus is split, these representations were well known and are given by parabolic induction of characters of the torus (extend the character to a Borel subgroup, then induce it up to G). The representations of parabolic induction can be constructed using functions on a space, which can be thought of as elements of a suitable zeroth cohomology group. Deligne and Lusztig's construction is a generalization of parabolic induction to non-split tori using higher cohomology groups. (Parabolic induction can also be done with tori of G replaced by Levi subgroups of G, and there is a generalization of Deligne–Lusztig theory to this case too.)
Drinfeld proved that the discrete series representations of SL2(Fq) can be found in the ℓ-adic cohomology groups
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