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 Pierre Deligne and George Lusztig (1976).
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.)
Vladimir Drinfeld proved that the discrete series representations of SL2(Fq) can be found in the ℓ-adic cohomology groups of the affine curve X defined by The polynomial
The case of compact groups is easier partly because there is only one conjugacy class of maximal tori.
The construction of Deligne-Lusztig characters uses a family of auxiliary algebraic varieties XT called Deligne–Lusztig varieties, constructed from a reductive linear algebraic group G defined over a finite field Fq.
If B is a Borel subgroup of G and T a maximal torus of B then we write for the Weyl group (normalizer mod centralizer) of G with respect to T, together with the simple roots corresponding to B.
If B1 is another Borel subgroup with maximal torus T1 then there is a canonical isomorphism from T to T1 that identifies the two Weyl groups.
In this case we say that B and B1 are in relative position w. Suppose that w is in the Weyl group of G, and write X for the smooth projective variety of all Borel subgroups of G. The Deligne-Lusztig variety X(w) consists of all Borel subgroups B of G such that B and F(B) are in relative position w [recall that F is the Frobenius map].
In other words, it is the inverse image of the G-homogeneous space of pairs of Borel subgroups in relative position w, under the Lang isogeny with formula For example, if w=1 then X(w) is 0-dimensional and its points are the rational Borel subgroups of G. We let T(w) be the torus T, with the rational structure for which the Frobenius is wF.
The F-stable conjugacy classes can be identified with elements of the non-abelian Galois cohomology group of torsors Fix a maximal torus T of G and a Borel subgroup B containing it, both invariant under the Frobenius map F, and write U for the unipotent radical of B.
The geometric conjugacy classes of pairs (T,θ) are parameterized by geometric conjugacy classes of semisimple elements s of the group G*F of elements of the dual group G* fixed by F. Two elements of G*F are called geometrically conjugate if they are conjugate over the algebraic closure of the finite field; if the center of G is connected this is equivalent to conjugacy in G*F. The number of geometric conjugacy classes of pairs (T,θ) is |Z0F|ql where Z0 is the identity component of the center Z of G and l is the semisimple rank of G. In this subsection G will be a reductive group with connected center Z.
If κ is a geometric conjugacy class of pairs (T,θ) then the character of the corresponding regular representation is given by and its dimension is the p′ part of the index of the centralizer of the element s of the dual group corresponding to it times the p-part of the order of the centralizer.
The number of unipotent characters depends only on the root system of the group and not on the field (or the center).
The representations Rθ(1) are given as follows: The Deligne-Lusztig variety X(w) for the non-split torus is 1-dimensional, and can be identified with the complement of X(1) in 1-dimensional projective space.
So it is the set of points (x:y) of projective space not fixed by the Frobenius map (x:y)→ (xq:yq), in other words the points with Drinfeld's variety of points (x,y) of affine space with maps to X(w) in the obvious way, and is acted on freely by the group of q+1th roots λ of 1 (which can be identified with the elements of the non-split torus that are defined over Fq), with λ taking (x,y) to (λx,λy).
(In this case the semisimple representations do not correspond exactly to geometric conjugacy classes of the dual group, as the center of G is not connected.)