Deligne–Lusztig theory

In mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ-adic cohomology (Étale cohomology) with compact support, introduced by Pierre Deligne and George Lusztig (1976).

Lusztig (1985) used these representations to find all representations of all finite simple groups of Lie type.

Motivation

Suppose that G is a reductive group defined over a finite field, with Frobenius map F.

Ian G. Macdonald conjectured that there should be a map from general position characters of F-stable maximal tori to irreducible representations of

G

F

{\displaystyle G^{F}}

(the fixed points of F). For general linear groups this was already known by the work of J. A. Green (1955). This was the main result proved by Pierre Deligne and George 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.)

Vladimir Drinfeld proved that the discrete series representations of SL2(Fq) can be found in the ℓ-adic cohomology groups

H

c

1

(

X

,

Q

ℓ

)

{\displaystyle H_{c}^{1}(X,\mathbb {Q} _{\ell })}

of the affine curve X defined by

x

y

q

−

y

x

q

=

1

{\displaystyle xy^{q}-yx^{q}=1}

.

The polynomial

x

y

q

−

y

x

q

{\displaystyle xy^{q}-yx^{q}}

is a determinant used in the construction of the Dickson invariant of the general linear group, and is an invariant of the special linear group.