Deligne-lusztig Varieties and Period Domains over Finite Fields

Let G0 be a reductive group over Fq. There are two classes of algebraic varieties over an algebraic closure F of Fq attached to G0. Let us recall their definition. We set G = G0×Fq F. To G there is associated the maximal torus, the Weyl group W and the set of fundamental reflections in W , cf. [DL] 1.1. Let X = XG be the set of all Borel subgroups of G. Then X is a smooth projective algebraic variety homogeneous under G. The set of orbits of G on X×X can be identified with W, and this defines the relative position map inv : X×X → W (associate to an element of X × X the G-orbit containing it). Let w ∈ W. The Deligne-