Following Green's successful combinatorial attack on the ordinary characters of GL(n, q), the further character computations cited in 16.11 dealt mainly with groups of very small rank. Even though Macdonald was able to predict the main features of the character theory for arbitrary groups of Lie type, finite group techniques alone seemed inadequate to prove his conjectures. Then a landmark paper by Deligne–Lusztig showed how to construct a large number of (virtual) characters from actions of a finite group of Lie type on the étale cohomology of certain subvarieties of the ambient algebraic group. All irreducible characters occur here as constituents. Further work, especially by Lusztig, has determined the irreducible constituents of these virtual characters. Most of the character values have by now been recursively determined; this too requires sophisticated homological and geometric techniques.

We shall not attempt to expose this theory in detail. Instead we rely on the accounts given by Carter and Digne–Michel; some of the highlights are sketched below. Our focus here is on the way in which Deligne–Lusztig characters (which we call for short DL characters) reduce modulo p. Combined with Lusztig's methods for relating ordinary characters to DL characters, this yields considerable insight into the decomposition patterns.

Broadly speaking, the ordinary character theory imitates the Harish-Chandra theory of infinite dimensional representations of Lie groups, with its organization of characters into “series” depending on different types of maximal tori.