This is a follow-up to a recent mathoverflow question 34387 about computing the orders of finite unitary groups and the comments made there.

Between 1955 (Chevalley's Tohoku paper) and 1968 (Steinberg's AMS Memoir 80 "Endomorphisms of linear algebraic groups"), the list of finite groups of Lie type was completed and the orders all computed by various methods: Chevalley (split) groups, twisted (quasi-split) groups of types ADE discovered independently by Hertzig, Steinberg, Tits, and the groups of Suzuki and Ree defined only in characteristic 2 or 3. This followed older work on classical and some exceptional groups. For example, it is easy to compute the order of GL$(n,q)$ and then SL$(n,q)$ by counting bases of an $n$-dimensional vector space over $\mathbb{F}_q$ (these being permuted simply transitively by the general linear group).

Chevalley developed a uniform method for split groups based on the BN-pair structure (Bruhat decomposition) together with knowledge of the degrees $d_i$ of fundamental invariants for the Weyl group $W$. For "universal type" groups like the special linear groups this yields a closed formula $$|G(\mathbb{F}_q)| = q^N \prod_i (q^{d_i}-1)$$ where $N$ is the number of positive roots $= \sum_i(d_i-1)$ and where $i=1, \dots, r$= rank.

Formulas for the other groups turn out to look similar, but with modifications. For instance, factors $q^{d_i}-1$ might occur with a plus rather than minus sign. For Suzuki or Ree groups, $q^2$ is an odd power of 2 or 3.

In his 1968 paper (11.16), following a related approach in his 1967-68 Yale
lectures, Steinberg unified everything by viewing the finite groups as fixed points
of a Frobenius-type morphism possibly incorporating a Dynkin diagram symmetry;
for Suzuki or Ree groups, the square of the morphism used is a usual Frobenius
morphism. Here the Weyl group may be replaced by a suitable subgroup, etc. The proof requires a more delicate version of Weyl group invariants, which
Springer approached later in his own way in "The order of a finite group of Lie
type" (*Algebraists' Homage*, AMS Contemp. Math. 13, 1982, pp. 81-89).

There is a geometric proof of Steinberg's unified formula based on the Weil-Deligne approach to counting points of a variety over a finite field using a Lefschetz fixed point formula in etale cohomology (the variety here being the algebraic group and the "Frobenius" morphism being enriched as above). Deligne sketches this at the end of "Applications de la formule des traces aux sommes trigonometriques" in SGA $4\frac{1}{2}$ (Springer Lecture Notes 569, 1977). Springer referred to this as "perhaps the best proof available nowadays ..."

Is there a more complete written account of this geometric approach?

`$W$`

via the Borel picture of cohomology of the flag variety. This note may be the only answer to my question – Jim Humphreys Aug 4 '10 at 18:07