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?

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    $\begingroup$ Jim, I have a 4-page .pdf file by Boyarchenko which gives the argument in the semisimple case (the essential case), referring in a couple of places to some things proved in SGA 4.5 and in the book of Kiehl-Weissauer. The main tool (apart from the Grothendieck-Lefschetz trace formula) is a computation of the $\ell$-adic cohomology algebra of a Borel subgroup. I should find some time to read through it carefully (meant to do it a while ago), but it looks quite complete. In particular, Boyarchenko was not aware of a complete account of this approach in the published literature. $\endgroup$ – BCnrd Aug 4 '10 at 15:05
  • $\begingroup$ Thanks very much for the file, which I've started to look at carefully (in spite of having too little background). Proving the unified formula by any method is nontrivial; each method has its own prerequisites. The formula itself mainly involves root system data plus combinatorics (Poincare series) of the "Weyl group" of the BN-pair. This isn't always a usual crystallographic group in extreme cases: Steinberg and Springer handle this in different ways. The note brings in $W$ via the Borel picture of cohomology of the flag variety. This note may be the only answer to my question $\endgroup$ – Jim Humphreys Aug 4 '10 at 18:07

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