# Applications (and source) of Bourbaki exercise on root systems with two root lengths?

In Chapters 4-6 of Bourbaki's Groupes et algebres de Lie, Exercise 20 for Section VI.1 concerns irreducible (reduced) root systems with roots of two lengths: in other words, systems of types $B_\ell, C_\ell, F_4, G_2$. The exercise is marked with their symbol for "challenging" but indicates no source. Once the root systems are classified, parts (a) and (b) of this exercise can be checked case-by-case, but without using the classification both are nontrivial. These are nice observations and make good exercises (which for some reason I didn't use in my books in spite of writing a marginal note to help locate the exercise again more easily). But I still don't know whether there are any interesting applications.

Start with simple roots $\alpha_1, \dots , \alpha_\ell$, numbered so that only the first $r$ are long. From the basic theory one knows that an irreducible root system has at most two root lengths, with all roots of a given length conjugate under the Weyl group $W$. Say the unequal squared euclidean lengths have ratio $p>1$ (in fact, $p$ is 2 or 3).

(a) If a root $\alpha = \sum_i c_i \alpha_i$ (where $c_i \in \mathbb{Z}$), then $\alpha$ is long if and only if $p | c_i$ for all $i>r$.

This can be proved directly from basic facts about root systems. Or, given the classification, one can check it easily in each case. (See Bourbaki's descriptions.) Similarly, in each case one finds that half of the roots are long (resp., short) for types $B_2 = C_2, F_4, G_2$, otherwise not. (See for example the lists at the end of Springer's IHES paper here.) Even without the classification, there is a simple formula, in terms of the Coxeter number $h$ of $W$:

(b) The number of long, resp. short, roots is $hr$, resp. $h \ell - hr$.

To prove this, one should go back to Steinberg's uniform proof here of Coxeter's observation that the total number of reflecting hyperplanes for $W$ (which can be any finite real reflection group) is $h \ell/2$. This can be found in Bourbaki V.6 or in Chapter 3 of my book on reflection groups.

Question: What if any applications do (a) and (b) have (and did they originate before the 1968 publication of Bourbaki's volume)?

• If I'm understandinf your question correctly, the proof of Lemma 7.1.1(1) in the book "Pseudo-reductive groups" gives a uniform proof of (a) and uses it, and this result in turn underlies the self-contained development given in section 7.1 there for the various exceptional isogenies in characteristic $p\in \{2,3\}$ with the simply connected absolutely simple types with an edge of multiplicity $p$ in the Dynkin diagram. – BCnrd Dec 23 '10 at 0:18
• Yes, part (a) is in the spirit of Chevalley's original construction of exceptional isogenies but seems not to be explicitly used there. Part (2) of your Lemma 7.1.1 depends only on basic facts about root systems and is the essence of (a). I've edited my question a bit. – Jim Humphreys Dec 23 '10 at 20:39
• Dear Jim: Thanks for the clarification. Hopefully the application to the exceptional isogenies in small characteristics will be considered a worthwhile application. (Applied over finite fields of small characteristics, it's the source of some of the exceptional finite groups, Suzuki etc., right?) – BCnrd Dec 23 '10 at 23:18