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Let $\Phi$ be an irreducible root system and $\Delta$ a simple system (base). Let $W$ be the Weyl group of $\Phi$. Let $\theta$ be the highest root and $h^\vee$ be the dual Coxeter number. Choose the shortest $w \in W$ such that $w(\alpha_i)=\theta$ for some simple root $\alpha_i$. I got the result that the length of $w$ can be determined by $l(w) = h^\vee −2$ (for instance see Lemma 5.1 in here). However I'm not given yet any explicit proof for that result. Can anyone give me proof or references where this formula is introduced and proved?

Thanks in advance.

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Apparently this isn't discussed in any of the published literature, even in the numerous exercises for Bourbaki's Chapter VI on root systems in Lie Groups and Lie Algebras. I'm not sure how strong the evidence is for the assertion here that the minimal length is always $h^\vee -2$. Even though it's true for some small rank cases, has it actually been verified for all types? For example, in the respective types $E_6, E_7, E_8$ we have $h = h^\vee = 12, 18, 30$, and in type $F_4$ we have $h=12$ but $h^\vee = 9$.

The underlying question here is whether an element $w$ of minimal length taking a (necessarily long) simple root to $\theta$ can be characterized uniformly in some way. (A further question is how interesting such a result would be: are there are useful consequences if $\ell(w)$ does turn out as predicted?) From a look at small rank examples, I don't yet see any uniform pattern here for the choice of $w$. So it's important to clarify how far the assertion about $\ell(w)$ has been verified case-by-case. Probably the length question might be studied more uniformly in two ways: using just the axiomatics of root systems, or using specific features of the adjoint representation of a simple Lie algebra. But neither approach looks straightforward.

[By the way, the dual Coxeter number has come up in a number of questions on MO. The geometric work by Coxeter treated arbitrary finite real reflection groups, but later it was seen that the order $h$ of a Coxeter element in a crystallographic group (such as the Weyl group of an irreducible root system) can be characterized as 1 plus the height of the highest root. Then it was useful in Kac-Moody theory to define $h^\vee$ to be 1 plus the height of the highest short root if there are two root lengths.]

UPDATE: As the comments below indicate, the length in question is treated by Cellini and Papi (Prop. 7.1 in their 2004 paper in Advances in Math. which I provided a link for) as well as in the 2011 thesis of a student of Papi. I had overlooked the paper, so I was overly pessimistic about finding an explicit discussion in the published literature.

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    $\begingroup$ Concerning Lemma 5.1 in Kac and Wang, there is a streamlined proof by Wang: see his Lemma 3 in the note ams.org/journals/proc/1999-127-03/S0002-9939-99-04946-1/… (but his concern here is with the dimension of the minimal nonzero nilpotent orbit in a simple Lie algebra). $\endgroup$
    – Jim Humphreys
    Commented Nov 6, 2015 at 19:49
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    $\begingroup$ @user: I guess I see better what you are trying to do with the formula for length of the reflection relative to $\theta$, but all I get from it is a rough estimate: $\ell(w) \geq h^\vee -2$. Am I missing something? (I was looking for an explicit recipe for $w$, but of course this depends on which long simple root you fix and isn't easy to approach except perhaps case-by-case.) $\endgroup$
    – Jim Humphreys
    Commented Nov 7, 2015 at 21:43
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    $\begingroup$ I found the answer which is proved in Proposition 1.3.4 in www1.mat.uniroma1.it/ricerca/dottorato/TESI/ARCHIVIO/… $\endgroup$
    – user
    Commented Nov 8, 2015 at 15:01
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    $\begingroup$ @user: This looks like an efficient general proof, though the cited paper of Cellini-Papi (ams.org/mathscinet-getitem?mr=2078340) involves a broader context and quotes some of the results of Panyushev. It might be instructive to use $F_4$ as an example when working through the details, to see precisely what is needed for the computation of the minimal length. (Is there an explicit statement in the published literature by the authors just mentioned?) $\endgroup$
    – Jim Humphreys
    Commented Nov 8, 2015 at 18:18
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    $\begingroup$ @user: Probably it's best to contact one of the authors directly. Note that Wang has become a professor at U. Virginia, while the joint paper followed up some of his thesis work with Kac two decades ago. $\endgroup$
    – Jim Humphreys
    Commented Nov 9, 2015 at 13:05
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Ok I'm late on this, but I've just seen the question! Another way to look at it: the W-orbit of the highest root consists of the long roots, so the long roots are in bijection with the minimal coset representatives for the stabilizer of the highest root (parabolic subgroup generated by the simple roots orthogonal to the highest roots, i.e. corresponding to nodes in the Dynkin diagram not connected to the affine root in the affine Dynkin diagram). One may ask: what is the relation between the length of a minimal coset representative and the height of the corresponding long root? Actually there is a simple relation between that length and the height of the corresponding coroot (hence the dual Coxeter number...). I needed this in my first article, "Cohomology of the minimal nilpotent orbit". Before submitting the paper, I found a paper of Carter ("Weyl groups and finite Chevalley groups") which dealt with precisely this problem (exchanging the role of roots and coroots, so he was talking about the highest short root); so I quoted it. This is the kind of problem which must have been worked out independently many times! :)

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