TITLE: Length of Weyl group element mapping highest root to a simple root
QUESTION [5 upvotes]: 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.

REPLY [3 votes]: 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! :)