Inner product of highest root.

Question

In Macdonalds book on affine Hecke algebras and orthogonal polynomials, it is stated, without proof, that $\langle \varphi^\vee,\alpha\rangle \in \{0,1\}$ for all positive roots $\alpha$ not equal to $\varphi$. Here $\varphi$ is the highest root of an fixed root system and we assume that the square length of this root equals 2, does someone have a reference for the statement or a poof? Thanks!

Answer 1

Well, if I am interpreting your notation/scaling correctly, $\langle \varphi^\vee, \alpha \rangle = p - q$ where $\alpha - p\varphi, \dots \alpha,\dots, \alpha + q\varphi$ is the $\varphi$ root string through $\alpha$ (see Humphreys, Introduction to Lie algebras, Section 9.4).

However, $\alpha+\varphi$ is not a root since $\varphi$ is the highest root which forces $q=0$. Similarly $\alpha - 2\varphi$ cannot be a root as otherwise $2\varphi - \alpha$ would also be a root and would be higher than $\varphi$ so that $p \leq 1$. Thus $\langle \varphi^\vee, \alpha \rangle \in \{0,1\}$.