Weyl function definition

Question

In Lie algebra book by Humphreys, in the section 24.1 (page number 136) he defines Weyl function $q$ as $ q = \Pi_{\alpha \gt 0}(\epsilon_{\frac{\alpha}{2}}-\epsilon_{-\frac{\alpha}{2}})$. I can't understand what is $\epsilon_{\frac{\alpha}{2}}$ here.

In the previous paragraph he defines $\epsilon_\lambda$, but I can't understand. Can anyone explain this little bit more clearly?

Thanks in Advance.

Answer 1

For each $\lambda\in H^*$, the function $\epsilon_\lambda:H^*\to F$ is defined to be the characteristic function of $\{\lambda\}$. That is,

$$\epsilon_\lambda(\mu) = \left\{\begin{array}{c}\mbox{ $1$ if }\lambda=\mu \\ \mbox{ $0$ if }\lambda\neq\mu.\end{array}\right.$$