Let G be a compact Lie group with semisimple Lie algebra $\mathfrak{g}$. Let $\mathfrak{t}$ be a Cartan subalgebra of $\mathfrak{g}$.
In the book compact Lie groups by Sepanski, the definition of Weyl chambers is the following
Definition: The connected components of ${(i\mathfrak{t})}^*$ \ $ \cup_{\alpha \in \Delta(\mathfrak{g}_\mathbb{C})} (\alpha^\perp )$ are called the Weyl chambers of ${(i\mathfrak{t})}^*$
I'm not sure what is the definition of $\alpha^\perp $, I would say it is the set $\lbrace \lambda \in {(i\mathfrak{t})}^*, B(\lambda, \alpha)=0\rbrace $, where B denotes the Killing form, am I right ?
In other references they define the Weyl chambers to be the Connected components of ${(i\mathfrak{t})}^*$ \ $\cup_{\alpha \in \Delta(\mathfrak{g}_\mathbb{C})} (H_\alpha )$, where $H_\alpha $ is the hyperplane perpendicular to the root $\alpha$, but even here I don't know what is the definition of $H_\alpha$ ?
Any help would be greatly appreciated!