Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 3 / 2 / ii

Choose a positive system $R^{+}$ and its simple roots $\Delta =\{{\alpha }_1,\dots ,{\alpha }_{\ell }\}$. The corresponding fundamental Weyl chamber is the open coneIts Closed dominant Weyl chamber replaces all strict inequalities by weak inequalities. The facts used here are that every positive root is a nonnegative integral combination of the simple roots, and that the Weyl chambers are the connected components left after removing the root-orthogonal hyperplanes. Hence the displayed inequalities describe one such component. The Weyl group permutes these components simply transitively, and every orbit in $E$ meets the Closed dominant Weyl chamber in exactly one point. A point on a wall may have a nontrivial stabilizer, which is why uniqueness of the chamber point does not always mean uniqueness of the group element carrying it there.