Longest element of Weyl group of a simple Lie algebra action on Weyl chambers

Question

Let, $\mathfrak{g}$ be a complex simple Lie algebra with Weyl group $W$,also let $\omega_0$ be the longest element of the Weyl group. We Know that Weyl group acts on the set of Weyl chambers freely and transitively and also $-\omega_0$ fixes the fundamental Weyl chamber(as a set). Now my question is, does it fixes all chambers,i.e Does the automorphism $-\omega_0$ is actually identity on the set of weyl chambers? Thanks in advance.