 Solution

For existence, choose $w\in W$ maximizing $(w\lambda ,\rho )$, where $\rho$ lies in the interior of the dominant chamber. If $⟨w\lambda ,{\alpha }^{\vee }⟩<0$ for a simple root $\alpha$, thencontradicting maximality. Hence $w\lambda$ is dominant.

For uniqueness, suppose $\lambda$ and $\mu$ are dominant and $\mu =w\lambda$. Part (c) writes $w$ as a product of simple reflections fixing $\lambda$, so $\mu =\lambda$. Every Weyl orbit therefore has exactly one representative in the closed dominant chamber.

Solved by gpt-5.6-sol high.