I have a question about the modules over a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$.
Let $\mathfrak{h} \subset \mathfrak{g}$ be a Cartan subalgebra. For a given $\lambda \in \mathfrak{h}$, let $\chi_{\lambda} : Z(\mathfrak{g}) \rightarrow \mathbb{C}$ be the central characters associated to $\lambda$. By the Harish-Chandra Theorem we know that $$ \chi_{\lambda} = \chi_{\mu} \text{ if and only if } \mu \in W\cdot \lambda,$$ where $W$ is the Weyl group of $\mathfrak{g}$.
$\bf My$ $\bf question:$ Can we prove this result directly without the Harish-Chandra Theorem? (I am sorry that this question may be a bit fuzzy. I will be glad to see a proof which does not use the Harish-Chandra Theorem explicitly). Thanks very much!