Let $\mathfrak{g}$ be a semisimple finite dimensional Lie algebra and $V_\lambda$, resp. $V_\mu$ its finite dimensional highest weight modules with highest weights $\lambda$, resp. $\mu$. Let $\chi_\lambda, \chi_\mu : C(\mathcal{U}(\mathfrak{g})) \rightarrow \mathbb{C}$ be the corresponding central characters. Harish-Chandra theorem asserts that $\chi_\lambda = \chi_\mu$ if and only if $w(\lambda+\delta)-\delta = \lambda$ for some $w$ in the Weyl group, where $\delta$ is the Weyl vector, i.e. the sum of all fundamental weights.
Is it also true in this setting, that $V_\lambda \simeq V_\mu$ if and only if $\chi_\mu = \chi_\nu$ ?
How is this version of the theorem related to the fact that Harish-Chandra homomorphism is an isomorphism ?
Thank you very much for your answers. (I am studying a program in mathematical physics and trying to figure out how general is the procedure of labeling irreducible representations by values of Casimir operators, as physicists do so often)