Every central character of a semisimple enveloping algebra arises from a weight

Define $\psi :=\chi \circ {\mathrm{HC}}_{\rho }^{-1}\text{\hspace{0.17em}⁣}:S(\mathfrak{h}{)}^W\to \mathbb{C}$. It is a unital $\mathbb{C}$-algebra homomorphism by [F1], and $\psi ({\mathrm{HC}}_{\rho }(z))=\chi (z)$ for every $z$. Its image contains $1$ and is closed under the $\mathbb{C}$-scalars, so the image is all of $\mathbb{C}$; hence $S(\mathfrak{h}{)}^W\mathrm{/}\ker \psi \cong \mathbb{C}$ is a field and $\ker \psi$ is a maximal, in particular proper, ideal of $S(\mathfrak{h}{)}^W$.

By [F2], $S(\mathfrak{h})$ is a finitely generated $S(\mathfrak{h}{)}^W$-module, so [F3] applies with $R=S(\mathfrak{h}{)}^W$, $I=\ker \psi$ and $M=S(\mathfrak{h})$: if $(\ker \psi )S(\mathfrak{h})=S(\mathfrak{h})$, then $(1-a)S(\mathfrak{h})=0$ for some $a\in \ker \psi$, and evaluating at $1\in S(\mathfrak{h})$ gives $1=a$, contradicting $\psi (1)=1$. Therefore $(\ker \psi )S(\mathfrak{h})\mathrm{\ne }S(\mathfrak{h})$, and this ideal is proper.

By [F4] the proper ideal $(\ker \psi )S(\mathfrak{h})$ is contained in a maximal ideal $\mathfrak{n}$ of $S(\mathfrak{h})$; in particular $\ker \psi \subseteq \mathfrak{n}$. The contraction $\mathfrak{n}\cap S(\mathfrak{h}{)}^W$ is proper because $1\mathrm{\notin }\mathfrak{n}$, and contains the maximal ideal $\ker \psi$; hence $\mathfrak{n}\cap S(\mathfrak{h}{)}^W=\ker \psi$. The quotient $S(\mathfrak{h})\mathrm{/}\mathfrak{n}$ is a field by [F5] containing the field $S(\mathfrak{h}{)}^W\mathrm{/}\ker \psi \cong \mathbb{C}$ of step 1.1, and it is finite-dimensional over it because $S(\mathfrak{h})$ is a finite $S(\mathfrak{h}{)}^W$-module.

A finite-dimensional field extension of the algebraically closed field $\mathbb{C}$ is $\mathbb{C}$ itself by [F6], so the quotient map of step 3.1 is a unital $\mathbb{C}$-algebra homomorphism $\phi \text{\hspace{0.17em}⁣}:S(\mathfrak{h})\to \mathbb{C}$ extending $\psi$: for $a\in S(\mathfrak{h}{)}^W$ the class $a+\mathfrak{n}$ depends only on $a+\ker \psi$ and equals $\psi (a)$.

The homomorphism $\phi$ is evaluation at a weight: fix a $\mathbb{C}$-basis $h_1,\dots ,h_r$ of $\mathfrak{h}$; since $S(\mathfrak{h})=\mathbb{C}[h_1,\dots ,h_r]$ with the $h_i$ as coordinate functions, $\phi$ is determined by the scalars $\phi (h_i)$, which define a unique $\lambda \in {\mathfrak{h}}^{\ast }$ by $\lambda (h_i):=\phi (h_i)$, and $\phi (p)=p(\lambda )$ for every polynomial $p$.

Combining the identities, for every $z\in Z(U(\mathfrak{g}))$: $\chi (z)=\psi ({\mathrm{HC}}_{\rho }(z))=\phi ({\mathrm{HC}}_{\rho }(z))={\mathrm{HC}}_{\rho }(z)(\lambda )=\mathrm{pr}(z)(\lambda -\rho )$ by [F1]; writing $\mu :=\lambda -\rho$ gives $\chi ={\chi }_{\mu }$. Thus every unital homomorphism $Z(U(\mathfrak{g}))\to \mathbb{C}$ is a ${\chi }_{\mu }$.

By [F2], $Z(U(\mathfrak{g}))\cong S(\mathfrak{h}{)}^W$ is a finitely generated commutative $\mathbb{C}$-algebra, so by [F5] every maximal ideal of $Z(U(\mathfrak{g}))$ is the kernel of a unital homomorphism to $\mathbb{C}$, hence by step 6.1 of the form $\ker {\chi }_{\lambda }$. Conversely each ${\chi }_{\lambda }$ is a surjective unital homomorphism onto the field $\mathbb{C}$, so $\ker {\chi }_{\lambda }$ is maximal by [F5], and $\ker {\chi }_{\lambda }=\ker {\chi }_{\mu }$ implies ${\chi }_{\lambda }={\chi }_{\mu }$ because both factor through the common quotient, which is $\mathbb{C}$ via the unital structure map. By [F7], ${\chi }_{\lambda }={\chi }_{\mu }$ exactly when $\mu \in W\cdot \lambda$; hence $\lambda \mapsto \ker {\chi }_{\lambda }$ induces a bijection ${\mathfrak{h}}^{\ast }\mathrm{/}W\to \mathrm{Max}Z(U(\mathfrak{g}))$, and the stated equivalences follow.