Primitive ideals are partitioned by their dot-orbit central character

By [F1], $I\cap Z(U(\mathfrak{g}))=\ker {\chi }_I$ for a central character ${\chi }_I$; any central character with the same kernel equals ${\chi }_I$, because the kernel determines the unital homomorphism through the quotient $Z(U(\mathfrak{g}))\mathrm{/}\ker {\chi }_I\cong \mathbb{C}$. Hence ${\chi }_I$ is unique, and it is determined by $I$.

By [F2] there is $\lambda \in {\mathfrak{h}}^{\ast }$ with ${\chi }_I={\chi }_{\lambda }$. If also ${\chi }_I={\chi }_{\mu }$, then ${\chi }_{\lambda }={\chi }_{\mu }$, so $\mu \in W\cdot \lambda$ by [F3]; hence the dot orbit $W\cdot \lambda$ is independent of the choice of $\lambda$ and determined by $I$.

For primitive ideals $I,J$, the identity $I\cap Z=J\cap Z$ is equivalent to $\ker {\chi }_I=\ker {\chi }_J$ by [F1], which is equivalent to ${\chi }_I={\chi }_J$ because a central character is determined by its kernel; and by step 2.1 with [F3] this is equivalent to $W\cdot \lambda =W\cdot \mu$ for any $\lambda ,\mu$ realizing ${\chi }_I,{\chi }_J$. Thus the fibres of $I\mapsto {\chi }_I$ on primitive ideals are exactly the sets of primitive ideals with a common dot-Weyl orbit, i.e. the primitive ideals are partitioned by their central characters.