Central characters are dot-Weyl orbits

If $\mu =w\cdot \lambda$ for some $w\in W$, then $\mu +\rho =w(\lambda +\rho )$. Under the stated AC, [F2] makes every central element $z$ determine a Weyl-invariant polynomial ${\mathrm{HC}}_{\rho }(z)\in S(\mathfrak{h}{)}^W$, and The Harish-Chandra projection computes the highest-weight scalar gives ${\chi }_{\lambda }(z)=\mathrm{pr}(z)(\lambda )={\mathrm{HC}}_{\rho }(z)(\lambda +\rho ),{\chi }_{\mu }(z)={\mathrm{HC}}_{\rho }(z)(\mu +\rho )\mathrm{.}$ Since ${\mathrm{HC}}_{\rho }(z)$ is ordinarily $W$-invariant, ${\mathrm{HC}}_{\rho }(z)(\mu +\rho )={\mathrm{HC}}_{\rho }(z)(\lambda +\rho )$. Hence ${\chi }_{\mu }(z)={\chi }_{\lambda }(z)$ for every central $z$, so ${\chi }_{\mu }={\chi }_{\lambda }$.

Let $a,b\in {\mathfrak{h}}^{\ast }$ have distinct $W$-orbits $O_a,O_b$. Their union is finite and disjoint by [F1]. For each $x\in O_a$ and each other $y\in O_a\cup O_b$, choose an affine linear polynomial $L_{x,y}$ with $L_{x,y}(x)=1$ and $L_{x,y}(y)=0$; a coordinate on which $x,y$ differ supplies one. Then $p_x={\prod }_{y\mathrm{\ne }x}{L}_{x,y}$ equals $1$ at $x$ and $0$ at every other point of the union. The polynomial $p={\sum }_{x\in O_a}{p}_x$ equals $1$ on $O_a$ and $0$ on $O_b$. Averaging $p$ over the finite group, $q(z)=\mathrm{\mid }W{\mathrm{\mid }}^{-1}{\sum }_{w\in W}p(wz)$, gives $q\in S(\mathfrak{h}{)}^W$ with $q(a)=1$ and $q(b)=0$.

Conversely, if ${\chi }_{\mu }={\chi }_{\lambda }$, step 1.1 and the surjectivity in [F2] under the stated AC show that every polynomial in $S(\mathfrak{h}{)}^W$ takes the same value at $\lambda +\rho$ and $\mu +\rho$. Step 1.2 rules out distinct ordinary $W$-orbits, so $\mu$ lies in the dot orbit of $\lambda$.

Therefore equal central characters are exactly the dot-Weyl orbits.