Harish-Chandra isomorphism for the center

Let $f_z(\mu ):=\mathrm{pr}(z)(\mu )$ for $z\in Z(U(\mathfrak{g}))$. Multiplicativity from [F1] shows that ${\mathrm{HC}}_{\rho }(z)(\lambda )=f_z(\lambda -\rho )$ defines an algebra homomorphism. To identify its image, fix a simple reflection $s_i$ and a weight $\mu$ with $⟨\mu +\rho ,{\alpha }_i^{\vee }⟩=m\in {\mathbb{Z}}_{>0}$. Work in the Verma module $M(\mu )$. By The simple-root singular vector in a Verma module, $f_i^m{v}_{\mu }$ is a nonzero highest-weight vector of weight $s_i\cdot \mu$. The central element $z$ acts on all of $M(\mu )$ by the scalar $f_z(\mu )$, while The Harish-Chandra projection computes the highest-weight scalar applied to the cyclic submodule generated by this singular vector gives the scalar $f_z(s_i\cdot \mu )$. Thus they are equal. The weights with $m\in {\mathbb{Z}}_{>0}$ are Zariski dense: a polynomial vanishing on all these infinitely many parallel hyperplanes vanishes identically. Hence $f_z$ is invariant under the dot action of every simple reflection, and therefore under $W$. By The rho-shift intertwines the dot and ordinary Weyl actions, ${\mathrm{HC}}_{\rho }(z)$ lies in $S(\mathfrak{h}{)}^W$.

The PBW filtration on $U(\mathfrak{g})$ induces a filtration on the center, and the Harish-Chandra projection does not increase PBW degree because it is defined by projecting the PBW decomposition onto the $U(\mathfrak{h})$ summand. Translation by $\rho$ preserves polynomial degree on $S(\mathfrak{h})$, so ${\mathrm{HC}}_{\rho }$ is a filtered algebra homomorphism.

The PBW theorem gives the symmetrization map $\mathrm{sym}:S(\mathfrak{g})\to U(\mathfrak{g}),x_1\cdots x_m\mapsto \frac{1}{m!}{\sum }_{\sigma \in S_m}{x}_{\sigma (1)}\cdots x_{\sigma (m)},$ as a filtration-preserving vector-space isomorphism whose associated graded map is the identity. The adjoint action is a derivation on both sides, so $\mathrm{sym}$ is $\mathfrak{g}$-equivariant. Since an element of $U(\mathfrak{g})$ is central exactly when it is killed by the adjoint action of the generators $\mathfrak{g}$, symmetrization restricts to invariant lifts and gives $\mathrm{gr}Z(U(\mathfrak{g}))=S(\mathfrak{g}{)}^{\mathfrak{g}}\mathrm{.}$ PBW gives an ordered monomial basis for the enveloping algebra

In the PBW decomposition, the top-degree part of $\mathrm{pr}(z)$ is obtained by discarding every monomial whose symbol contains a factor from ${\mathfrak{n}}^{-}$ or ${\mathfrak{n}}^{+}$. Translation by $\rho$ changes only lower-degree terms. Thus the associated-graded Harish-Chandra map is exactly restriction $S(\mathfrak{g}{)}^{\mathfrak{g}}⟶S(\mathfrak{h}{)}^W,$ which is an isomorphism by [A1, F2].

A filtered algebra homomorphism whose associated graded map is an isomorphism is itself an isomorphism: lift target elements degree by degree for surjectivity, and take the leading symbol of a kernel element for injectivity. Applying this to steps 2.1 and 4.1 shows that ${\mathrm{HC}}_{\rho }$ is an algebra isomorphism $Z(U(\mathfrak{g}))\stackrel{\sim }{\to }S(\mathfrak{h}{)}^W$.