Theorem 2

Let U be the enveloping algebra of $\mathfrak{g}$, Z its centre, $V\subset U$ the enveloping algebra of $\mathfrak{h}($identified with $\mathbf{S}(\mathfrak{h}))$, $U^0$ the commutant of V in U, $\varphi$ the Harish-Chandra homomorphism (§6, no. 4) from $U^0$ to $V$ relative to $B$.

Let $S(\mathfrak h)^W$ be the set of elements of $S(\mathfrak h)$ invariant under the action of $W$. Then $(\delta\circ\varphi)|_Z$ is an isomorphism from $Z$ to $S(\mathfrak h)^W$, independent of the choice of $B$.

a) Let $P_{++}$ be the set of dominant weights of $R$, $w\in W$, $\lambda\in P_{++}$, $\mu=w\lambda$. Then $Z(\mu-\rho)$ is isomorphic to a submodule of $Z(\lambda-\rho)$ (§6, no. 3, Cor. 2 of Prop. 6), and $\varphi(u)(\lambda-\rho)=\varphi(u)(\mu-\rho)$ for all $u\in Z$ (§6, no. 4, Prop. 7). Thus, the polynomial functions $(\delta\circ\varphi)(u)$ and $(\delta\circ\varphi)(u)\circ w$ on $\mathfrak h^*$ coincide on $P_{++}$. But $P_{++}$ is dense in $\mathfrak h^*$ in the Zariski topology: this can be seen by identifying $\mathfrak h^*$ with $k^B$ by means of the basis consisting of the fundamental weights $\varpi_\alpha$, and by applying Prop. 9 of Algebra, Chap. IV, §2, no. 3. Hence

which proves that $(\delta\circ\varphi)(Z)\subset S(\mathfrak h)^W$.

b) Let $\eta$ be the isomorphism from $I(\mathfrak g)$ to $S(\mathfrak h)^W$ defined in no. 3, Cor. 2 of Th. 1. Consider the canonical isomorphism from the $\mathfrak g$-module $U$ to the $\mathfrak g$-module $S(\mathfrak g)$ (Chap. I, §2, no. 8), and let $\theta$ be its restriction to $Z$. Then $\theta(Z)=I(\mathfrak g)$. Let $z$ be an element of $Z$ with filtration $\leqslant f$ in $U$.

Introduce the notations of §6, no. 4, and put

Let $v((q_i),(m_i),(p_i))$ be the monomial $X_{\alpha_1}^{q_1}\cdots X_{\alpha_n}^{q_n}H_1^{m_1}\cdots H_l^{m_l}X_{\alpha_1}^{p_1}\cdots X_{\alpha_n}^{p_n}$ calculated in $S(\mathfrak g)$. Denoting by $S_d(\mathfrak g)$ the sum of the homogeneous components of $S(\mathfrak g)$ of degrees $0,1,\ldots,d$, we have

so

and consequently

c) We show that $\delta\circ\varphi:Z\longrightarrow S(\mathfrak h)^W$ is bijective. The canonical filtrations on $U$ and $S(\mathfrak g)$ induce filtrations on $Z$, $I(\mathfrak g)$ and $S(\mathfrak h)^W$, and $\theta$, $\eta$ are compatible with these filtrations, so that gr($\eta \circ \theta$ ) is an isomorphism from the vector space gr(Z) to the vector space gr($\mathbf{S}(\mathfrak{h})^W$). By (3), gr($\varphi$ ) $=$ gr($\eta \circ \theta$ ), and it is clear that gr($\delta$ ) is the identity. Hence gr($\delta \circ \varphi$ ) is bijective, so

is bijective (Commutative Algebra, Chap. III, §2, no. 8, Cor. 1 and 2 of Th. 1).

d) Recall the notations in a). Let E be a simple $\mathfrak{g}$-module of highest weight $\lambda$, and $\chi$ its central character (§6, no. 1, Def. 2). Let $\varphi '$ and $\delta '$ be the homomorphisms analogous to $\varphi$ and $\delta$ relative to the basis $w(B)$. The highest weight of E relative to $w(B)$ is $w(\lambda )$. By §6, no. 4, Prop. 7,

for all $u\in Z$, so, by a),

Thus, the polynomial functions $(\delta \circ \varphi )(u)$ and $(\delta '\circ \varphi ')(u)$ coincide on $w(P_{++}) +w\rho$, and hence are equal.