Theorem 2

(i) Let $(\varpi_{\alpha})_{\alpha\in B}$ be the family of fundamental weights relative to B. Let $(T_{\alpha})_{\alpha\in B}$ be a family of indeterminates. The map $f \rightarrow f(([\varpi_{\alpha}])_{\alpha\in B})$ from $\mathbf{Z}[(T_{\alpha})_{\alpha\in B}]$ to $\mathscr{R}(\mathfrak{g})$ is an isomorphism of rings.

(ii) The homomorphism ch from $\mathscr{R}(\mathfrak{g})$ to $\mathbf{Z}[P]$ induces an isomorphism from the ring $\mathscr{R}(\mathfrak{g})$ to the ring $\mathbf{Z}[P]^W$.

(iii) Let E be a finite dimensional $\mathfrak{g}$-module. If ch $E =\sum_{\lambda\in P_{++}}m_{\lambda}$ch[$\lambda ]$, the isotypical component of E of highest weight $\lambda$ has length $m_{\lambda}$.

The family $([\lambda ])_{\lambda\in P_{++}}$ is a basis of the $\mathbf{Z}$-module $\mathscr{R}(\mathfrak{g})$, and the family (ch[$\lambda ]$)$_{\lambda\in P_{++}}$ is a basis of the $\mathbf{Z}$-module $\mathbf{Z}[P]^W$ (Lemma 6, and Chap. VI, §3, no. 4, Prop. 3). This proves (ii) and (iii). Assertion (i) follows from (ii), Lemma 6 and Chap. VI, §3, no. 4, Th. 1.