Proposition 2

The homomorphism Ch induces an isomorphism from R(G) to $Z[X(T)]^W$.

For $\lambda \in X_{++}$, denote by $[\lambda ]$ the class in R(G) of the irreducible representation of highest weight $\lambda$. Since the family $([\lambda ])_{\lambda\in X_{++}}$ is a basis of the $\mathbf{Z}$-module R(G), it suffices to prove the following assertion:

The family (Ch[$\lambda ]$)$_{\lambda\in X_{++}}$ is a basis of the $\mathbf{Z}$-module $\mathbf{Z}[X(T)]^W$.

For every element $u=\sum_{\lambda}a_{\lambda}e^{\lambda}$ of $\mathbf{Z}[X(T)]$, a term $a_{\lambda}e^{\lambda}$ in $u$ is called maximal if $\lambda$ is a maximal element of the set of $\mu\in X(T)$ such that $a_\mu\not= 0$. Th. 1 implies that Ch[$\lambda ]$ has a unique maximal term, namely $e^{\lambda}$. Thus, the proposition follows from the following lemma.