Lemma 3

For each $\lambda \in X_{++}$, let $C_{\lambda}$ be an element of $\mathbf{Z}[X(T)]^W$ having unique maximal term $e^{\lambda}$. Then the family $(C_{\lambda})_{\lambda\in X_{++}}$ is a basis of $\mathbf{Z}[X(T)]^W$.

The proof is identical to that of Prop. 3 of Chap. VI, §3, no. 4, replacing A by $\mathbf{Z}, P$ by X(T) and $P\cap \overline{C}$ by $X_{++}$.

Let $\Theta (G)$ (resp. $\Theta (T)$) be the $\mathbf{C}$-algebra of continuous representative functions on G (resp. T), and let $Z\Theta (G)$ (resp. $\Theta (T)^W$) be the subalgebra consisting of the central (resp. W-invariant) functions. The restriction map $\Theta (G)\rightarrow \Theta (T)$ induces a homomorphism of rings $r: Z\Theta (G)\rightarrow \Theta (T)^W$. On the other hand, the map that associates to a representation $\tau$ its character (that is, the function $g \rightarrow$ Tr $\tau (g)$) extends to a homomorphism of $\mathbf{C}$-algebras Tr$:\mathbf{C}\otimes_{\mathbf{Z}}R(G)\rightarrow Z\Theta (G)$ which, by Spectral Theories, is an isomorphism. Similarly, the canonical injection $X(T)\rightarrow \Theta (T)$ induces an isomorphism of $\mathbf{C}$-algebras $\iota :\mathbf{C}[X(T)]\rightarrow \Theta$(T), which induces an isomorphism $\iota :\mathbf{C}[X(T)]^W\rightarrow \Theta (T)^W$. The diagram

is commutative: indeed, for every representation $\tau : G\rightarrow \mathbf{G}\mathbf{L}(V)$ and all $t\in T$,

Tr $\tau (t) =\sum_{\lambda\in X(T)}$(dim $V_{\lambda}(T)$)$\lambda (t) =\iota$(Ch$\tau$ )$(t)$,

that is, $(r\circ$ Tr)[$\tau ] = (\iota \circ$ Ch)[$\tau ]$.

We can now deduce from the proposition the following result.