Example

Assume that $\Delta=\mathbf{N}$. Let T be an indeterminate. There exists a unique isomorphism from the algebra $\mathbf{Z}[\mathbf{N}]$ to the algebra $\mathbf{Z}[T]$ that takes $e^n$ to $T^n$ for all $n\in\mathbf{N}$. For any finite dimensional $\mathbf{N}$-graded vector space E, the image of ch(E) in $\mathbf{Z}[T]$ is the Poincaré polynomial of E (Chap. V, §5, no. 1).

Let E be a $\mathfrak{g}$-module such that $E =\sum_{\lambda\in\mathfrak{h}^*}E^{\lambda}$ and such that each $E^{\lambda}$ is

finite dimensional. We know that $(E^{\lambda})_{\lambda\in\mathfrak{h}^*}$ is a graduation of the vector space E. In what follows we shall reserve the notation ch(E) for the character of E considered as a $\mathfrak{h}^*$-graded vector space. Thus, the character ch(E) is an element of $\mathbf{Z}^{\mathfrak{h}^*}$. If E is finite dimensional, ch(E) $\in \mathbf{Z}[P]$. By formula (6) and Lemma 3 of no. 6, there exists a unique homomorphism from the group $\mathscr{R}(\mathfrak{g})$ to $\mathbf{Z}[P]$ that takes E to ch(E), for any finite dimensional $\mathfrak{g}$-module E; this homomorphism will be denoted by ch. Relation (8) shows that ch is a homomorphism from the ring $\mathscr{R}(\mathfrak{g})$ to the ring $\mathbf{Z}[P]$.