Lemma 2

Let $\lambda \in \mathfrak{h}^*$. The module $Z(\lambda ) ($§6, no. 3) admits a character that is an element of $\mathbf{Z}\langle P\rangle$, and we have $d$. ch $Z(\lambda ) =e^{\lambda+\rho}$.

Let $\alpha_1, . . . , \alpha_q$ be distinct elements of $R_+$. The $X_{-\alpha_1}^{n_1}X_{-\alpha_2}^{n_2}. . . X_{-\alpha_q}^{n_q}\otimes 1$ form a basis of $Z(\lambda ) ($§6, Prop. 6 (iii)). For $h\in \mathfrak{h}$, we have

Thus, the dimension of $Z(\lambda )^{\lambda-\mu}$ is $\mathfrak{P}(\mu)$. This proves that ch $Z(\lambda )$ is defined, is an element of $\mathbf{Z}\langle P\rangle$, and that

ch $Z(\lambda ) =\sum_\mu\mathfrak{P}(\mu)e^{\lambda-\mu}= Ke^{\lambda}$.

It now suffices to apply Lemma 1.