Theorem 2

Let E be a finite dimensional simple $\mathfrak{g}$-module, $\lambda$ its highest weight and $(\cdot | \cdot )$ a W-invariant non-degenerate positive symmetric bilinear form on $\mathfrak{h}^*$. Then:

dim E = = 1 +.

Let T be an indeterminate. For all $\nu \in P$, denote by $f_{\nu}$ the homomorphism from $\mathbf{Z}[P]$ to $\mathbf{R}[[T]]$ that takes $e^\mu$ to $e^{(\nu|\mu)T}$ for all $\mu\in P$. Then dim E is the constant term of the series $f_{\nu}$(ch E).

For all $\mu, \nu \in P$, we have

In particular, in view of Chap. VI, §3, no. 3, formula (3),

Hence, setting Card(R$_+$) $= N$,

$f_{\rho}(J(e^\mu))\equiv T^N\prod_{\alpha\in R}(\mu|\alpha )$ (mod $T^{N+1}\mathbf{R}[[T]]$).

The equality $J(e^{\lambda+\rho}) =$ ch(E)$.J(e^{\rho})$ (Th. 1) thus implies that

$T^N\prod_{\alpha\in R_+}(\lambda +\rho |\alpha )\equiv f_{\rho}$(ch $E$)$.T^N\prod_{\alpha\in R_+}(\rho |\alpha )$ (mod $T^{N+1}\mathbf{R}[[T]]$)

so

dim $E =((\prod_{\alpha\in R_+}(\lambda +\rho |\alpha ))/((\prod_{\alpha\in R_+}(\rho |\alpha ))=\prod_{\alpha\in R_+}(1 +\frac{(\lambda|\alpha)}{(\rho|\alpha)})$.

Now, if $\alpha \in R_+,\alpha$ can be identified with an element of $\mathfrak{h}_{\mathbf{R}}$ proportional to $H_{\alpha}$, so