Exercise 1, § 6

Let R be a reduced root system in a real vector space V. Give the space V a scalar product invariant under W(R), and the space $\mathbf{S}(V)$ the corresponding scalar product (Topological Vector Spaces, Chap. V, §3, no. 3, formulas (13) and (14)); thus, for $x_1, . . . , x_n, y_1, . . . , y_n$ in V,

Choose a chamber C of R; put N = Card(R$_+$)$,\rho =\frac{1}{2}\sum_{\alpha\in R_+}\alpha$ and $w(R) =$

Card(W(R)). Let P be the element $\prod_{\alpha\in R_+}\alpha$ of $\mathbf{S}^N(V)$.

a) Show that $P =\frac{1}{N!}\sum_{w\in W(R)}\varepsilon (w)(w\rho )^N$ (cf. Chap. VI, §3, no. 3, Prop. 2).

b) Deduce the equality $(P|P) =w(R)\prod_{\alpha\in R_+}(\rho |\alpha )$.

c) Prove that $(P|P) = 2^{-N}w(R)\prod_{i=1}^lm_i$!$\prod_{\alpha\in R_+}(\alpha |\alpha ) = 2^{-N}\prod_{i=1}^l(m_i$+1)!$\prod_{\alpha\in R_+}(\alpha |\alpha )$,

where $m_1, . . . , m_l$ are the exponents of W(R) (use Exerc. 3 of Chap. VIII, §9).