Exercise 5, § 3

Lie Groups and Lie Algebras, V

Bourbaki marks this one as harder than the rest.

Let V be a finite dimensional real vector space, W a finite subgroup of $\mathbf{GL}(V)$ generated by reflections, C a chamber of W and S the set of walls of C. If $J \subset S$ , let $W_J$ be the subgroup of W generated by the $s_H$ for $H \in J$ , and let $\varepsilon(J) = (-1)^{\mathrm{Card}(J)}$ . Show that

\frac{1}{\mathrm{Card}(W)} = \sum_{J \subset S} \frac{\varepsilon(J)}{\mathrm{Card}(W_J)}.

(*)

(Let $(x|y)$ be a scalar product on V invariant under W, $\Sigma$ the unit sphere of V, and $\mu$ a positive measure on $\Sigma$ invariant under W and of total mass 1. If $H \in S$ , let $D_H$ be the open half-space determined by H and containing C. Let $E_H = D_H \cap \Sigma$ . Then

(-\overline{C}) \cap \Sigma = \bigcap_{H \in S} (\Sigma - E_H),

\frac{1}{\mathrm{Card}(W)} = \mu(\overline{C} \cap \Sigma) = \int_{\Sigma} \prod_{H \in S} (1 - \varphi_{E_H}) d\mu

= \sum_{J \subset S} \varepsilon(J) \mu \left( \bigcap_{H \in J} E_H \right).

Conclude by remarking that $\bigcap_{H \in J} E_H$ is the intersection with $\Sigma$ of a chamber of $W_J$ , and hence has measure equal to $1/\mathrm{Card}(W_J)$ .)

Recover formula (*) by means of Exerc. 26 e) of Chap. IV, § 1 (put $t = 1$ in the identity proved in the exercise in question).

Solution

No solution has been written yet.