Exercise 20, § 4

Lie Groups and Lie Algebras, V

Bourbaki marks this one as harder than the rest.

For every subset {s, s'} of S such that m(s, s') = \infty, let r(s, s') be a real number \leq -1. Equip E with the bilinear form B_r such that

B_r(e_s, e_{s'}) = B_M(e_s, e_{s'}) = - \cos \frac{\pi}{m(s, s')} \quad \text{if } m(s, s') \neq \infty

B_r(e_s, e_{s'}) = r(s, s') \qquad \qquad \qquad \text{if } m(s, s') = \infty.

We define, as for B_M, the reflections $\sigma_s$ with vector $e_s$ , leaving the form B_r invariant.

a) Show that there exists a unique homomorphism $\sigma_r : W \to \mathbf{GL}(E)$ such that $\sigma_r(g_s)$ is equal to the reflection $\sigma_s$ defined above.

b) Show that the assertions in Prop. 4, Th. 1, its Corollaries, and Lemma 1 remain true for $\sigma_r$ .

Solution

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