Exercise 6, § 4

Lie Groups and Lie Algebras, VI

Bourbaki marks this one as harder than the rest.

Let $R$ be an irreducible reduced root system, $(\alpha_1, \ldots, \alpha_l)$ a basis of $R$ , $p_1 \alpha_1 + \cdots + p_l \alpha_l$ the highest root, $a_1 \alpha_1 + \cdots + a_l \alpha_l$ the sum of the positive roots, $m_1, \ldots, m_l$ the exponents of $W(R)$ , $g$ its order, $f$ the connection index.

a) Verify that, in each case,

l! p_1 p_2 \cdots p_l m_1 m_2 \cdots m_l = a_1 a_2 \cdots a_l.

b) Show that

g m_1 m_2 \cdots m_l = f a_1 a_2 \cdots a_l.

(Use $a$ and Prop. 7 of $§ 2$ , no. 4.)

c) For any positive root $\alpha = \sum_i c_i \varphi_i$ , let $e(\alpha) = \sum_i c_i$ . Calculate in each case the polynomial

P(t) = \sum_{\alpha > 0} t^{e(\alpha)}.

Verify $^5$ that $P(t) = t \sum_{i=1}^l (1 + t + \cdots + t^{m_i-1})$ .

$^5$ For a proof that does not use the classification, see: B. KOSTANT, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. of Math.s., Vol. LXXXI (1959), p. 973-1032.

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