Exercise 22, § 1

Lie Groups and Lie Algebras, VI

Bourbaki marks this one as harder than the rest.

With the assumptions and notations of Prop. 33 of no. 11, denote by $\omega_1, \ldots, \omega_l$ the fundamental weights of $R$ .

a) Show that $c^{-1}(\omega_i) = \omega_i - \theta_i$ . Deduce that $1-c$ maps $P(R)$ to $Q(R)$ .

b) Let $f$ be the connection index of $R$ , and let $m_1, \ldots, m_l$ be the exponents of $W(R)$ . Show that

f = \det(1-c) = \prod_{j=1}^l (1 - w^{m_j}) = 2^l \prod_{j=1}^l \sin(m_j \pi / h)

with $w = e^{2 \pi i / h}$ .

c) Let $p$ be a prime number. Write $h$ in the form $h = p^n H$ , with $H$ not divisible by $p$ . Show that $p$ divides $f$ if and only if $H$ divides one of the $m_j$ .²

Solution

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