Exercise 10, § 4

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Lie Groups and Lie Algebras, VI

a) Let $(W, S)$ be an irreducible finite Coxeter system. Put

W(t) = \sum_{w \in W} t^{l(w)} \quad (\text{cf. Chap. IV, } § 1, \text{ Exerc. 26.})

Let

m_1, \ldots, m_l

be the exponents of

W

(Chap. V, § 6, no. 2). Verify the formula

W(t) = \prod_{i=1}^l (1 + t + \cdots + t^{m_i})

for small values of

l

(use Th. 1 and Exerc. 26 of Chap. IV, § 1)\footnote{For a proof of this formula that does not use the classification and is valid for all values of

l

, see: L. SOLOMON, The orders of the finite Chevalley groups, Journal of Algebra, Vol. III (1966), p. 376-393.}

b) Let R be a reduced, irreducible root system and W (resp. $(W_a)$ ) the Weyl group (resp. the affine Weyl group) of R, equipped with the Coxeter group structure determined by the choice of an alcove. Define $W(t)$ and the exponents $m_i$ as above and put

W_a(t) = \sum_{u \in W_a} t^{l(u)}.

Verify the formula

W_a(t) = W(t) \prod_{i=1}^l \frac{1}{1 - t^{m_i}} = \prod_{i=1}^l \frac{1 + t + \cdots + t^{m_i}}{1 - t^{m_i}}

for small values of

l

(use Th. 2 and Exerc. 26 of Chap. IV, § 1)\footnote{For a proof of this formula that does not use the classification and is valid for all values of

l

, see: R. BOTTE, An application of the Morse theory to the topology of Lie groups, Bull. Soc. Math. France, Vol. LXXXIV (1956), p. 251-281; N. IWAIHORI and H. MATSUMOTO, On some Bruhat decomposition and the structure of the Hecke rings of

p

-adic Chevalley groups, Publ. Math. Inst. Hautes Et. Sci., no. 25 (1965), p. 5-48.}.

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Exercise 10, § 4

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