Exercise 26, § 1

Lie Groups and Lie Algebras, IV

Let $(W, S)$ be a Coxeter system, with $S$ finite. For any subset $H$ of $W$ , denote by $H(t)$ the formal power series with integer coefficients defined by

H(t) = \sum_{w \in H} t^{l(w)}.

a) Assume that $\mathrm{Card}(S) = 2$ . Show that

W(t) = (1 + t - t^m - t^{m+1})/(1 - t) \quad \text{if } W \text{ is of finite order } 2m,

W(t) = (1 + t)/(1 - t) \quad \text{if } W \text{ is infinite}.

b) Assume that W is finite. Let $w_0$ be the element of W of greatest length (Exerc. 22) and let $m = l(w_0)$ . Show that

W(t) = t^m W(t^{-1})

(use Exerc. 22).

c) Let X be a subset of S. Denote by $A_X$ the set of $(X, \varnothing)$ -reduced elements of W (Exerc. 3) and $W_X$ the subgroup of W generated by X. We know (Exerc. 3) that an element $w \in W$ belongs to $A_X$ if and only if $l(xw) = l(w) + 1$ for all $x \in X$ , that every element $w \in W$ can be written uniquely in the form $w = uv$ with $u \in W_X$ and $v \in A_X$ , and that in that case $l(w) = l(u) + l(v)$ . Deduce the formula

W(t) = W_X(t) \cdot A_X(t).

d) Retain the above notation and denote by $B_X$ the set of $w \in A_X$ such that $l(sw) = l(w) - 1$ for all $s \in S - X$ . Show that

A_X(t) = \sum_{X \subset Y \subset S} B_Y(t).

Deduce that

B_X(t) = \sum_{X \subset Y \subset S} \varepsilon(Y - X) A_Y(t) \quad \text{with} \quad \varepsilon(Z) = (-1)^{\operatorname{Card}(Z)}

(use Exerc. 25).

e) Assume that W is finite and define m and $w_0$ as in b). Show that $B_\varnothing = \{ w_0 \}$ and that

t^m = \sum_{Y \subset S} \varepsilon(Y) \frac{W(t)}{W_Y(t)}

(use c) and d)).

f) Assume that W is infinite. Show that $B_\varnothing = \varnothing$ and that

0 = \sum_{Y \subset S} \frac{\varepsilon(Y)}{W_Y(t)}.

g) Show that the formal power series $W(t)$ is a rational function of t (use f) and argue by induction on Card(S)). Show that this rational function vanishes whenever t is a root of unity, and that $1/W(\infty)$ is an integer. Show that $\frac{1}{W(t^{-1})} \in \mathbf{Z}[[t]]$ .

Solution

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