Exercise 3, § 2

Let G be a group, B a subgroup of G and (C(w))_{w∈W} the family of double cosets of G with respect to B. Then B is called a Tits subgroup of G if there exists a subset S of W such that the following conditions are satisfied:

(1) the union of the C(s) for s ∈ S generates G; (2) for all s ∈ S, the set B ∪ C(s) is a subgroup of G and the index of B in B ∪ C(s) is ≥ 3; (3) for all s ∈ S and all w ∈ W, there exists an element w' ∈ W such that C(s).C(w) ⊂ C(w) ∪ C(w').

From now on we assume that B is a Tits subgroup of G and that we are given a subset S of W satisfying conditions (1), (2) and (3).

a) Show that C(s)^{-1} = C(s) and that C(s).C(s) = B ∪ C(s) for all s ∈ S. Show that, for all s ∈ S and all w ∈ W, there exists an element w'' ∈ W such that C(w).C(s) ⊂ C(w) ∪ C(w'').

b) If w ∈ W, the length of w, denoted by l(w), is the smallest integer n ≥ 0 for which there exist s_1, ..., s_n ∈ S with C(w) ⊂ C(s_1)...C(s_n). Show that l(w) is finite for all w ∈ W.

Let u, v ∈ W with l(u) < l(v) and let s ∈ S. Show that, if C(v) ⊂ C(u).C(s) (resp. C(v) ⊂ C(s).C(u)), then C(v) = C(u).C(s) (resp, C(v) = C(s).C(u)) (argue by induction on the length of u. If C(v) ≠ C(u).C(s), then C(u).C(s) = C(u) ∪ C(v). By using the induction hypothesis, show that there exist t ∈ S and w ∈ W such that

From the relations C(v) ⊂ C(u).C(s) = C(t).C(w).C(s) and C(t).C(w) = C(u) ≠ C(v), deduce the existence of an element w' ≠ w such that

$C(w') \subset C(w).C(s)$ and $C(v) \subset C(t).C(w')$, so $l(w') \geq l(v) - 1 > l(u) - 1 = l(w)$. The induction hypothesis now implies that

Moreover,

and also

which is a contradiction since $w \neq u, v, w'$.

c) Show that, for all $w \in W$ and all $s \in S$, there exists a unique element, denoted by $s.w$ (resp, $w * s$), distinct from $w$ and such that

(Show by induction on $l(w)$ that $C(s).C(w) \neq C(w)$. For this, write

with $t \in S, u \in W$ and $l(w) = l(u) + 1$. If $C(s).C(w) = C(w)$, then

and multiplying on the right by $C(t)$, we obtain that $C(u) \cup C(w) = C(s).C(u) \cup C(w)$. Since $C(u) \neq C(s).C(u)$ by the induction hypothesis, we have, by b),

and hence

which is absurd).

d) Let $s \in S$. Show that the map $p_s : w \mapsto s.w$ (resp. $q_s : w \mapsto w * s$) is a permutation of $W$ and that $p_s^2 = \mathrm{Id}$ (resp. $q_s^2 = \mathrm{Id}$). Show that $p_s \circ q_t = q_t \circ p_s$ for all $s, t \in S$ (study the product $C(s).C(w).C(t)$ for $w \in W$ and show that $(s.w)*t \in \{w, s.w, w*t, s.(w*t)\}$; show that $(s.w)*t \notin \{s.w, w*t\}$ and that if $(s.w)*t = w$ then $s.w = w*t$ and $w = s.(w*t)$).

e) Show that the group of permutations $P$ (resp. $Q$) generated by the $p_s$ (resp. $q_s$) for $s \in S$ acts simply-transitively on $W$ (to show that $P$ is transitive, use (2) and argue as in Lemma 1 of no. 1; to show that $P$ is simply transitive, use d)). Deduce that $W$ has a unique group structure such that the map $p \mapsto p(e)$ (where e denotes the element of W such that B = C(e)) is an isomorphism from P to W. The map q ↦ q(e) is then also an isomorphism; moreover, s.w = w \* s = sw for all s ∈ S and all w ∈ W and C(w)^{-1} = C(w^{-1}).

f) Show that the pair (W, S) is a Coxeter system and generalize the results of no. 4.

g) Let X be a subset of S and WX the subgroup of W generated by X. Show that the union G_X of the C(w) for w ∈ WX is a subgroup of G and that Th. 3 of no. 5 is still true. Show that B is a Tits subgroup of G_X. Generalize Props. 2 and 3 of no. 5, and Def. 2, Prop. 4 and Th. 4 of no. 6. Show that S is the set of w ∈ W such that B ∪ C(w) is a subgroup of G distinct from B. The Coxeter system (W, S) and the group W (called the Weyl group of (G, B)) thus depend only on (G, B).

h) Let N be a subgroup of G such that B ∩ N is normal in N and such that the intersection with N of every double coset C(w) with respect to B is a double coset with respect to B ∩ N. Show that the group N/(B ∩ N) can be identified with W and that (G, B, N, S) is a Tits system.