Exercise 10, § 2

Let $G$ be a group and $B$ a Tits subgroup of $G$. We use the notation of Exerc. 3. For $s \in S$, denote by $G^{(s)}$ the subgroup $G_{S-\{s\}}$ (Exerc. 3 g)). Let $I$ be the set of subsets of $G$ of the form $gG^{(s)}$ (for $g \in G$ and $s \in S$), and $\mathscr{C}$ the set of subsets of $I$ of the form $C_g = \{ gG^{(s)} \mid s \in S \}$ for $g \in G$. The $C_g$ are called the chambers of $I$ (§ 1, Exerc. 15). Let $G$ act on $I$ by left translations.

a) Let $\mathfrak{F}$ be the set of facets of $I$ (that is, the subsets of the chambers, cf. § 1, Exerc. 15). Let $F \in \mathfrak{F}$. Show that there exist a unique subset $X$ of $S$ and an element $g \in G$ such that $gG_X = \bigcap_{a \in F} a$; $F$ is said to be of type $X$. Show that $F$ is then the set of $gG^{(s)}$ for $s \in S - X$ and is of codimension $\mathrm{Card}(X)$ in any chamber containing it. Show that the map $j : F \mapsto \bigcap_{a \in F} a$ is a strictly decreasing bijection, compatible with left translations, from $\mathfrak{F}$ to the set of subsets of $G$ of the form $gG_X$ for $g \in G$ and $X \subset S$. Show that, if $X \subset Y \subset S$, every facet of type $X$ contains a unique facet of type $Y$.

b) Show that $G$ acts transitively on the set $\mathscr{C}$ of chambers and that the stabilizer of the chamber $C_g$ ($g \in G$) is equal to $gB g^{-1}$; the map $g \mapsto C_g$ thus defines a bijection from $G/B$ to $\mathscr{C}$.

c) Show that two chambers $C_g$ and $C_{g'}$ ($g, g' \in G$) are neighbouring (§ 1, Exerc. 15) if and only if there exists $s \in S$ such that $g' \in g(B \cup BsB)$.

d) Let $C_1, \ldots, C_n$ be chambers of $I$ and put $C_0 = C_e$. Show that the following conditions are equivalent: (i) the sequence $\Gamma = (C_0, C_1, \ldots, C_n)$ is an injective gallery; (ii) there exist a sequence $s = (s_1, \ldots, s_n)$ of elements of $S$ and a sequence $(b_1, \ldots, b_n)$ of elements of $B$ such that $C_j = b_1 \bar{s}_1 b_2 \bar{s}_2 \ldots b_j \bar{s}_j (C_0)$ (where $\bar{s}_j$ denotes a given element of the double coset $Bs_j B$) for $1 \leq j \leq n$.

Show that, if these conditions are satisfied, the sequence s is unique: it is then called the type of $\Gamma$ and denoted by $s(\Gamma')$. Show that an injective gallery is minimal if and only if its type is a reduced decomposition. Show that the following conditions are satisfied:

(WI 1) For any two chambers C and C' of I, there exists a unique element t(C, C') of W such that the set of types of minimal galleries with extremities C and C' is the set of reduced decompositions of t(C, C').

(WI 2) For any chamber C, the map C' $\mapsto t(C, C')$ from the set of chambers of I to W is surjective.

e) Show that two minimal galleries of the same type and with the same extremities are identical. (Reduce to proving that, if $(s_1, \ldots, s_n)$ is a reduced decomposition and if $b_1, \ldots, b_n, b'_1, \ldots, b'_n$ are elements of B such that $b_1 \bar{s}_1 \ldots b_n \bar{s}_n \in b'_1 \bar{s}_1 \ldots b'_n \bar{s}_n B$, then $b_1 \bar{s}_1 \in b'_1 \bar{s}_1 B$. For this, remark that if $\bar{s}_1^{-1} b_1^{-1} b'_1 \bar{s}_1 \notin B$, then this element would belong to Bs_1B and we would have $b_2 \bar{s}_2 \ldots b_n \bar{s}_n \in b \bar{s}_1 b' b'_2 \bar{s}_2 \ldots b'_n \bar{s}_n B$ with $b, b' \in B$, contradicting Cor. 1 of Th. 2 of no. 4.)

f) Show that I equipped with $\mathscr{C}$ is a building, said to be associated to the pair (G, B). Show that there exists a unique numbering ($§ 1$, Exerc. 20) of I for which the type of a facet is that defined in a).

g) Show that I is spacious, namely that every panel of I is contained in at least three chambers (cf. $§ 1$, Exerc. 24). Show that the following condition is satisfied:

(G) Given a panel F, a chamber C, a gallery $\Gamma = (C_0, \ldots, C_n)$ such that $F \subset C_n$ and of smallest possible length, and chambers $C'$ and $C''$ containing F and distinct from $C_n$, there exists an element $g \in G$ such that $g(C_i) = C_i$ for $0 \leq i \leq n$ and $g(C') = C''$.

(Reduce to the case $C_0 = C$; let $u \in S$ be such that F is of type $S - \{u\}$ and let s be the type of $\Gamma$. Show that $(s, u)$ is a reduced decomposition. Take $h \in G$ such that $C_n = h(C)$; there exist $b'$ and $b'' \in B$ such that $C' = hb'u(C)$ and $C'' = hb''u(C)$. Then use Cor. 1 of Th. 2 of no. 4 generalized to the case of a Tits subgroup (cf. Exerc. 3 f)) to show that there exists $b \in B$ such that $bb'b'uB = hb''uB$; then $b \in hBuBu^{-1}Bh^{-1} \subset (hBh^{-1}) \cup (hBuBh^{-1})$. If $b \in hBuBh^{-1}$, we would have $BhB \subset BhBuB = BhuB$ (loc. cit.), which is impossible. Hence $b(C_{n'}) = C_n$ and e) implies that $b(C_i) = C_i$ for all $i$.)