Exercise 20, § 2

We use the notation of no. 2, and assume that $k$ is commutative. Consider the following cases:

(B_l) $n = 2l + 1$ (with $l \geq 1$), $k$ is of characteristic $\neq 2$ and $k^n$ is equipped with the quadratic form $Q = x_1x_n + \cdots + x_lx_{l+2} + x_{l+1}^2$; (C_l) $n = 2l$ (with $l \geq 1$) and $k^n$ is equipped with the alternating form $\Phi$ such that $\Phi(e_i, e_j) = 0$ for $1 \leq i \leq n, 1 \leq j \leq n$ and $i \leq j$, except when $i \leq l, j = n + 1 - i$, when $\Phi(e_i, e_j) = 1$; (D_l) $n = 2l$ (with $l \geq 2$), $k$ is of characteristic $\neq 2$ and $k^n$ is equipped with the quadratic form $Q = x_1x_n + \cdots + x_lx_{l+1}$.

Denote by $G_1$ the special orthogonal group $\mathbf{SO}(Q)$ in cases (B_l) and (D_l), and the symplectic group $\mathbf{Sp}(\Phi)$ in case (C_l). Put

a) Show that the action of $N_1$ on the set of lines $ke_i$ defines a homomorphism from $N_1$, with kernel $T_1$, onto the subgroup $W_1$ of $\mathfrak{S}_n$, allowing us to identify $W_1$ with $N_1/T_1$. Show that $W_1$ is the subgroup of $\mathfrak{S}_n$ generated by the $\sigma_j = s_j s_{n-j}$ for $1 \leq j < l$ and by $\sigma_l = s_{l-1} s_l s_{l-1}$ in case (B_l); the $\sigma_j = s_j s_{n-j}$ for $1 \leq j < l$ and by $\sigma_l = s_l$ in case (C_l); the $\sigma_j = s_j s_{n-j}$ for $1 \leq j < l$ and by $\sigma_l = s_{l-1} s_l s_{l-1} s_l s_{l+1} s_l$ in case (D_l).

b) Let $S_1$ be the set of $\sigma_j$ for $1 \leq j \leq l$. Show that $(G_1, B_1, N_1, S_1)$ is a Tits system (to prove that the subgroup H of $G_1$ generated by $B_1$ and $N_1$ is the whole of $G_1$, argue as in Algebra, Chap. II, § 10, no. 13), remarking that H contains the lower triangular subgroup of $G_1$ and that, for any $\xi_i \in k$ ($2 \leq i \leq n$), there exists a matrix $b = (b_{ij}) \in B_1$ such that $b_{11} = 1, b_{1i} = \xi_i$ for $2 \leq i \leq n-1$ and $b_{1n} = \xi_n$ in case (C_l), and $b_{1n} = 0$ in cases (B_l) and (D_l). Then argue as in no. 2, introducing the subgroups $G_{1,j} = G_1 \cap (G_j; G_{n-j})$ for $1 \leq j < l$ and the subgroup $G_{1,l}$ of elements of $G_1$ leaving fixed

the $e_i$ for $i \neq l, l+1, l+2$ and the subspace generated by $e_l, e_{l+1}$ and $e_{l+2}$ in case (B_l);

the $e_i$ for $i \neq l, l+1$ and the plane generated by $e_l$ and $e_{l+1}$ in case (C_l);

the $e_i$ for $i < l-1$ or $i > l+2$ and the two planes generated by $e_{l-1}$ and $e_{l+1}$ and by $e_1$ and $e_{l+2}$ in case (D_l).

Show that $G_{1,j}$ can be identified, respectively, with $\mathbf{GL}(2, k), \mathbf{SL}(2, k)$ or the special orthogonal group of Exerc. 19).

c) Show that the Coxeter graph of the group $W_1$ is of type (Bl), (C_l) or (D_l) in the three cases (Chap VI, § 4, no. 1).*

d) Show that, for any subset X of $S_1$, the subgroup $G_{1X}$ consists of the $g \in G_1$ such that $g(V_i) = V_i$ for all $i$ such that $\sigma_i \notin X$, except in case (D_l) where the same assertion is valid provided that $V_{r-1}$ denotes the subspace generated by $e_1, \ldots, e_{r-1}$ and $e_{r+1}$. Deduce, as in Exerc. 18 b), that there exists a bijection $j$ from the building associated to $(G_1, B_1)$ to the set of totally isotropic subspaces $\neq 0$ in cases (B_l) and (C_l), and to the set of totally isotropic subspaces of dimension $\neq 0$ and $\neq r-1$ in case (D_l). Show that points $a_1, \ldots, a_k$ of I belong to the same facet if and only if $\{ j(a_1), \ldots, j(a_k) \}$ is a flag.