Lemma 3

Let V be a finite dimensional real vector space, S a root system in $V^*,\mathscr{P}$ the set of parabolic subsets of S; let $\mathscr{H}$ be the set of Ker $\alpha$ for $\alpha \in S$, and $\mathscr{F}$ the set of facets of V relative to $\mathscr{H}$ (Chap. V, §1, no. 2, Def. 1).

If $P\in \mathscr{P}$, let F(P) be the set of $v\in V$ such that $\alpha (v)\geq 0$ for all $\alpha \in P$. If $F\in \mathscr{F}$, let P(F) be the set of $\alpha \in R$ such that $\alpha (v)\geq 0$ for all $v\in F$.

Then $F \rightarrow P(F)$ is a bijection from $\mathscr{F}$ to $\mathscr{P}$; for all $F\in \mathscr{F}$, F(P(F)) is the closure of F.

a) Let $P\in \mathscr{P}$. There exists a chamber C of S and a subset $\Sigma$ of the basis B(C) such that $P = S_+(C)\cup Q$ where Q is the set of linear combinations of elements of $\Sigma$ with non-positive integer coefficients (Chap. VI, §1, no. 7, Prop. 20). Put

If $v\in V$, we have the following equivalences:

$\alpha (v)\geq 0$ for all $\alpha \in P$

so F(P) is the closure of the set

a set which is a facet F relative to $\mathscr{H}$ since every element of S is a linear combination of $\alpha_1, . . . , \alpha_l$ in which the coefficients are either all $\geq 0$ or all $\leq 0$. Moreover, if $\beta =u_1\alpha_1+\cdots +u_l\alpha_l\in S$,

$\Leftarrow \Rightarrow \beta \in S_+(C)$ or $(-\beta \in S_+(C)$ and $u_{m+1}=. . .=u_l= 0)$

so P(F) = P.

b) Let $F\in \mathscr{F}$. It is clear that $P(F)\in \mathscr{P}$. On the other hand, F is contained in the closure of a chamber relative to $\mathscr{H}$ (Chap. V, §1, no. 3, formulas (6)), and so is a facet relative to the set of walls of this chamber (Chap. V, §1, no. 4, Prop. 9). Consequently, $\overline{F}$ is of the form $\{v\in V|\alpha (v)\geq 0$ for all $\alpha \in T\}$, where T is a subset of S which we can clearly take to be equal to P(F). Thus, $\overline{F}= F(P(F))$. Q.E.D.

If $P\in \mathscr{P}$, the facet F such that P = P(F) is said to be associated to P; we denote it by F(P). We extend these conventions to the case in which $(\mathfrak{g},\mathfrak{h})$ is split reductive.