1.2. ROOT DATA 3

with the monomorphisms ϕi,j (i, j ∈ I) coincides with the simply connected Kac-

Moody group GD(K) of type D over K (see Theorem A and its application in

[Cap05a]).

1.2. Root data

1.2.1. Definition. Let (W, S) be a Coxeter system, let Φ be the associated

root system (viewed either as a set of half-spaces in the chamber system associated

with (W, S) or as a subset of the real vector space

RS

) and let Π be a basis of Φ.

Let Φ+ (resp. Φ−) be the subset of positive (resp. negative) roots. We refer to

[Wei03, Chapter 3] (resp. [Bou81]) for general facts on root systems from the

combinatorial (resp. algebraic) viewpoint.

A pair of roots {α, β} ⊂ Φ is called prenilpotent if there exist w, w ∈ W such

that {w(α), w(β)} ⊂ Φ+ and {w (α), w (β)} ⊂ Φ−. In that case, we set

[α, β] :=

w ∈ W

∈ {+, −}

{γ ∈ Φ| {w(α), w(β)} ⊂ Φ ⇒ w(γ) ∈ Φ }

and

]α, β[:= [α, β]\{α, β}.

A twin root datum of type (W, S) is a system Z = (G, (Uα)α∈Φ) consisting

of a group G together with a family of subgroups Uα indexed by the root system Φ,

which satisfy the following axioms, where H :=

α∈Φ

NG(Uα), U+ := Uα| α ∈ Φ+

and U− := Uα| α ∈ Φ− :

(TRD0): For each α ∈ Φ, we have Uα = {1}.

(TRD1): For each prenilpotent pair {α, β} ⊂ Φ, the commutator group

[Uα, Uβ ] is contained in the group U]α,β[ := Uγ | γ ∈]α, β[ .

(TRD2): For each α ∈ Π and each u ∈ Uα\{1}, there exists elements

u , u ∈ U−α such that the product µ(u) := u uu conjugates Uβ onto

Usα(β) for each β ∈ Φ.

(TRD3): For each α ∈ Π, the group U−α is not contained in U+ and the

group Uα is not contained in U−.

(TRD4): G = H Uα| α ∈ Φ .

This definition was first given in [Tit92]; more details can be found in

[R´ em02b, Chapter 1] (see also [Abr96, § I.1]). The following two lemmas are

well known. The first one shows in particular that the product u uu in (TRD2) is

uniquely determined by the element u, as suggested by the notation µ(u).

Lemma 1.1. Let Z = (G, (Uα)α∈Φ) be a twin root datum and set H :=

α∈Φ

NG(Uα) . Let α ∈ Φ and set Xα := Uα ∪ U−α . We have the following.

(i) There are unique elements v , v ∈ U−α such that conjugation by v uv

swaps Uα and U−α.

(ii) An element h ∈ H centralizes Xα if and only if it centralizes Uα.

Proof. The existence part of Assertion (i) follows from (TRD2). The uni-

queness part is well known and follows from the fact, easy to deduce from (TRD2),

that Xα has a split BN-pair of rank one. Since H normalizes Xα, Assertion (ii)

follows from (i).

Lemma 1.2. Let N := H µ(u)| u ∈ Uα\{1}, α ∈ Π . Then H is normal in N

and N/H is isomorphic to W .