4 1. THE OBJECTS: KAC-MOODY GROUPS, ROOT DATA AND TITS BUILDINGS

Proof. This follows (for example) from [Tit92, Proposition 4].

The Coxeter group W is called the Weyl group of Z.

Groups endowed with a twin root datum include isotropic semisimple algebraic

groups, split and quasi-split Kac-Moody groups, as well as some other more exotic

families of groups, including those constructed in [RR06]. Here, we will focus on

split Kac-Moody groups, but a great deal of our discussion will apply to the slightly

more general class of groups endowed with a locally split twin root datum. We now

recall this notion.

1.2.2. Locally split twin root data. A twin root datum (G, (Uα)α∈Φ) of

type (W, S) is called locally split over (Kα)α∈Φ (or over K if Kα K for all α)

if the following conditions are satisfied:

(LS1): The group T :=

α∈Φ

NG(Uα) is abelian.

(LS2): For each α ∈ Φ+, there is a homomorphism

ϕα : SL2(Kα) → Uα ∪ U−α

which maps the subgroup of upper (resp. lower) triangular unipotent

matrices onto Uα (resp. U−α).

Notice that the second condition holds for all α ∈ Φ as soon as it holds for all

roots α in some basis of Φ (this is a direct consequence of (TRD2) and the fact

that W.Π = Φ for each basis Π of Φ).

Lemma 1.3. Let (G, (Uα)α∈Φ) be a twin root datum which is locally split over

(Kα)α∈Φ, let T :=

α∈Φ

NG(Uα) and let α ∈ Φ. Set Xα := Uα ∪ U−α . We have

the following.

(i) If |Kα|≥ 4, then the derived subgroup of T ∪ Xα coincides with Xα.

(ii) The normalizer of Uα in T ∪ Xα is solvable.

Proof. Note that if |Kα|≥ 4 then SL2(Kα) is perfect. Thus (i) and (ii) follow

from (LS1) and (LS2) and the fact that T normalizes Uα and U−α.

1.2.3. Twin root data for Kac-Moody groups. Let D =

(I, A, Λ, (ci)i∈I , (hi)i∈I ) be a Kac-Moody root datum and let M(A) = (mij )i,j∈I be

the Coxeter matrix over I defined as follows: mii = 1 and for i = j, mij = 2, 3, 4, 6

or ∞ according as the product Aij Aji is equal to 0, 1, 2, 3 or ≥ 4. Let (W, S) be

a Coxeter system of type M(A) with S = {si| i ∈ I}, Φ be its root system and

Π = {αi| i ∈ I} be a basis of Φ such that for each i ∈ I, the reflection associated

with αi is si.

Let F = (G, (φi)i∈I , η) be the basis of a Tits functor of type D and K be a field.

Let G := G(K), T := η(TΛ(K)) and for each i ∈ I let ¯i s := φi(K)

0 1

−1 0

and

Uαi (resp. U−αi ) be the image under φi(K) of the subgroup of upper (resp. lower)

triangular unipotent matrices.

Lemma 1.4. One has the following:

(i) The assignments ¯i s → si extend to a surjective homomorphism onto W ,

denoted ζ, whose kernel is contained in T .

(ii) For each w ∈ W and i ∈ I, the group Uw(αi

)

:= ζ(w)Uαi

ζ(w)−1

depends

only on the element α := w(αi) of Φ (and not on αi or on w).

(iii) The group T coincides with

α∈Φ

NG(Uα).