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
{+, −}
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 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 = {1}.
(TRD1): For each prenilpotent pair {α, β} Φ, the commutator group
[Uα, ] is contained in the group U]α,β[ := | 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 onto
Usα(β) for each β Φ.
(TRD3): For each α Π, the group U−α is not contained in U+ and the
group 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 := U−α . We have the following.
(i) There are unique elements v , v U−α such that conjugation by v uv
swaps and U−α.
(ii) An element h H centralizes 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 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 .
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