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 K for all α)
if the following conditions are satisfied:
(LS1): The group T :=
NG(Uα) is abelian.
(LS2): For each α Φ+, there is a homomorphism
ϕα : SL2(Kα) U−α
which maps the subgroup of upper (resp. lower) triangular unipotent
matrices onto (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 :=
NG(Uα) and let α Φ. Set := U−α . We have
the following.
(i) If |Kα|≥ 4, then the derived subgroup of T coincides with Xα.
(ii) The normalizer of in T 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 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
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
only on the element α := w(αi) of Φ (and not on αi or on w).
(iii) The group T coincides with
Previous Page Next Page