1.2. ROOT DATA 5
(iv) The system ZD(K) := (G, (Uα)α∈Φ) is a twin root datum of type (W, S)
which is locally split over K.
(v) Given α Φ and a subgroup H of T , if H normalizes a conjugate of
contained in := U−α and different from and U−α, then H
centralizes Xα.
Proof. This statement is implicitly contained in [Tit87b]. See also [Tit92,
§3.3] and [R´ em02b, Proposition 8.4.1]. The technical assertion (v) follows from
the fact that T acts on by diagonal automorphisms (and never by field auto-
morphisms).
The twin root datum ZD(K) is called the standard twin root datum associated
with F and K. The basis Π = {αi| i I} of Φ is called standard (with respect to
F).
1.2.4. Isomorphisms of root data. Let Z := (G, (Uα)α∈Φ) and Z :=
(G , (Uα)α∈Φ ) be twin root data of type (W, S) and (W , S ) respectively.
Let S1, S2, . . . , Sn be the irreducible subsets of S. In other words, S = S1
··· Sn is the finest partition of S such that [Si, Sj ] = 1 whenever 1 i j n.
An ordered pair (ϕ, π) consisting of an isomorphism ϕ : G G and an iso-
morphism π : W W is called an isomorphism of Z to Z if the following
condition hold:
(ITRD1): π(S) = S and, hence, π induces an equivariant bijection Φ Φ
again denoted π.
(ITRD2): There exists x G and a sign
i
for each 1 i n such that
xϕ(Uα)x−1
= U
i
π(α)
for every α Φ such that WSi .
Thus, if (W, S) is irreducible, then either
xϕ(Uα)x−1
= Uπ(α) or
xϕ(Uα)x−1
=
U
−π(α)
for all α Φ(W, S). In particular, this means that ϕ maps the union of
conjugacy classes
{gU+g−1|g
G}
{gU−g−1|g
G}
to
{gU+g−1|g
G }
{gU−g−1|g
G },
with the notation of §1.2.1.
A crucial fact on isomorphisms between twin root data we will need later is the
following.
Theorem 1.5. Let Z := (G, (Uα)α∈Φ(W,S)) and Z := (G , (Uα)α∈Φ(W
,S
)) be
twin root data with S and S finite and let ϕ : G G be an isomorphism. Assume
that
(∗) {ϕ(Uα)| α Φ(W, S)} =
{xUαx−1|
α Φ(W , S )}
for some x G . Then there exists an isomorphism π : W W such that (ϕ, π)
is a twin root data isomorphism of Z to Z .
Proof. See [CM05a, Theorem 2.2].
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