REDUCTIVE GROUPS
7
homomorphism a
v
: GLx - Ga such that T = (Im a
v
)r
a
, a,
av
= 2. These a
v
make up
0V.
The axiom (RD1) is built into the definition of a
v
. We use 1.4 to establish (RD2).
Let na eGa - Ta normalize Ta. Then n2a G Ta and, sa being as in 1.1, we have, for
x e X, t 6 T,
In fact, working in Ga one shows that there is w e JTV such that the left-hand side
equals **-*•"a. One then shows that a, w = 2 and that x, w = 0 if x, av = 0.
It follows that (RD2') holds. So 0(G) is a root datum. The root system 0 is reduced
(for all these facts see [3] or [14]).
2.3. To e a c h a e 0 there is associated a unique homomorphism of algebraic
groups xa: Ga -* Ga such that
txa(u)t~l = xa(faw) (/ G T, u G fi).
Put C/a = Im(xa) and let Xa e g be a nonzero tangent vector to C/a. Then
9
= Lie(JT) 0 2 flZa.
Let B be a Borel subgroup containing T. There is a unique ordering of 0 (as in 1.6)
such that B is generated by T and the Ua with a 0, and any B = T is so ob-
tained. It follows that we can associate to the triple (G, B, T) a based root system
p0(G, B, T) = (X*(T), J, X+{T),
Ay)
(or ^o(G)) where A is the basis of 0 deter-
mined by the ordering associated to B.
2.4. Isogenics. An isogeny f: G - G' of algebraic groups is a surjective rational
homomorphism with finite kernel.
EXAMPLES,
(i) The canonical homomorphism SL(2) -• PSL(2) (PSL(2) is to be
viewed as the group of linear transformations of the space of 2 x 2-matrices of the
form x H+
gxg~l,
where g G SL(2)). If char 0 = 2 this is an isomorphism of abstract
groups, but not of algebraic groups.
(ii) Let G be defined over the finite field Fq. The Frobenius isogeny G - G raises
all coordinates to the #th power. It is again an isomorphism of groups, but not of
algebraic groups.
Let G and G' be connected reductive and let T be a maximal torus of G. A central
isogeny j\ G - G' is an isogeny which (with the notations of 2.3) induces an iso-
morphism in the sense of algebraic groups of Ua onto its image, for all a G 0.
Equivalently, d(j(Xa) ^ 0 for all a G 0 (where dcj) is the induced Lie algebra ho-
momorphism). The image T = f(T) is a maximal torus of G'. We shall say that
f is a central isogeny of (G, T) onto (G\ T).
Let/(0) be the homomorphism X*{T) - X*(T) defined by 0.
2.5.
PROPOSITION,
(i) If (pis a central isogeny thenf(fr) is an isogeny ofcp(G\ T')
into 0(G, T)\
(ii) iff and $' are central isogenics of(G, T) onto (G\ T') such thatf(f) = f(f')
then there is t e T with f' = j ° Int(r).
That/(0) has property (a) of 1.7 is equivalent to the fact that j induces a sur-
jection T- T' with finite kernel. There is a bijection a *-* a of root systems such
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