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