10

T. A. SPRINGER

be as in 2.3. One shows that there is a bijection a »-* a of 0 onto 0' and a function

q:0 -+ {/?n|ne JV} (/? the characteristic exponent) such that the jca and xa, can

be so normalized that j(xa(t)) = xa,{ti{a)). It follows that

/(a') = q(a)a9 '/(av) = ?(a)(a')v.

If ^ is a central isogeny then all q(a) are 1. If f is the Frobenius isogeny of 2.4 then

/ i s multiplication by # and all q(a) are #.

Such an/i s called ap-morphism in [17, Expose XXI]. The analogue in question

of 2.9(ii) is obtained by assuming / to be a /?-morphism and admitting in the

conclusion an arbitrary isogeny j. The proof can be given along similar lines,

reducing to the case of an isomorphism. For semisimple G the result is due to

Chevalley [10, Expose 23].

EXAMPLE OF A /?-MORPHISM.

p = 2, G is semisimple of type B2 and, with the

notations of the example in 1.5, we have f(ex) = ex + e2if(e2) = ex — e2.

A classification of the possible p-morphisms can be found, e.g., in [17, Expose

XXI, p.71].

2.11. Let (j: G - G' be a homomorphism of connected reductive algebraic

groups. Let ran d T be maximal tori in G, G' with f(T) c T'. Assume that Im tj is

a normal subgroup of G'. We shall briefly describe the relation between the root

data 0(G, T) = (X, 0, Z

v

,

0V)

and 0(G', T) =

(^,,$,(Z,)V,((P,)V).

Let/: * ' -+ Z

be the dual of 0: T - T \ In general,/is neither injective nor surjective.

Put 0! = 0 n Im/, 0

2

= 0 ~ 0i- Then 0 = 0! (J 02 is a decomposition into

orthogonal subsets (i.e., 0l9 0^ = 02, 0^ = 0).

Likewise, if 0'2 = 0' fl Ker/, 0i = 0' - 02, then 0' = 0i U 02 is a decompo-

sition into orthogonal subsets. There is a bijection a »-• a' of 0! onto 0J and a

function ? : 0X -

{/?n|/a

e TV}, such that for ct e 0i we have

/(«') = q(a)a, '/(a*) = tfa)

(a')v-

Moreover/(a) = 0 if a e 02 and /(a) = 0 if a e 02.

2.12. It follows readily from 2.9(i) that if W0 is a based root datum with reduced

root system there exists a triple (G, 5, T) as in 2.3 with (J)0(G, B, T) = W0, which is

unique up to isomorphism. There is no canonical isomorphism of one such triple

onto another one. In fact, based root systems have nontrivial automorphisms.

In this connection the following results should be mentioned. Letp0(G, B T) =

0O(G). For each a e A fix an element ua ~& e in the group Ua. The following is then

an easy consequence of 2.5(H).

2.13.

PROPOSITION.

Aut (p0(G) is isomorphic to the group Aut(G, B, T, {ua}a, j)

of automorphisms ofG which stabilize Bf Tand the set ofua.

2.14. COROLLARY. There is a split exact sequence

{1} Int(G) Aut(G) Aut 0O(G) • {!}•

In fact, an isomorphism as in 2.13 defines a splitting. Any two such splittings

differ by an automorphism Int(f) (t e T).

2.15. Let G be a connected reductive group, with a maximal torus T. Put p(G, T)

= (X, 0, XV, 0V).