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).
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