REDUCTIVE GROUPS

9

Let G" cz G' x G{ be the image of the homomorphism g H- {j(g\ fi(g)). Let

0: G - G" be the induced homomorphism. Then 0 is a central isogeny of con-

nected reductive groups and so are the projections %i'. G" - G', %z\ G" -• G[. A

straightforward check (working in tori) shows that Ker %{ = {e}9 Ker rf^ = {0}

(/ = 1, 2). Hence %\ and ^r2

a r e

isomorphisms, whence the lemma.

2.8.

LEMMA.

Let W be a root datum and let f be an isogeny ofW into f(G, T). Then

there exist a pair (G', T') and a central isogeny f of (G, T) onto ((?', 7") such that

T' = MG\T')9fW)=f.

From the knowledge of/ we can recover successively the groups G, figuring in

(1). This allows one to define G'. We omit the details.

2.9.

THEOREM,

(i) For any root datum W with reduced root system there exist a

connected reductive group G and a maximal torus T in G such that W = p(G, T). The

pair (G, T) is unique up to isomorphism]

(ii) let ¥ = 0(G, T), W = 0(C, T). Iffis an isogeny ofW into V there exists a

central isogeny $ of (G, T) onto (G', 7") with f($) = /. Two such § differ by an

automorphism Int(/) (r e T) ofG.

Let/be the canonical isogeny W -• W of 1.7. Using 2.8 we see that it suffices to

prove the existence statement of (i) for the two cases that W is semisimple or toral.

The second case (0 = 0) is easily dealt with: take for G the torus T — Wom{X, A*).

In the semisimple case the statement follows from the existence theorem of the

theory of semisimple groups which can be dealt with using the theory of Chevalley

groups. (See [18] or [6, part A]. The uniqueness statement of (i) is part of (ii). To

prove (ii) one first reduces to the case that / is an isomorphism (using 2.7 and

2.8).) In the case that G is semisimple the statement of (ii) is Chevalley's fun-

damental isomorphism theorem, proved, e.g., in [10, Expose 24], or in [14, Chapter

XI]. The case of a torus G is easy. In the general case, there are central isogenics

Gx x S - G, G{ x S' - G\ where Gx and G{ are the derived groups of G and G',

and where S and S' are tori, such that the corresponding isogenics of root data

are just the canonical ones of 1.7 (see also 2.15).

Now/defines an isogeny/i of ^(Gi x S")into0(G x S) and we may assume that

there exists a central isogeny 0!: Gx x S -• G[ x S' w i t h / ^ ) =f.

We can then complete the diagram like (2), with (?i x S, G[ x S', G, G' instead

of G, G, G', G\ respectively, and with fi as first arrow. The right-hand arrow, which

is uniquely determined by ^ , is then the required isomorphism. The last point of

(ii) follows from 2.5 (ii).

2.10.

REMARKS,

(i) In the semisimple case the existence statement of 2.9(i) is due

to Chevalley [Seminaire Bourbaki, Expose 219,1960-1961]. He constructs a group

scheme G0 over Z such that G — G$ xzk. This construction is also discussed in

[6, part A].

A generalization of 2.9, where the field k is replaced by a base scheme, is con-

tained in [17, Expose XXIV].

(ii) The result on the existence of central isogenics of reductive groups contained

in 2.9(ii) is a special case of one on arbitrary isogenics, which we shall briefly indi-

cate. Let j: G - G' be an isogeny of connected reductive groups w i t h e r ) = T'.

Let/ be the induced homomorphism X*(T) e X*(T). Let xa, xa, (a e 0, a' e 0')