4

T. A. SPRINGER

the subgroup of X orthogonal to 0X. Put V = Q ® g, V0 = X0 ® Q. Define sim-

ilarly subgroups

Qv,

X^ of

Xy

and vector spaces F

v

, K^.

We say that W is semisimple if A^ = {0} and toral if 0 is empty.

1.2. LEMMA. Q f] X0 = {0} awd g + X0 has finite index in X.

This is contained in [17]. We sketch a proof. Define a homomorphism

p\X-+ A^by

Since x,/?(*) = Sae0 *,

av 2

we have Z0 = Ker/7.

Next observe that if a e 0 we have p{a) = i a, /?(a)av, as follows by summa-

tion over /3 E 0 from the identity

a,

£v2av

= a, /3V^v

+ a

^

( / 3

v

) K v

(/3v).

This shows that p ® id is a surjection F -» Kv, whence dim F v g dim K By sym-

metry we then have dim V = dim

Kv,

whence Q f| Ker p = {0}. The assertion

now follows readily.

1.3. Root systems. It follows from the proof of 1.2 that

Vy

can be identified with

the dual of the vector space V. We write again , for the pairing. Also identify

0 with 0 ® \ a V and assume that 0 ^ 0 . We then see that 0 is a root system in

V in the sense of [7]. Recall that this means that the following conditions are

satisfied:

(RSI) 0 is finite and generates V, moreover O$0;

(RS2) for allae0 there is

ay

e

VV

such that x,

av

= 2 and that sa (defined as

before) stabilizes 0;

(RS3)/or allcce0 we have av($) c Z.

The sa then generate a finite group of linear transformations of V, the Weyl group

W(0) of 0.

If W = (A", 0, Xw, 0V) is a root datum which is not toral, we call the root system

0 c Kthe root system ofW. The Weyl group W(0) is identified with the group of

automorphisms of X generated by the ^a of 1.1 and with the group of automorph-

isms of

Xv

generated by the saV.

The following observation is sometimes useful.

1.4.

LEMMA.

Axiom (RD2) is equivalent to:

(RD2')(a) For allae0 we have sa(0) a 0\

(b) the sa (a e 0) generate a finite group.

It suffices to prove that (RD2') implies the second assertion of (RD2). Let a,

j8 e 0. Then saspsa and sSa^ are involutions in the group generated by the sa. We

have by an easy computation,

\ W W « W =

x

+ «*»

%(Py)

- *

J«(j8)vK(j8),

where sa is the transpose of sa. Since ja(j8), ^a(/3v) - *a(j8), sa(/3)v =

j6,

|3V

- sa(/3),

*ya(]3)v

= 0, we see that the above automorphism of X is

unipotent. Since it lies in a finite group it must be the identity. Hence .ya(/3)v =

*.sa(/3v),

and the assertion follows by observing that ^aV =

lsa.