6

T. A. SPRINGER

(b)/induces a bijection of 0' onto 0 and its transpose '/'induces a bijection of 0V

onto (0')v.

Notice that then ' / is also an injection Xy -+ (A")v with finite cokernel. Also,

coker/and coker(f/) are in duality.

EXAMPLE.

Given W, we shall construct a W and an isogeny of X into X', which

we shall call the canonical isogeny associated to W.

If L is a subgroup of X we denote by L the largest subgroup containing L such

that L/L is finite. Then L = L if and only if L is a direct summand.

Let

Ar0

and ? be as in 1.1. By 1.2 we can view 0 as a subset of X/XQ. It follows that

W{ = {XIX^ 0, g

v

,

0V)

is a semisimple root datum. Likewise, W{ = (Ayg, 0 , A"0, 0 )

is a toral root datum. Put W = &{ ® W2. Then the canonical isogeny f: X -

(X/XQ)

® {X/Q) is the canonical homomorphism of A' into the right-hand side.

1.8. Let 0 be a root system in the g-vector space V. The a v (a e 0) in the dual Vs"

of V also form a root system 0V (the dual or inverse root system). There are finitely

many semisimple root data (X, 0, Xv, 0V) where X a V, Xy c Kv. In fact, let Q

and g v be the lattices in V and Kvgenerated by 0 and 0V, respectively, and

define P = {* e K|*,

av

e Zfor all a e 0}. F

v

c

Kv

is defined similarly. Then

Q =. P and //(? is a finite group, in duality with

Pv/Qv.

An A" as above is then

contained between P and Q and for each such X there is a unique

A^v

between

/v

and

Qw

such that (A", 0,

A^v, 0V)

is a root datum.

1.9. Let W = (A\ 0,

Afv, 0V)

be a root datum. Assume that its root system 0 c K

is reduced. Let J be a basis of 0. Then J

v

=

{ccv\a

e J} is a basis of the dual root

system

0V

c

Kv.

We call A a ^ roo/ datum SL sextuple ¥0 = (A", 0, J,

A'v, 0V,

J

v

), where

(Ar,

0,

Ary, 0V)

is a root datum with reduced root system 0 and where J is a basis

of 0. However, since J and J

v

determine 0 and

0V

uniquely, it also makes sense

to view a based root system as a quadruple Wo = (Ar, J, Z v , J v ). This we shall do.

2. Reductive groups (absolute theory).

2.1. Let G be a connected reductive linear algebraic group. In this section we

consider the absolute case, where fields of definition do not come in. So we can view

G as a subgroup of some GL(«, Q), 0 an algebraically closed field (see [2]). Let S be

a subtorus of G. We define the root system 0(7, S) of G with respect to S to be the

set of nontrivial characters of » S which appear when one diagonalizes the represen-

tation of S in the Lie algebra g of G, S operating via the adjoint representation.

2.2. The root datum ofG. Fix a maximal torus T of G. We shall associate to the

pair (C, T) a root datum 6{G, T) =

(Ar,

0,

A'v, 0V)

(also denoted by u{G)).

X is the group of rational characters X*(T) of T. This is a free abelian group of

finite rank.

ArVis

the group X*(T) of 1-parameter multiplicative subgroups of T, i.e.,

the group of homomorphisms (of algebraic groups) GLX - T. Then

A^v

can be put

in duality with X by a pairing , defined as follows: if x e

Ar*(7'),

w e

A^JT),

thenx(w(0) =

tx u

{t eQ*).

We take 0 = 0(G, T), the root system of G with respect to 7. To complete the

definition we have to describe

0V.

If a e 0, let Ta be the identity component of the

kernel of a. This is a subtorus of codimension 1. The centralizer Za of Ta in G is a

connected reductive group with maximal torus T9 whose derived group Ga is semi-

simple of rank 1, i.e., is isomorphic to either SL(2) or PSL(2). There is a unique