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
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