Proceedings of Symposia in Pure Mathematics

Vol. 33 (1979), part 1, pp. 3-27

REDUCTIVE GROUPS

T. A. SPRINGER

This contribution contains a review of the theory of reductive groups. Some

knowledge of the theory of linear algebraic groups is assumed, to the extent

covered in §§1-5 of BorePs report [2] in the 1965 Boulder conference.

§§1 and 2 contain a discussion of notion of the "root datum" of a reductive

group. This is quite important for the theory of L-groups. Since the relevant

results are not too easily accessible in the literature (they are dealt with, in a more

general context, in the latter part of the Grothendieck-Demazure seminar [17]), it is

shown how one can deduce these results from the theory of semisimple groups

(which is well covered in the literature). In §§3 and 4 we review facts about the

relative theory of reductive groups. There is more overlap with [2, §6], which

deals with the same material.

§5 contains a discussion of a useful class of Lie groups (the "selfadjoint" ones).

We indicate how the familiar properties of these groups can be established, assum-

ing the algebraic theory of reductive groups.

I am grateful to A. Borel for valuable suggestions and to J. J. Duistermaat for

comments on the material of §5.

1. Root data and root systems. The notion of root datum (introduced in [17,

Expose XXI] under the name of "donnee radicielle") is a slight generalization of

the notion of root system, which is quite useful for the theory of reductive groups.

Below is a brief discussion of root data. For more details see [loc. cit.]. For the

theory of root systems we refer to [7].

1.1. Root data. A root datum is a quadruple W = (X, 0,

Xy, 0V)

where: X and

Xv

are free abelian groups offinitetype, in duality by a pairing X x

Xy

-• Z denoted

by X $ and

0V

arefinitesubsets of A" and

Xy

and there is a bijection a *-*

ay

of

0 onto

0V.

If a e 0 define endomorphisms sa and saV of X,

Xy,

respectively, by

sa(x) = x - x,

av

a, saV(u) = u — a, u)

ay.

Then the following two axioms are imposed:

(RD1) For allae0 we have a,

av

= 2;

(RD2) For allae0 we have sa(0) c 0, say

(0v)

c

0v.

It follows from (RD1) that si = id, sa(a) = — a (and similarly for «saV). It is clear

from the definition of a root datum that if W = {X, 0,

Xy, $v)

is one, then

Wy

=

(Xy, 0V,

X9 0) is also one, the dual of W.

Let W be as above. Let Q be the subgroup of Xgenerated by 0 and denote by X0

AMS (MOS) subject classifications (1970). Primary 20G15, 22E15.

© 1979, American Mathematical Society

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http://dx.doi.org/10.1090/pspum/033.1/546587