REDUCTIVE GROUPS

17

Before continuing with local and global fields, we must say a little about group

schemes over rings.

4.5. Groups over rings. If G is a Agroup, the product and inversion are described

(see [2, p. 4]) by morphisms of algebraic varieties [x'.G x G -+ G, p: G -• G, which

in turn are given by homomorphisms of ^-algebras ju*: k[G] - k[G] ®k k[G] and

p*: k[G] -+ k[G]. These have a number of properties (which we will not write down)

expressing the group axioms. We thus obtain a description of the notion of linear

algebraic group in terms of the coordinate algebra.

The fact that k is a field does not play any role in this description.

Replacing k by a commutative ring o, we get a notion of "linear algebraic group

G over o", which is habitually called "affine group scheme G over o", which we

abbreviate to o-group. It can be viewed as a functor, cf. [2, p. 4]. We write G(o) for

the group of o-points of G (i.e., the value of the functor at o).

Let o[G] be its algebra. If o' is an o-algebra we have, by base extension, an o'-

group G x

0

o\ with algebra o[G] ®

0

o'.

Let m be a maximal ideal of o and put k(m) = o/m; this is an o-algebra.

DEFINITION.

The o-group G has good reduction at m if G x0 k(m)is a fc(m)-group.

G is smooth if it has good reduction at all maximal ideals m.

EXAMPLE OF BAD REDUCTION, O

= Z, G is the group of matrices

[lb a)

with a2 - 2b2 = 1. Then Z[G] = Z[X, Y]/(X* - 2Y* - 1) and Z[G] ® F2 ^

f2[X, y]l(X2) which cannot be the coordinate ring of a linear algebraic group

over F2, since it contains nilpotent elements.

Now let G be a &-group and let o be a subring of k. We shall say that G is de-

finable over o if there exists a smooth o-group G0 such that G ^ G0 x

0

k. By abuse

of notation, we sometimes write, if o' is an o-algebra, G(o') for G0(o'). One can also

define when an algebraic variety over k is definable over o; it is clear how to do

this.

EXAMPLE.

By a theorem of Chevalley a complex connected semisimple group is

definable overZ [6, A, §4].

4.6. Local fields. Let k be a local field. We denote by o its ring of integers and by

in the maximal ideal of o. The residue field o/m is denoted by F.

A profound study of reductive groups over local field has been made by Bruhat

and Tits. So far, only part of this has been published [9]. For a resume see [8].

In Tits' contribution [21] in these

PROCEEDINGS

more details are given about the

Bruhat-Tits theory. In particular, he discusses the building of a reductive k-

group and the theory of maximal compact subgroups. Here we mention only a few

results.

4.7.

LEMMA.

Let G be a connected reductive k-group. Then there is an unramified

extension lofk such that G is quasi-split over I.

This can be deduced from the fact that a maximal unramified extension of k is a

field of dimension g 1 (see [16, p. II-11]).

The group G(k) of ^-rational points is a locally compact topological group

(even a Lie group over k). It is a compact group if and only if the identity com-