REDUCTIVE GROUPS

19

G(A) = SL(«, R) • (I] SL(«, Z,)). Gg,

from which one sees that there is a surjective continuous map G(A)/G(Q) -»

SL(n, R)/SL(n, Z). More precisely, if one defines, for a positive integer N, the con-

sequence subgroup r(N) of SL(«, Z)by T(N) = {y e SL(«, Z) 17 * = 1 (mod JV)},

then

G(A)/G(Q) = proj lim SL(«, * ) / / W

(c) Let G be a g-group and let G - GL(«) be an embedding (over g). Fix a lattice

L in Qn and let /"be the subgroup of G(Q) of elements stabilizing L.

There exists a connection, similar to that of the previous examples, between

G(A)/G(Q) and G(R)/r(see [1]).

The main results about G(A)/G(k) are as follows (G a connected reductive

&-group). Let X be the group of ^-rational characters of G. For each %^X define

a character |

z

| : G(^() - rt* by |;c I ((&,)) = lit, l%(s»)L

w h e r e

I I *

i s a n

absolute

value, normalized so as to satisfy the product formula. Let G(A)° be the intersection

of kernels of the | % |, for x e X.

The product formula shows that G(k) c G(A)°.

4.11. THEOREM, (i) G(A)°/G(k) has finite invariant volume;

(ii) (G semisimple) G(A)°IG(k) is compact if and only ifG is anisotropic over k.

This is a consequence of reduction theory, due to Borel and Harish-Chandra for

number fields and to Harder for function fields (see [1] and [11]). Notice, that by

restriction of the ground field, it suffices to prove this for k = Q or k = Fq(T).

5. A class of Lie groups. In this section we discuss a class of Lie groups close to

the groups of real points of reductive l£-groups. This is the class of groups occurring

in Wallach's paper in these

PROCEEDINGS

(see also [12]). We shall indicate briefly

how the properties of these groups can be deduced from the algebraic properties

of reductive groups, discussed above.

We shall say that an algebraic group G defined over a field of characteristic zero

is reductive if its identity component G° (in the Zariski topology) is so.

5.1. Let G be a Lie group, with Lie algebra g. Its identity component is denoted

by G°. We denote by °Gthe intersection of the kernels of all continuous homomor-

phisms G -» Rf. Then °G is a closed normal subgroup and G/°G is a vector group.

A split component of G is a vector subgroup V of G such that G = °G • V,

O G n V = {e}.

We assume henceforth that G possesses the following properties:

(1) There is a reductive R-group G andamorphism v: G -+ G(R) with finite kernel

whose image is an open subgroup of G(R).

It follows that v induces an isomorphism of g onto the set of real points of Lie

G. We shall often identify g and y(g). It also follows that v(G)° = G(R)° and that

G° has finite index in G (since this is so for G(R), see 4.2(H), and ker v is finite).

(2) The image of G in the automorphism group of gc = g ®R C lies in the image

of the identity component G° ofG.

The main reason to allow for finite coverings of linear groups is to include the

metaplectic group and all connected semisimple groups with finite center. The