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
