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)},
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 |
| : G(^() - rt* by |;c I ((&,)) = lit, l%(s»)L
w h e r e
I I *
i s a n
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
(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 (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
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 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
Previous Page Next Page