REDUCTIVE GROUPS
23
Let W be the Weyl group of the root system 0. This is the relative Weyl group
of(G\ S')(see3.5), i.e.,
W = NGo(R) (S)/ZG0(R)(S).
5.17.
LEMMA.
M meets all components ofG.
It suffices to prove this for the case that v(G) = G°(R). In that case it follows
from Bruhat's lemma that the connected components of G°(R) all meet N =
NGo(R)(S). Let a e 0 be a simple root (for the order defined by P), U{a) the uni-
potent subgroup whose Lie algebra is (ga)c + (g2a)c (where the second term is
zero if 2a $0\ and 7(_a) = 0(U{a)). It is known that Uia)(R)'U(-a)(R)-U(a)(R)
contains an element of N representing the reflection in W defined by a. It follows
that N c Z0o(B)(5)G°, which implies the assertion.
It follows from the lemma that W = NG(A)/ZG(A).
5.18. LEMMA. W ^ (K f\ NG(A))/M.
Let g = kan e NG(A). Then also (Qn)~la2n e NG(A). Let w0 e NGo(R)(S) represent
the element of maximal length of W. Then 0n~l =
U'O^VTQ
1
9
for some n^ e N. We
then have niw^ahi e
WQ1NG(A).
The uniqueness statement of Bruhat's lemma then
implies that nx = e, whence n = e. This implies the assertion.
From 5.18 we see that W is the Weyl group of the symmetric spaces GjK ([13,
p. 244]).
5.19.
PROPOSITION (CARTAN DECOMPOSITION).
We have G = KAK.
This follows from 5.5 and the following lemma. Here S is as in 5.3 (observe that
K normalizes S).
5.20. LEMMA. S = \JkcK kAk~K
It suffices to prove this for y(G), i.e., when v = id. Let s e S. Then s lies in a 0-
stable i?-split torus Sx = G°. Since S is a maximal l£-split torus in (because P
is a minimal parabolic /^-subgroup of G°, see 3.5) we have that Si is conjugate to
a subtorus of S by an element of G°(R). By 5.17 we maj take this element to be in
G°(J?)°, hence in G. So there is g e G with ax = g
- 1
^
e
A. Writing g = kan we
obtain
n~la~2 On ax = a^nr1 ar2 On.
Using again the uniqueness statement of Bruhafs lemma, as in the proof of 5.18,
we see that a.\ commutes with n. It follows that s is conjugate to an element of A via
K, which is what had to be proved.
5.21. We finally give a brief elementary discussion of the geometric properties of
the symmetric space G/K. We identify it with S (cf. 5.5). It is a homogeneous space
for G, the action being given by (x, s) •-• x-s =
xs(0x)~~l.
If x e % define
\\X\\2
=
Tr(A"2). This defines a AT-invariant Euclidean distance on §. The exponential map
exp defines a diffeomorphism of § onto S. Its inverse is denoted by log.
Define a Euclidean metric d(,) on A by d(a, b) = ||log a log i||. This deter-
mines a structure of Euclidean affine space on A.
We may and shall assume that v=id.
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