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 G° (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.