24

T. A. SPRINGER

5.22.

LEMMA.

Let s,teS.

(i) There is xeG such that x • s and x • t lie in A;

(ii) ifx' is another element with the property of(i) then there is neG normalizing

A such that

x~lnx'

fixes s and t.

(i) follows from 5.20. To prove (ii) it is sufficient to assume x' = e. Then s, t e A.

Put xs = a, x-t = b. It now follows that

arl/2xsin andfe~1/2^1/2liein^r,

from

which one concludes that

st~l

and

ab~l

are conjugate in G. The uniqueness part of

Bruhat's lemma then implies that these elements are conjugate by an element of

the Weyl group W, from which (ii) follows.

5.23.

LEMMA.

Ifx eG,x-A=A then x normalizes A in G.

Apply 5.22(H), taking x' = e, s a regular element of A, t' = e. It follows that we

may assume xeK and xs(dx)~1 = xsx'1 = s. Since s is regular, x centralizes A.

The assertion follows.

5.24. The translates x A of A in S are called apartments in S. It follows from

5.23 that for any apartment J*/ there is a unique structure of Euclidean affine space

on s/ such that any bijection A -+ A/ of the form a *- x-a is an isomorphism of

such spaces.

5.22(i) shows that for any two elements s, t eS there is an apartment stf con-

taining them. It follows from 5.22(H) that, if s # t, the line in ja/ containing s and t,

together with its structure of 1-dimensional affine space, is independent of the choice

of jaf. We call such lines geodesies in S.

It now also makes sense to speak of the geodesic segment [st], and of the midpoint

of [st].

It also follows that there is a unique G-invariant function don S x S whose re-

striction to A x A is the function of 5.21.

5.25.

PROPOSITION,

(i) d is a distance on S;

(ii) ifs, t,ueS, d(s, t) = d(s, u) + d(u, t) then u lies on the segment [st];

(iii) a closed sphere {x e X \ d(x, a) g r} is compact in X.

It suffices to prove this for the case G = GL(n, R).

A proof of (i) and (ii) is given in the appendix to this section. The proof of (iii)

is easy.

5.26.

PROPOSITION.

For each seS there is a unique involutorial analytic diffeo-

morphism a5 ofS with the following properties:

(a) as is an isometry {for d),

(b) s is the only fixed point ofas,

(c) as stabilizes all geodesies through s.

We have ax.s = x © as ° x-1.

We may take x = e. The geodesic through e and exp X consists of the exp(^A')

(£ e R). Observing that d(e, exp(£X)) is proportional to |£| it follows that the only

possibility for ae is the map / *-

t~~l.

That this satisfies our requirements is clear.

The final statement follows from the rest.

5.27.

LEMMA.

Let s, s' be distinct points ofS, let m be the midpoint [ss']. Let t be