20

T. A. SPRINGER

main use of (2) is to insure that G acts trivially on the center of the universal en-

veloping algebra of g.

5.2. Let 0O be the automorphism of GL(«, R) which sends g to tg~l. Let § (resp.

SQ) be the set of real symmetric (resp. and positive nondegenerate) n x ^-matrices.

Then exp: x i-» exp x = 1 + x + x2/2! + ••• is an isomorphism of § onto 50.

Given s e S0, there is a unique analytic subgroup of GL(«, /?), isomorphic to R,

contained in S0, and passing through s. It is contained in any 0o-stable Lie

subgroup of GL(«, R) with finitely many connected components which contains s.

5.3.

LEMMA.

Let G c= GL(w, C) be an R-subgroup stable under 0Q.

(i) Let s G S0 P| G(R). Then there exists a QQ-stabIe R-split torus SofG such that

seS(R);

(ii) let X G § p| Lie(G). There is a do-stable R-split subtorus of G whose Lie

algebra contains X;

(iii) G is reductive.

The element s generates an infinite subgroup of G, whose Zariski closure is a

torus with the required properties, (ii) follows from (i), applied to exp x. Let U be

the unipotent radical of G, let s e G(R). Then s and

(OQS)S~1

are unipotent. By (i) the

last element is also semisimple, which implies s= 1. Hence U={\}. This proves (iii).

5.4. By definition, a Cartan involution of GL(w, R) is an automorphism conju-

gate to do by an inner automorphism. Let G and G be as in 5.1. Let G a GL(n, C)

be an embedding over R. Then v(G) is stable under some Cartan involution 0 of

GL(/i, R). In other words, we may assume, after conjugation, that v(G) is stable

under do (in which case it is said to be selfadjoint) [1], [15]. Let f (resp. g) be the

fixed point (resp. — 1 eigenspace) of 0 in g, and K the inverse image in G of the

fixed point set of 0 in v(G).

5.5.

PROPOSITION.

The automorphism 0of$ extends uniquely to an automorphism

ofG whose fixed point set is K. The map n'.{k,x)^k • exp x is an isomorphism of

analytic manifolds ofK x § onto G.

The automorphisms of G thus defined are the Cartan involutions of G. They

form one conjugacy class with respect to inner automorphisms by elements of G°.

The decomposition G = K • S (S = exp §) is a Cartan decomposition of G.

After conjugation, we may assume that 0 = 0O. If G — GL(«, R), then K =

0(ri), S = So and our assertion follows from the polar decomposition of real

matrices. Assume now that v is the identity. Write ^ e G a s a product g = k • s

where k e 0{n)9 seSQ. Then

s2

= (0O

g)~l

• ge G, and the unique 1-parameter

subgroup in S0 through

s2

(see 5.2) is contained in G. In this group, there is a uni-

que element with square

s2,

which must then be equal to s. Thus s G G, hence also

keG. This implies that ju is surjective. Injectivity follows from the uniqueness of

the polar decomposition. The decomposition g = ! + § implies that the tangent

map at any point is bijective; hence p. is an analytic isomorphism. Thus G is the

direct product of K and a euclidean space.

This proves the proposition when v is the identity. Let G be the simply connected

group with Lie algebra g, K the analytic subgroup of G with Lie algebra t and

%\ G -* G° the natural projection. Since the fundamental group of A^is that of G°,

the group Kis the universal covering of K; hence ker % a K. The automorphism 0