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 -* 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
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