REDUCTIVE GROUPS

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of g extends to one of G, which fixes K pointwise, hence acts trivially on ker ic,

and goes down to an automorphism of G°. The result for y(G) implies that G =

K • S = K • G°. Since ker v acts trivially on g, hence on G°, d:G° - G° extends

obviously to an automorphism of G which fixes A T pointwise. The remaining asser-

tions are then obvious.

5.6. COROLLARY, (i) K is a maximal compact subgroup ofG;

(ii) K meets every connected component of G.

G is the topological product of K by a connected space S, whence (ii). The first

assertion follows from the fact that every s ^ 1 in S generates an infinite discrete

subgroup.

Fix a Cartan involution 0 of GL(n, R) stabilizing v{G). We also denote by 0

the Cartan involution of G defined in 5.5.

5.7. Let C be the center of G. It has again the properties (1), (2) and it is 0-stable.

The group corresponding to G is the center of G. The subset corresponding to

S is now a vector group V. It is, in fact, the maximal 0-stable vector subgroup

contained in C. Let G2 be the derived group of G.

5.8. LEMMA, (i) °G = KGX and V is a split component ofG;

(ii) °G has the properties (1), (2) and is 6-stable.

KGi is a 0-stable closed normal subgroup of G, contained in °G. The Lie algebra

g is the direct sum of those of KGX and of V, which implies (i). As to (ii), for the

algebraic group of (1) we take the Zariski closure of v(°G) in G. Its identity com-

ponent differs from G° only in its center. This implies (2), and the final assertion

is clear.

5.9. Parabolic subgroups. A parabolic subalgebra p of g is a subalgebra such that

pc is the Lie algebra of a parabolic /^-subgroup of G°. A parabolic subgroup P of

G is the normalizer in G of a parabolic subalgebra (which then is the Lie algebra of

P). The parabolic subgroups of G correspond to the parabolic J?-subgroups of G°.

Let P be a parabolic ^-subgroup of G°. Let N be its unipotent radical. Put

L = P ftdP.

5.10.

LEMMA,

(i) L is a Levi subgroup ofP;

(ii) the Lie algebra ofG is the direct sum of those ofN, ON and L.

5.3(iii) shows that L is reductive. LN'is a parabolic /^-subgroup ofG0 contained

in P (see 3.6) with unipotent radical N9 hence equal to P. This proves (i). Then

(ii) follows by using that P and OP are opposite parabolics.

5.11. Let S be the maximal /?-split torus in the center of L. Put A = v~l(S(R))°9

N = v~l(N(R))°. These are subgroups of G. Since A is a 0-stable vector group, we

have da = a~l for all a e A. Let P be the parabolic subgroup of G defined by P and

put L = P n OP, M = °L. Then L is the centralizer of A in G. Also, L and M are

0-stable.

5.12. PROPOSITION, (i) L satisfies (I), (2) of 5.1. A is a split component ofL and ofP;

(ii) (m, a, n) •-» man defines an analytic diffeomorphism ofMxAx N onto P.

Let H be the centralizer of S in G. Then v{L) c H. Moreover, the identity com-

ponent H° is reductive and is equal to H f] G° (the last point because centralizers