22

T. A. SPRINGER

of tori in Zariski-connected algebraic groups are connected, see [3, p. 271]). It fol-

lows that (1) and (2) hold for L. The last point of (i) is easy. From P = ZJVwe

conclude that P = LN = MAN. Now (ii) readily follows.

The decomposition of 5.12(ii) is called the Langlands decomposition of P. There

is a similar decomposition of the Lie algebra of P: p = m 4- n 4- n.

"A parabolic pair in G is a pair (G, A) where P i s a parabolic subgroup of G and

A is as above.

5.13. Minimal parabolic subgroups. Now assume that P is a minimal parabolic

subgroup. Then jPis a minimal parabolic /^-subgroup of G°. In that case the derived

group of L is an anisotropic semisimple i?-group. It follows that M is compact.

We then must have M c K (recall that M is 0-stable), so M = K f| P. Let 0 be

the root system of (G, S) (see 3.5). If a e 0 put ga = {* e 9 | Ad(flf)^r = a**,

a e ^4}. Then we also have

9a

= {*e

9

I [H9 X] = fa(#)*, He a}

(a being the Lie algebra of A). Also, there is an ordering of 0 such that 1 1 =

2J«O

ga» On = La o ga (n and dn are the Lie algebras of A T and ON).

5.14.

LEMMA,

(i) We have direct sum decompositions

g = a 4- m 4- it 4- dn, g = t 4- a 4- it;

(ii) a is a maximal commutative subalgebra of%.

The first decomposition follows by using 5.10(ii). It then follows that f is the

direct sum of m and the space of all X 4- OX(Xen). Hence f fl " = {0}, f + u =

m + n + On. This gives the second decomposition. We also get that § is the direct

sum of a and the space of all X — 0X(Xe n). Since n commutes with no nonzero

element of n + On, the assertion of (ii) follows.

From the above we see that 0 is also the root system of the symmetric pair

(G, A:) (see [13, Chapter VII]).

5.15. PROPOSITION (IWASAWA DECOMPOSITION). (£, a, ri) •-» kan is an analytic

diffeomorphism of K x A x N onto G.

Let f be the map of the statement.

(a) im (f is closed. AN is a closed subgroup of G and G/AN is compact (because

G/P and M are compact). Let % be the projection G - G/AN. Since K is compact,

im(^ © (j)) is closed. Hence so is im f =

%~l

\m{it ° 0).

(b) im cj) meets all components of G, since A: does (see 4.6(H)).

(c) The tangent map d(j is bijective at any point (k, a, n). This follows from the

direct sum decomposition g = ! + a + n.

(a), (b) and (c) imply that im f is open and closed and meets all components.

Hence (j) is surjective. To finish the proof, it suffices to show that j is injective. It is

enough to prove that kan = ax implies a = al9 k = n = e. Now if this is so we

have On = c^na{2. The image under y of the last element is unipotent. It then

follows that

a2

= a\, a = au whence On e N f| ON = {e}.

5.16.

COROLLARY.

For any parabolic subgroup PofGwe have G = KP.