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.
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