18 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPSrepresentation C f-t a.c.a-^1. To determine them, it is convenient to write C as a
3 x 3 block
C=(Cij) (l:s;i,j:s;3)
corresponding to the subsets { e 1 , ... , ep}, { ep+ 1 , ... , en-p}, and
{ en-p+ 1 , ... , en}. We leave it as an exercise to find the conditions on the Cij given
by (39) and to see the roots are
Ai - Aj ( i -=f. j) with multiplicity one±(.Ai+ ,\j) (i -=f. j) with multiplicity one
and, if n -=f. 2p,( 40) ±,\i with multiplicity n - 2p.If n -=f. 2p, then a system of simple roots is given by a1 = >-1 - .A2, a2 = >-2 - .-\3,
... , ap-1 = Ap-l - .AP, and ap = Ap· Thus F is of type BP if n -=f. 2p, and Dp if
n = 2p. The group is split over F if (and only if) n - 2p = 0, 1.
Again, let n -=f. 2p. A standard isotropic flag is an increasing sequence of
isotropic subspaces
( 41) 0=VoCViC···CV 8 =V
where V; is of dimension d(i), spanned by e 1 , ... , ed(i) (1 :::; d(l) < d(2) < · · · <
d(s) = p). Let ni = d(i)-d(i-l) (i = 1,... , s). The standard parabolic subgroups
are the stabilizers of standard isotropic flags. The stabilizer of the flag (41) is the
group PJ, where J = {ad(l)> ad( 2 ), ... , ad(s-l)}· It is represented by matrices which
are block upper triangular, with the diagonal blocks consisting of matrices of the
form
t -1 t -1
91, · · · ,9s,90, 9s '· · ·' 91 '
where 9i E GLn, (IR) (1 :::; i :::; s) and 9o E SO(Fo). The entries of an elements
9 E PJ are zero below the diagonal blocks; those above the blocks are subject only
to conditions derived from the fact tha t 9 E C = SO(F).
If n = 2p, there is a slight modification in the description of maximal proper
parabolic subgroups (cf. e.g. [7], 7.2.4).
4.6. We go back to the notation of Section 3.4 and sketch the proof of formula (8)
in Section 3.4. Let c^0 be the identity component of C. It is of finite index in C, so
we can write C = C -C^0 for some finite subset C of C, and (n2) in Section 2.1 shows
that it suffices to prove (8) for c^0. We assume therefore that C is connected and
use the notation reviewed in Sections 4.1-4.5 for F = R Let A= JRAp, where Pis
a minimal parabolic IR-group and = (C, A). Fix an ordering on and let qi+
(respectively 6.) be the set of positive (respectively simple) roots. As is well-known
(see e.g. [15]) the transforms under K of the image of A in X cover X, whence the
equality
(42) C = K.A.K
which can be refined to
(43) C = K.A+.K