34 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS
the parabolic subgroup PJ, where J is the set of o: for which ma = 0. We fix
a scalar product on V which is invariant under K, let 11. 11 be the corresponding
Euclidean norm, and write CJ for Ciw· Let ew be a basis vector of the highest weight
line.
Let x E G. It has the decomposition x = nxmxa(x)k. The elements nxmx
leave the highest weight line invariant and k is isometric. Therefore
(88)11.2. Proposition. Fix a Siegel set 6 with respect to P 0. Then for any w E C,
we have
(i) a(!)w --< 1 (! E f)
(ii) a(yx)w --< a(y)wa(x)w (y E G, x E 6).
Proof. w is also a positive linear combination of the Wa, so it suffices to consider
the case where w = l:a mawa (ma E N) (or even w = wa)· We use the notation
of Section 11.1. The element ew is rational, so some multiple may be assumed
to belong to a lattice in Vw(Q) that is stable under r. Therefore the set of 1ew
is discrete in Vw, and does not contain zero, so that there exists c > 0 so that
llrewll ?'.: c for all r E f. In view of (88), this proves (i).
Fix an orthonormal basis { ei} of Vw ( Q) consisting of eigenvectors of Ao, so that
e1 = ew. We have
and hence, by (88),
(89)On the other hand,
(90)We have
where /Ji is the weight of ei. It is of the form
/Ji= w - L ma(/Ji)o: ma(/3i ) EN
aEtl.therefore
(91)
from (90) and (91) we get
I ICJ(x-ly-^1 )ew11^2 ~ L Ci (y )^2 a(x )-^2 ,6' >--a(x )-^2 w (L Ci (y )^2 )
Together with (89), this yields (ii). D
Remark. Let D be a compact set containing the identity. Then for any .A E X(A)
(92) a(dx)>. ~ a(x)>. (d ED, x E 6)