36 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPSC. The map which associates to >. E X(A)c its differential at 1 is an isomorphism
of X(A)c onto af. c·
Fix a scalar product on X(Ao) invariant under the Weyl group. The (open)
Weyl chamber Co c X(Ao) is the set of elements A E X(A) such that (.A, a) > 0
for a E .6.(A 0 , G). It is also the set of strictly positive linear combinations of the
fundamental highest weights (see Section 11.1). Thus ao -Ap is a set of elements
on which the af3"' are bounded above, but not below, and -A= { alaf3"' :::; 1, a E .6. }.
By restriction, C 0 defines a Weyl chamber Cp C X(Ap).
12.2. Given PE P<Q, we let Yp = fpNp\G. If Q E P<Q contains P, f E C(Yp)
ands E X(Ap)c, we define formally the series EQp(f,s) by
EQp(f, s) = L f(rx)a(rx)s+p
rp\ r QIf convergent and continuous, it defines an element of C(YQ); if Q = G, it is called
an Eisenstein series.
12.3. An automorphic form f on Yp is an element of C(Yp) which is K-finite on
the right and such that for any x E Yp, the function y __, f(y.x) is an automorphic
form on fp \MpAp. In particular, it is of moderate growth on Yp, and in fact of
uniform moderate growth with respect to Z(m EB a).12.4. Theorem. Let f be an automorphic form on Yp. The Eisenstein series(98) Ecp(f, s)(x) = L f(rx )a(rx)8+P
,,Erp\ rconverges absolutely in the region ~( s) E so + Cp for some so E X ( Ap) and
represents an automorphic form on Y.Remark. In the literature I know, this theorem is stated for f E c=(NpApfp \G),
right K-finite, automorphic and in L^1 (fMp \Mp). As formulated here, it is due to
J .Bernstein, as well as the reduction given below to the case f = l.
We first show that f.a(x)s+p is Z(g)-finite, of a type determined by that off
with respect to Z(m EB a). In the notation of Section 6.4(c) we have, for z E Z(g)-
finite
z(f.as+p) = v(z)(fas+plMA)
The assumption on f implies that f.a(x)s+p is annihilated by an ideal J' of Z(mEBa)
of finite codimension, hence f.a(x)s+p on G is annihilated by v-^1 (J), which is an
ideal of finite codimension of Z(g). On the other hand, f.a(x)s+p has the same
right K-type as f. Therefore, if the series converges, it is Z(g)-and K-finite. The
issues are thus convergence and moderate growth. We first treat a special case in
Sections 12.5 and 12.6 and then reduce the general case to it in Section 12. 7.
12.5. Assume in this section that P =Po and f = l. It follows from the definition
that Ec,P 0 (1,s) = Eoc,P~(l,s') where P6 =Pon ° G ands' is the projection of s
on X(Ap 0 n °G). We may assume therefore that G =^0 G. The proof, which gives
so = p, is due to R. Godement and can be found in many places. We sketch it.
Since