1549380232-Automorphic_Forms_and_Applications__Sarnak_

12. EISENSTEIN SERIES 35

Proof. Let w EC. By t he proposition

(93) a(dx)w--< a(d)wa(x)w (d E D , x E 6).

Since Dis compact, a(d)w;:::: 1 (d ED). Hence

(94) a(dx)w --< a( x )w (d ED, x E 6).

On the other hand, since 1 ED, we have

(95) a(x)w --< a(dx)w (d ED, x E 6),

whence (92) for A EC. But thew<> form a basis of X(A) and (92) follows for any

.A. 0

In particular, we can let x run through a standard fundamental set w = D. 6 for
r , with DC G(Q) finite and containing 1.
This was for P 0. The extension of (ii) to a gener al P is easy. Assume that
Po C P so P = PJ for some J C b.. The restrictions of the {3<>, for a ~ J , to
X(Ap) form a b asis dual to b.(P, Ap ). One h as rational representations CJw, where
w is an integral linear combination of thew<> (for a~ J). (These are highest weights
of rational irreducible representations wh ose highest weight lines are stable under
P.) Let 6p be a Siegel set with respect to P. The previous argument yields

11.3. Corollary. We have
ap(yx)w--< ap(y)wap(x)w (y E G, x E 6p)
where w is a positive linear combination of thew<>, a~ J.
11.4. It will be useful to express some of the previous results in terms of t he
elements of Ao or Ap. As usu al, A+ is the posit ive Weyl chamber in A , the set
of elements on which t he a E b.(P, Ap ) are 2 1. The dual cone +A is the set
of elem ents on which the {3<> are 2 1. We let -A = {a E A, a-^1 E+ A}. Then
Proposition 11.2 gives

11.5. Proposition. There exists ao E A p such that

(i) ap(t) E ao -Ap (t Er)

(ii) ap(tx) E a 0 ap(x)-Ap (t Er, x E 6p)

12. Eisenstein series

12.1. We fix a minimal parabolic Q -subgroup P 0 , and let

(96) Po = NoAoMo

be the Langlands decomposition of P 0. We consider the parabolic
Q -subgroups containing P 0. The choice of P 0 determines an ordering on the Q -
roots ( G, A 0 ), with respect to which the elements of (P, Ap) are the non-zero
restrictions of positive roots. Let further