1549380232-Automorphic_Forms_and_Applications__Sarnak_

6. CONSTANT TERMS. THE BASIC ESTIMATE 27

mk varies in a relatively compact set as x E 6. There exists therefore c > 0 such

that lcji((mk)-^11 :Sc, and we get:

I y;' XjfJ+1 (y)I :Sc L a(x)-f3J L IYi!J+i (y)I

Using (73), (74), we get from (71), multiplying both sides by a(x)->-:

(75) l!J(x) - !J+1(x)la(x)-A = ca(x)-,6j f

1

dt r t ds L IYi!J+1(Y)la(y)-A

lo lo i

and (i) follows. Under the assumption of (ii), the last sum on the right-hand side

is bounded by a constant, whence (ii). [It would be enough to assume that all first

derivatives of f be bounded by .A.] 0

6.8. Corollary. Let f be as in the second part of Proposition 6. 7. Let N E N.

Then

(76)

(77)

l(fj - !J+i)(x)I-< a(x)>--N,ej (x E 6)

l(f-fp)(x)I-< La(x)>--N,BJ (x E 6).
j
Clearly, (fj - !J+i)j+l = 0 and (fj)j = fj. We can apply (68) to fj - !J+ 1 which
yields l(fj - !J+1)(x)I -< a(x)>--^2 t3J. Now Assertion (i) follows by iteration, and
then (ii) is obvious, in view of (66).
Assume now that P = P°' is proper maximal (a E Qb.). Then {3j = mj.a with
mj E N, mj > 0. So we get

(78) l(f - fp)(x)I-< a(x)-N°' (x E 6pa )

for any NE N.
Let now P not be maximal. Assume that fq = 0 for any proper maximal
parabolic Q containing P (there are only finitely many, as follows from 3.3d.)
Given .A E X(A 1 ), we have to show that
(79) lf(x) I-< a(x)>- (x E 6 = 6t,w)

Since f p = 0, (77) shows the existence oft' ~ t such that (79) is satisfied whenever

a(x)°' ~ t' for all a Eb.-I , i.e. for x in the smaller Siegel set 6t',w· There remains

to consider the elements x E 6 such that a(x)°' :S t' for some a. For J strictly
containing I , such that PJ is a proper subgroup of G, let U 1 ,J = {x E 6, a(x)°' :S
t' for a E J - I}. By 5.2.3, U 1 ,J is contained in a Siegel set with respect to PJ.
Therefore, by the induction assumption, f is fast decreasing on U1,J, so that (79)
holds on U 1 ,J. Since 6 is covered by 6t',w and the U1,J, (79) is established. This
proves:

6.9. Theorem. Assume AG= 1. Let f be an automorphic form on r\G and p a
proper parabolic IQ-subgroup of G. If fq = 0 for every proper parabolic IQ-subgroup
containing P, then f is fast decreasing on any Siegel set of G with respect to P. In
particular, a cusp form on G is fast decreasing.

6.10. Let Yp = rp.Np\G and F(Yp) be a G-invariant space of functions on
Yp, which are at least measurable, locally L^1. By definition, its cuspidal part
° F(Yp) is the space of functions all of whose constant terms with respect to the