1549380232-Automorphic_Forms_and_Applications__Sarnak_

3. FIRST PROPERTIES OF AUTOMORPHIC FORMS 11

which proves (9).

If C is a compact subset of G, then

(10)

The property ( nl) of JJ. JJ with respect to product implies the existence of a constant

c > 0 such that xCy C Ccllxll-llYll, so that (9) implies (10).

3.5. We shall need the following lemma:

Lemma. Let a E C~ ( C). There exists n E N such that

Ju* a(x)J--< JJxllnllulJ1.

([14], Lemma 8 and corollary). We sketch the argument. We have

(11) Ju* a(x)I s; 1 dyJu(y)Jla(y-^1 x)I = r Ju(y)J L Ja(y-^1 ,,-^1 x)Jdy

G lr\G "I

so that it suffices to show the existence of N such that

(12) L Ja(y-^1 1x)J--< IJxlJN

"(

We may assume that a is the characteristic function of some compact symmetric

set C. Then the sum on the left of (12) is equal to #(r n yCx-^1 ). Fix bin that

set. Then, for any I' in it, we have J-^1 ,, E xc-^1 y-^1 ycx-^1 = xC^2 x-^1 , so that the

left hand side of (12) is equal to #r n xC^2 x -^1 , and our assertion follows from (10)
and (nl).

3.6. Proposition. Assume that vol(r\C) < oo. Let u be a function on G which
satisfies (Al), (A2), (A3) and belongs to £P(r\C) for some p ;::=: 1. Then f has
moderate growth, i.e. is an automorphic farm.

Proof. Since r\C has finite volume, £P(r\C) c L^1 (r\C), so we may assume

p = 1. By (A2) and (A3), there exists a E C':'(C) so that u = u *a. The
proposition now follows from Section 3.5. D

3. 
   
   
   
   7. We now recall some formalism to describe more precisely the notion of a
      K-finite function. Let dk be the Haar measure on K of mass 1. As usual, let
      k be the set of isomorphism classes of (finite dimensional) continuous irreducible
      representations of K. For v E k, let d(v) be its degree, Xv its character, and
      ev = d(v)xvdk, viewed as a measure on G with support on K. Let u E C(C) be
      K-finite on the right, and more precisely, belonging to an irreducible K-module,
      under right translations, of type v. Then we leave it as an exercise to deduce from
      t he Schur orthogonality relations that:
   
   
   
   

(13)

{

u μ = v

U*eμ = Q μ -j. V

and therefore t hat ev is an idempotent, a projector of C( C) on the isotypic subspace

of type v, and we have eμ * ev = 0 ifμ #-v. Consequently, u is K-finite on the

r ight if and only if there exists an idempotent ~ which is a finite sum of ev, such

that u * ~ = u. The element ~ will be called a standard idempotent.