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§6.2. The Projection of Lj^(H) onto //j[(H)
227
For a function /(z) in L|f(H), we put /*(a)=/(aO;(a,0-'
(aeSL^W).
Then Theorem 1.4.5 imphes ||/||p = ||/*||p. In particular, we have \\K,{zJ)\\,=
\\Kno^n,
for all /? (1 < p < 00). For a function /eLjf(H), we put g{z) = J K,{z, Z2)fiz2)lm(z2fdv(z2)
(zeH).
H
Then we see by (6.2.3), g*(oc)= J
Knr'oi)f*{P)dp.
SL 2iU)
Since SLjC"^) is unimodular, we have II ^* II p ^ II i^.* Ill-II n i p and therefore
by [Weil
5, pp. 54-55],
||^||,^||K,(z,Olli-||/llp. This implies that K is a continuous Hnear mapping of Ljf(H) into itself. Next we are going to prove that the image of K is included in H|f(H). Let H„ (n = 1, 2,. . . .) be compact subsets of H satisfying H„ c= H„+i and IJ^H„ = H. We put (K«/)(z)= i K,(z,Z2)f(z2)lm{z,fdv{z,)
(/6L«H)).
H,
Then X'"*/ is a holomorphic function by Lemma 6.1.5 and lim (X<">/)(z) = (Kf)(z). Let M be a compact subset of H. Then by Theorem 6.1.8, there exist a point ZQEM and a constant C such that \K,iz,,Z2)\ ^ C|iC,(zo,Z2)|
(zieM,Z2 6H).
Hence for zeM, |(X/)(z)^(X(«>/)(z)|gC
J |K,(zo,Z2)/(z2)|Im(z2)*dt;(z2). H-H„
Therefore {K^"^f)(z) converges to {Kf){z) uniformly on any compact subsets of H and Kf is holomorphic on H. In the end, we shall prove that if/eHjf(H), then Kf=f. We define a function Kfc(wi, W2) on K X K by K,(wi, W2) = K,{p-'w,,p-'w2){l
- wi)-*(l - 1.2)-^
(-[: -;])
Then Kfc(wi, W2) isjhe kernel function of H^iK) by the isomorphism given in Theorem 6.2.1. For/eLf(K), we also put (Kf)M
= j X,(W, K
W2)f(W2)i\-\W2ndVKiW2y
228
6. Traces of Hecke Operators
Put g„(w) = w" for n^O. Since ^„e/ffc (K), we have Kg„ = g„. Then for a function/ in HJ(K), we have Therefore (Kf—f, ^„)K = 0 for all integer n(>0). Denote the Taylor expansion of K 7 - / a t w = 0 by ( Kf-f){w) = X"=o ««vv". Then 0
This implies a„ = 0 for all n >0 and Kf = f. By the isomorphism in Theorem 6.2.1(1), we obtain Kf=f for all fe HJ(H). D
§6.3. Function Spaces Consisting of Automorphic Forms Let r be a Fuchsian group of the first kind, x ^ character of F of finite order and fc(> 2) an integer. We assume /(— 1) = (— 1)* if/" contains — 1. For any measurable function/(z) on H satisfying (6.3.1)
if\,y) (z) = x W / ( 4
for any y e r ,
we put i/p
I J \fiz)lm{zr''\^dviz)\ lr\H J
(l