2

STEVEN ZELDITCH

deal more involved than in the compact case of [Z.2]. On the other hand, its

study leads to results of some interest in their own right. For example, we

obtain a weak (signed and averaged) version of the Lindelof hypothesis for

Rankin-Selberg zeta functions.

To state our results more precisely, let us recall some basic facts

2 2

regarding the spectral decompositions of L (r\G) and L (r\h), and on the

dynamics of the geodesic flow G on the unit tangent bundle T\G of T\h.

Unexplained terminology or notation can be found in the appendix.

First,

L2(r\G)

=

°L2(r\G)

e B, where

°L2

is the discretely occurring

subspace of cusp forms and where 8 is the subspace of incomplete 6- series.

2 2

In turn, 6 = L . ©

I2

i © C,'where L . is the space of wave packets of

' ei s

re„

s ei s

v Y

o

Eisenstein series (of all K-weights; see below), where L is the discretely

res

occurring subspace of residues of Eisenstein series, and where C is of course

the trivial representation.

Next, let V ~ I .

Q

denote the generator of K = SO(2), and let fl

denote the Casimir operator of G. An element a € L (r\G) is said to have

weight m if ^ Ma = ma. It is a Casimir eigenform of parameter s if

da - s(l-s)r. Ve will reserve the terminology "automorphic form of weight mff

for a e C0D(r\G) which are joint (V,fl) eigenforms.

Automorphic forms of weight 0 are thus eigenfunctions of the Laplacian A

2 2

on L (r\h). The previous decompositionof L (r\G) obviously determines a

2 0 2 2 2

corresponding decomposition L (r\h) = L © L . © L © C in weight 0. Each

ei s re0s

term has a further spectral decomposition into eigenspaces of A. Cuspidal

eigenfunctions of weight 0 will be denoted by u., so that Au. = A-u.. Thus,

0 2

L =© Cu-. Aside from residual eigenfunctions (which we will temporarily

ignore for simplicity), the remaining eigenfunctions are Eisenstein series

1 1 2

E('?2+ir) of A-eigenvalue j+r . Here, E is really a vector, with one