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
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.
e B, where
is the discretely occurring
subspace of cusp forms and where 8 is the subspace of incomplete 6- series.
In turn, 6 = L . ©
i © C,'where L . is the space of wave packets of
' ei s
s ei s
Eisenstein series (of all K-weights; see below), where L is the discretely
occurring subspace of residues of Eisenstein series, and where C is of course
the trivial representation.
Next, let V ~ I .
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
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,
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