Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 80, Number 4
(2004), 28-33.
The aim of the present note is to develop a study on the feasibility
of a unified theory of mean values of automorphic $L$-functions,
a desideratum in the field. This is an outcome of the investigation
commenced with the part XII ([14]), where a framework was laid
on the basis of the theory of automorphic representations,
and a general approach to the mean values was envisaged. Specifically,
it is shown here that the inner-product method, which was initiated
by A. Good [7] and greatly enhanced by M. Jutila [9], ought
to be brought to perfection so that the mean square of the
$L$-function attached to any cusp form on the upper half-plane
could be reached within the notion of automorphy. The Kirillov
map is our key implement. Because of its geometric nature,
our method appears to extend to bigger linear Lie groups.
This note is essentially self-contained.
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