Affiliation:
1. 119496 HUN-REN Alfréd Rényi Institute of Mathematics , Budapest , Hungary
2. 119496 HUN-REN Alfréd Rényi Institute of Mathematics , Budapest , Hungary ; and Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Budapest, Hungary
Abstract
Abstract
A certain generalization of the Selberg trace formula was proved by the first named author in 1999. In this generalization instead of considering the integral of
K
(
z
,
z
)
{K(z,z)}
(where
K
(
z
,
w
)
{K(z,w)}
is an automorphic kernel function) over the fundamental domain, one considers the integral of
K
(
z
,
z
)
u
(
z
)
{K(z,z)u(z)}
, where
u
(
z
)
{u(z)}
is a fixed automorphic eigenfunction of the Laplace operator. This formula was proved for discrete subgroups of
PSL
(
2
,
ℝ
)
{\mathrm{PSL}(2,\mathbb{R})}
, and just as in the case of the classical Selberg trace formula it was obtained by evaluating in two different ways (“geometrically” and “spectrally”) the integral of
K
(
z
,
z
)
u
(
z
)
{K(z,z)u(z)}
.
In the present paper we work out the geometric side of a further generalization of this generalized trace formula: we consider the case of discrete subgroups of
PSL
(
2
,
ℝ
)
n
{\mathrm{PSL}(2,\mathbb{R})^{n}}
where
n
>
1
{n>1}
. Many new difficulties arise in the case of these groups due to the fact that the classification of conjugacy classes is much more complicated for
n
>
1
{n>1}
than in the case
n
=
1
{n=1}
.