On the geometric trace of a generalized Selberg trace formula

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} .