On the geometric trace of a generalized Selberg trace formula

TL;DR

This paper extends a generalized Selberg trace formula to higher-dimensional groups, addressing new complexities in classifying conjugacy classes for discrete subgroups of PSL(2,R)^n.

Contribution

It develops the geometric side of a further generalized trace formula for discrete subgroups of PSL(2,R)^n, expanding the scope beyond the classical case.

Findings

Formulated the geometric side of the generalized trace formula for n>1

Addressed classification challenges of conjugacy classes in higher dimensions

Extended the applicability of trace formulas to more complex groups

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 $K(z,z)$ (where $K(z,w)$ is an automorphic kernel function) over the fundamental domain, one considers the integral of $K(z,z)u(z)$, where $u(z)$ is a fixed automorphic eigenfunction of the Laplace operator. This formula was proved for discrete subgroups of $PSL(2,\mathbb{R})$, 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 $K(z,z)u(z)$. 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 $PSL(2,\mathbb{R})^n$ where $n>1$. 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 $n>1$ than in the case $n=1$.