$\Gamma$-Cohomology and the Selberg Zeta Function

TL;DR

This paper introduces a novel approach to analyze $ $- and $ $-cohomology of Harish-Chandra modules' globalizations for rank one semisimple Lie groups, linking Selberg zeta functions with cohomology.

Contribution

It proves Patterson's conjecture connecting Selberg zeta function singularities with $ $-cohomology of principal series representations for cocompact, torsion-free $ $.

Findings

Established a new method for studying cohomology of Harish-Chandra modules.

Proved Patterson's conjecture relating Selberg zeta functions to $ $-cohomology.

Linked singularities of Selberg zeta functions with $ $-cohomology in the rank one case.

Abstract

We propose a new method for studying $n$- and $\Gamma$-cohomology of globalizations of Harish-Chandra modules, where $G=KAN$ is a rank one semisimple Lie group, $\Gamma$ is a discrete subgroup of $G$ and $n=Lie(N)$. We prove a conjecture of Patterson relating the singularities of Selberg zeta functions with the $\Gamma$-cohomology of principal series representations if $\Gamma$ is cocompact and torsion free.