Ergodic boundary representations

TL;DR

This paper establishes an ergodic theorem for unitary operators associated with the Furstenberg-Poisson boundary of lattices in non-compact semisimple Lie groups, extending classical ergodic results to a broader algebraic setting.

Contribution

It proves a von Neumann type ergodic theorem for boundary representations of lattices in non-compact semisimple Lie groups, a novel extension in the field.

Findings

Proves ergodic convergence for boundary representation averages.

Extends classical ergodic theorems to semisimple Lie group lattices.

Provides new tools for analyzing unitary representations of these groups.

Abstract

We prove a von Neumann type ergodic theorem for averages of unitary operators arising from the Furstenberg-Poisson boundary representation (the quasi-regular representation) of any lattice in a non-compact connected semisimple Lie group with finite center.