Representations of nilpotent Lie groups via measurable dynamical systems

TL;DR

This paper explores how measurable dynamical systems relate to unitary representations of nilpotent Lie groups, showing that ergodicity conditions can determine irreducibility of these representations.

Contribution

It establishes a general criterion linking ergodicity of dynamical systems to the irreducibility of associated unitary representations, with applications to nilpotent Lie groups.

Findings

Ergodicity implies irreducibility in certain unitary representations.

Applicable to finite-dimensional nilpotent Lie groups.

Extended to infinite-dimensional Heisenberg groups.

Abstract

We study unitary representations associated to cocycles of measurable dynamical systems. Our main result establishes conditions on a cocycle, ensuring that ergodicity of the dynamical system under consideration is equivalent to irreducibility of its corresponding unitary representation. This general result is applied to some representations of finite-dimensional nilpotent Lie groups and to some representations of infinite-dimensional Heisenberg groups.