Unitary Representations of Lattices of Free Nilpotent Lie Groups of Step-Two

TL;DR

This paper investigates the irreducibility of unitary representations of lattices in step-two free nilpotent Lie groups, revealing a parity-dependent dichotomy in their reducibility properties.

Contribution

It establishes a parity-based criterion for the irreducibility of representations of lattices in step-two free nilpotent Lie groups, extending previous theoretical results.

Findings

Almost every irreducible representation is irreducible when the Lie algebra dimension is odd.

Almost every unitary irreducible representation is reducible when the dimension is even.

The results depend on a theorem by Bekka and Driutti regarding representation restrictions.

Abstract

Using a theorem proved by Bekka and Driutti, we show that if $\mathfrak{f}$ is a freely generated nilpotent Lie algebra of step-two, then almost every irreducible representation of the corresponding Lie group restricted to some lattice $\Gamma$ is an irreducible representation of $\Gamma$ if the dimension of the Lie algebra is odd. However, if the dimension of the Lie algebra is even, then almost every unitary irreducible representation of the Lie group restricted to $\Gamma$ is reducible.