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

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.