Harmonic cocycles, von Neumann algebras, and irreducible affine isometric actions

Abstract

Let $G$ be a compactly generated locally compact group and $(\pi ,\mathcal{H})$ a unitary representation of $G.$ The $1$-cocycles with coefficients in $\pi$ which are harmonic (with respect to a suitable probability measure on $G$) represent classes in the first reduced cohomology ${\bar{H}}^1(G,\pi ).$ We show that harmonic $1$-cocycles are characterized inside their reduced cohomology class by the fact that they span a minimal closed subspace of $\mathcal{H}.$ In particular, the affine isometric action given by a harmonic cocycle $b$ is irreducible (in the sense that $\mathcal{H}$ contains no non-empty, proper closed invariant affine subspace) if the linear span of $b(G)$ is dense in $\mathcal{H}.$ The converse statement is true, if $\pi$ moreover has no almost invariant vectors. Our approach exploits the natural structure of the space of harmonic $1$-cocycles with coefficients in $\pi$ as a Hilbert module over the von Neumann algebra $\pi (G{)}^{\prime },$ which is the commutant of $\pi (G)$. Using operator algebras techniques, such as the von Neumann dimension, we give a necessary and sufficient condition for a factorial representation $\pi$ without almost invariant vectors to admit an irreducible affine action with $\pi$ as linear part.

13 pages