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

The author acknowledges the partial support of the French Agence Nationale de la Recherche (ANR) through the projects Labex Lebesgue (ANR-11-LABX-0020-01) and GAMME (ANR-14-CE25-0004).