Abstract. We characterise the actions, by holomorphic isometries on a Kähler manifold with zero first Betti number, of an abelian Lie group of dim ≥ 2 , for which the moment map is horizontally weakly conformal (with respect to some Euclidean structure on the Lie algebra of the group).Furthermore, we study the hyper-Kähler moment map ϕ induced by an abelian Lie group T acting by triholomorphic isometries on a hyper-Kähler manifold M , with zero first Betti number, thus obtaining the following:• If dim T = 1 then ϕ is a harmonic morphism. Moreover, we illustrate this on the tangent bundle of the complex projective space equipped with the Calabi hyper-Kähler structure, and we obtain an explicit global formula for the map.• If dim T ≥ 2 and either ϕ has critical points, or M is nonflat and dim M = 4 dim T then ϕ cannot be horizontally weakly conformal.