Harmonic morphisms from solvable Lie groups

In this paper we study the existence of complex valued harmonic morphisms from a Lie group G. We establish an algebraic condition on the Lie algebra $\mathfrak{g}$ of G ensuring the existence of left-invariant Riemannian metrics on G admitting complex valued harmonic morphisms. It is then shown that this condition is satisfied in many important cases of solvable Lie groups, in particular, whenever G is nilpotent or a non-compact Riemannian symmetric space of rank at least 3. We then give a continuous family of 3-dimensional solvable Lie groups not admitting any complex valued harmonic morphisms, not even locally.