Conformal foliations on Lie groups and complex-valued harmonic morphisms

We study left-invariant foliations F on Riemannian Lie groups G generated by a subgroup K. We are interested in such foliations which are conformal and with minimal leaves of codimension two. We classify such foliations F when the subgroup K is one of the important SU(2)*SU(2), SU(2)*SL_2(R), SU(2)*SO(2) or SL_2(R)*SO(2). By this we yield new multi-dimensional families of Lie groups G carrying such foliations in each case. These foliations F produce local complex-valued harmonic morphisms on the corresponding Lie group G.