r-Harmonic and complex isoparametric functions on the Lie groups R^m⋉Rⁿ and R^m⋉H²ⁿ⁺¹

In this paper we introduce the notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper r-harmonic functions. We then apply this to construct the first known explicit proper r-harmonic functions on the Lie group semidirect products R^m⋉ Rⁿ and R^m⋉ H ²ⁿ⁺¹, where H ²ⁿ⁺¹ denotes the classical (2 n+ 1) -dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.