Classification of Certain Rational Isoparametric Functions on Damek-Ricci Spaces

We classify isoparametric functions on Damek-Ricci spaces which can be written in terms of the standard coordinates $(v,z,t)$ on the half-space model as a polynomial function divided by t. Regular level sets of the functions in our classification encompass almost all previously known examples of isoparametric hypersufaces in Damek-Ricci space and also yield new ones. For the new examples, the focal varieties are determined and the mean curvatures of the regular level sets are expressed as a function of the distance from the focal variety. We also study the exceptional case of tubes about R Hᵏ in C Hᵏ, which are isoparametric, but cannot be obtained as the level sets of any function in our classification. We show that these tubes are level sets of a polynomial function divided by t², and that analogous functions on Damek-Ricci spaces can be isoparametric only in the case of C Hᵏ.