Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth

We prove that on Iwasawa AN groups coming from arbitrary semisimple Lie groups there is a Laplacian with a nonholomorphic functional calculus, not only for L^1(AN), but also for L^p(AN), where 1<p<∞. This yields a spectral multiplier theorem analogous to the ones known for sublaplacian on stratified groups.