Lp-spectral multipliers for the hodge laplacian acting on 1-forms on the heisenberg group

We prove that, if Delta(1) is the Hodge Laplacian acting on differential 1- forms on the ( 2n + 1)- dimensional Heisenberg group, and if m is a Mihlin - Hormander multiplier on the positive half- line, with L-2- order of smoothness greater than n+ 1/2, then m(Delta(1)) is L-p- bounded for 1 < p < infinity. Our approach leads to an explicit description of the spectral decomposition of Delta(1) on the space of L-2- forms in terms of the spectral analysis of the sub- Laplacian L and the central derivative T, acting on scalar- valued functions.