  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-H\"ormander 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<\infty.
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.
