Flat Manifolds Isospectral on p-Forms

We study isospectrality on p-forms of compact flat manifolds by using the equivariant spectrum of the Hodge-Laplacian on the torus. We give an explicit formula for the multiplicity of eigenvalues and a criterion for isospectrality. We construct a variety of new isospectral pairs, for instance, pairs of flat manifolds of dimension n=2p, p>1, not homeomorphic to each other, which are isospectral on p-forms but not on q-forms for q different from p. Also, we give manifolds isospectral on p-forms if and only if p is odd, one of them orientable and the other not, and a pair of 0-isospectral flat manifolds, one of them Kahler, and the other not admitting any Kahler structure. We also construct pairs, M, M' of dimension n>5, which are isospectral on functions and such that the Betti numbers of M are less than those of M' for every 0<p<n; and pairs isospectral on p-forms for every p odd, and having different holonomy groups, Z_4 and Z_2+Z_2 respectively.