Comparison of Twisted P-Form Spectra for Flat Manifolds with Diagonal Holonomy

We give an explicit formula for the multiplicities of the eigenvalues of the Laplacian acting on sections of natural vector bundles over a compact flat Riemannian manifold MG = G\ \mathbb Rn, G a Bieberbach group. In the case of the Laplacian acting on p-forms, twisted by a unitary character of G, when G has diagonal holonomy group F \mathbb Z2k, these multiplicities have a combinatorial expression in terms of integral values of Krawtchouk polynomials and the so called Sunada numbers. If the Krawtchouk polynomial Kpn(x) does not have an integral root, this expression can be inverted and conversely, the presence of such roots allows to produce many examples of p-isospectral manifolds that are not isospectral on functions. We compare the notions of twisted p-isospectrality, twisted Sunada isospectrality and twisted finite p-isospectrality, a condition having to do with a finite part of the spectrum, proving several implications among them and getting a converse to Sunada's theorem in our context, for n= 8. Furthermore, a finite part of the spectrum determines the full spectrum. We give new pairs of nonhomeomorphic flat manifolds satisfying some kind of isospectrality and not another. For instance: (a) manifolds which are isospectral on p-forms for only a few values of p? 0, (b) manifolds which are twisted isospectral for every ?, a nontrivial character of F, and (c) large twisted isospectral sets. [PUBLICATION ABSTRACT]