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

Abstract

We give an explicit formula for the multiplicities of the eigenvalues ofthe Laplacian acting on sections of natural vector bundles over acompact flat Riemannian manifold MΓ = Γ\ℝn,Γ a Bieberbach group. In the case of the Laplacian acting onp-forms, twisted by a unitary character of Γ, when Γ hasdiagonal holonomy group F ≃ ℤ2k, these multiplicities have acombinatorial expression in terms of integral values of Krawtchoukpolynomials and the so called Sunada numbers. If the Krawtchoukpolynomial Kpn(x)does not have an integral root, this expressioncan be inverted and conversely, the presence of such roots allows toproduce many examples of p-isospectral manifolds that are notisospectral on functions. We compare the notions of twistedp-isospectrality, twisted Sunada isospectrality and twisted finitep-isospectrality, a condition having to do with a finite part of thespectrum, proving several implications among them and getting a converseto Sunada's theorem in our context, for n ≤ 8. Furthermore, a finitepart of the spectrum determines the full spectrum. We give new pairs ofnonhomeomorphic flat manifolds satisfying some kind of isospectralityand not another. For instance: (a) manifolds which are isospectral onp-forms for only a few values of p ≠ 0, (b) manifolds which aretwisted isospectral for every χ, a nontrivial character of F, and(c) large twisted isospectral sets.