p-forms on d-spherical tessellations

The spectral properties of p-forms on the fundamental domains of regular tesselations of the d-dimensional sphere are discussed. The degeneracies for all ranks, p, are organised into a double Poincare series which is explicitly determined. In the particular case of coexact forms of rank (d-1)/2, for odd d, it is shown that the heat--kernel expansion terminates with the constant term, which equals (-1)^{p+1}/2 and that the boundary terms also vanish, all as expected. As an example of the double domain construction, it is shown that the degeneracies on the sphere are given by adding the absolute and relative degeneracies on the hemisphere, again as anticipated. The eta invariant on a fundamental domain is computed to be irrational. The spectral counting function is calculated and the accumulated degeneracy give exactly. A generalised Weyl-Polya conjecture for p-forms is suggested and verified.